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Università degli Studi di Milano

Corso di Dottorato in

Scienze Matematiche

Ciclo XXXIII

MAT-05: Analisi Matematica

[t]0.4Sorbonne Univeristé

École doctorale de
Sciences Mathématiques
de Paris Centre

Specialité : Mathématiques

Dipartimento di Matematica

[t]0.47Centre d'analyse et de
mathématique sociales

Evolution equations
with applications
to population dynamics

Doctoral thesis in conjoint program

Elisa Affili

Matr. R12038

Prof. Enrico Valdinoci

Prof. Luca Rossi

Coordinatore del Corso di Dottorato

Prof. Vieri Mastropietro

[b]0.47Directeur de l'école doctorale

Prof. Elisha Falbel

Anno accademico 2019-2020

CHAPTER: ABSTRACT

The main topic of this thesis is the analysis of evolution equations reflecting issues in ecology and population dynamics. In mathematical modelling, the impact of environmental elements and the interaction between species is read into the role of heterogeneity in equations and interactions in coupled systems. In this direction, we investigate
three separate problems, each corresponding to a chapter of this thesis.

The first problem addresses the evolution of a single population living in an environment with a fast diffusion line.
From a mathematical point of view, this corresponds to a system of two coupled reaction-diffusion equations working on domains of different dimensions, which is called as in [20] a “road-field model”. We introduce a periodic dependence of the reaction term in the direction of the fast diffusion line; in the ecological interpretation, this corresponds to the presence of more and less favourable zones for the growth of the population.
Necessary and sufficient conditions for persistence or extinction of the population and the effects of the presence of the road are analysed through the study of a suitable generalised principal eigenvalue, originally defined in [16]. By comparison with the literature about reaction-diffusion equations in periodic media, we show that the presence of the road has no impact on the survival chances of the population, despite the deleterious effect that is expected from fragmentation.

The second investigation regards a model describing
the competition between two populations in a situation of asymmetrically aggressive interactions – one is the attacker and the other the defender.
We derive a system of ODEs from basic principles, obtaining a modified Lotka-Volterra model relying on structural parameters as the fitness of the population and the frequency and effectiveness of the attacks.
The evolution progresses through two possible scenarios, where only one population survives.
Then, the interpretation of one of the parameters as the aggressiveness of the attacker population naturally raises questions of controllability. With the aid of geometrical arguments we characterise the set of initial conditions leading to the victory of the attacker through a suitable (possibly time-dependant) strategy.
Indeed, we prove that bang-bang strategies are sufficient and sometimes necessary over constant controls.
Finally, we treat a time minimization question.

The third and last part of this thesis analyses the time decay of some evolution equations with classical and fractional time derivatives. Carrying on an analysis started in [43], we deal with evolution equations with a possibly mixed Caputo and classical time derivative.
By using energy methods, we prove quantitative estimates of polynomial or exponential type; the different behaviour depends heavily on the choice of the time derivative.
The decay results apply to a large class of diffusion operators, comprehending local, nonlocal, real, complex, and even nonlinear ones, of which we provide concrete examples.

CHAPTER: RIASSUNTO

Il principale argomento di questa tesi è l'analisi delle equazioni dell'evoluzione che riflettono questioni di ecologia e di dinamica della popolazione. Nell'ambito della modellizzazione matematica, l'impatto degli elementi ambientali e delle interazioni tra le specie viene studiato mediante il ruolo dell'eterogeneità nelle equazioni e nelle interazioni nei sistemi accoppiati. In questa direzione, indaghiamo
tre problemi distinti corrispondenti a tre capitoli di questa tesi.

Il primo problema riguarda l'evoluzione di una singola popolazione che vive in un ambiente con una linea di diffusione rapida.
Dal punto di vista matematico, lo studio riguarda un sistema di due equazioni di reazione-diffusione accoppiate, che lavorano su domini di dimensioni diverse, chiamato come in [20] un modello “campo-strada”. Introduciamo una dipendenza periodica in direzione della linea di diffusione per il termine di reazione, che nell'interpretazione ecologica corrisponde alla presenza di zone più e meno favorevoli alla crescita della popolazione.
Le condizioni necessarie e sufficienti per la persistenza o l'estinzione della popolazione e gli effetti della presenza della strada sono analizzati attraverso lo studio di un adeguato autovalore principale generalizzato, recentemente definito in [16]. Tramite il confronto con la letteratura in mezzi periodici, si mostra che la presenza della strada non ha alcun impatto sulle possibilità di sopravvivenza della popolazione, nonostante l'effetto deleterio che ci si aspetta dalla frammentazione.

La seconda indagine riguarda un modello che descrive la competizione tra due popolazioni in una situazione di aggressione asimmetrica, in cui una popolazione aggredisce una seconda.
Deriviamo un sistema di ODE da alcune assunzioni fondamentali, ottenendo un modello Lotka-Volterra modificato che si basa su parametri strutturali come la fitness della popolazione e la frequenza e l'efficacia degli attacchi.
L'analisi della dinamica mostra due possibili scenari, in cui una sola delle due popolazioni sopravvive.
Dopodiché, l'interpretazione di uno dei parametri come l'aggressività della prima popolazione solleva in modo naturale un problema di controllabilità. Tramite argomentazioni geometriche caratterizziamo l'insieme delle condizioni iniziali permettendo, con un'adeguata strategia eventualmente variabile nel tempo, la vittoria della popolazione che attacca. Infatti, dimostriamo che le funzioni di tipo bang-bang sono sufficienti a raggiungere l'obiettivo e talvolta sono necessarie rispetto a funzioni costanti.
Infine, trattiamo una questione di minimizzazione nel tempo.

La terza e ultima parte analizza il decadimento nel tempo in equazioni di evoluzione con una possibile derivata temporale frazionaria. Proseguendo un'analisi iniziata in [43], trattiamo equazioni d'evoluzione con una combinazione di derivata temporale di Caputo e classica.
Utilizzando metodi d'energia, dimostriamo stime quantitative di tipo polinomiale o esponenziale; il diverso comportamento dipende principalmente dalla scelta della derivata temporale.
I risultati di decadimento si applicano ad una vasta classe di operatori di diffusione, comprendendone alcuni locali, non locali, reali, complessi e anche non lineari, di cui forniamo esempi concreti.

CHAPTER: RÉSUMÉ

Le sujet principal de cette thèse est l'analyse des équations d'évolution reflétant les questions d'écologie et de dynamique des populations.  En modélisation, la compréhension de l'impact des éléments environnementaux et de l'interaction entre les espèces dépend de la compréhension du rôle de l'hétérogénéité dans les équations et les interactions dans les systèmes couplés. Dans cette direction, nous étudions
trois problèmes indépendents correspondant à trois chapitres de cette thèse.

Le premier problème concerne l'évolution d'une seule population vivant dans un environnement avec une ligne de diffusion rapide.
L'analyse porte sur un système de deux équations de réaction-diffusion couplées, travaillant sur des domaines de dimensions différentes, qui est appelé comme dans [20] un modèle  “champ-route”. Nous introduisons une dépendance périodique dans la direction de la ligne de diffusion pour le terme de réaction, qui, dans l'interprétation écologique, correspond à la présence de zones plus ou moins favorables à la croissance de la population.
Les conditions nécessaires et suffisantes pour la persistance ou l'extinction de la population et les effets de la présence de la route sont analysés par l'étude de la valeur propre principale généralisée appropriée, définie pour la première fois dans [16]. Par comparaison avec des études similaires dans des environnements périodiques, nous prouvons que la présence de la route n'a aucun impact sur les chances de persistence de la population, malgré l'effet délétère attendu lié à la fragmentation.

La deuxième étude porte sur un modèle décrivant l'interaction compétitive et agressive entre deux populations. Nous dérivons un système d'EDO à partir de principes de base, en obtenant un modèle Lotka-Volterra modifié reposant sur des paramètres structurels comme la fertilité de la population et la fréquence et l'efficacité des attaques.
L'analyse de la dynamique donne deux scénarios possibles, où une seule population survit.
Ensuite, l'interprétation d'un des paramètres comme étant l'agressivité de la première population soulève tout naturellement des questions de contrôlabilité. Grâce à des arguments géométriques, nous caractérisons l'ensemble des conditions initiales permettant la victoire de la première population avec une stratégie appropriée éventuellement dépendante du temps. En effet, nous prouvons que les stratégies de bang-bang sont suffisantes et parfois nécessaires face à des contrôles constants.
Enfin, nous traitons une question de minimisation du temps.

La troisième et dernière partie de la thèse analyse la décroissance dans le temps pour des solutions d'une classe d'équations d'évolution avec dérivées temporelles fractionnaires et classiques. Poursuivant une analyse commencée dans [43], nous traitons des équations d'évolution avec une combinaison linéaire des dérivées temporelles Caputo et classiques.
En utilisant des méthodes d'énérgie, nous prouvons des estimations quantitatives de type polynomial ou exponentiel ; le comportement différent dépend fortement du choix de la dérivée temporelle.
Les résultats de la décroissance s'appliquent à une large classe d'opérateurs de diffusion, comprenant des opérateurs locaux, non locaux, réels, complexes et même non linéaires, dont nous fournissons des exemples concrets.

CHAPTER: RINGRAZIAMENTI

Per primi vorrei ringraziare i miei relatori, Luca Rossi ed Enrico Valdinoci, senza i quali questo lavoro non sarebbe stato possibile.
Luca mi segue già da molti anni e le esperienze positive con lui sono state determinanti sulla mia scelta di intraprendere il dottorato. Ha continuato a seguirmi con attenzione e meticolosità durante questi tre anni.
Enrico fin dal primo giorno ha avuto fiducia in me e mi ha suggerito le migliori possibilità per lo sviluppo della mia carriera, incoraggiandomi, finanziandomi e aiutandomi quando il compito mi risultava troppo difficile.
A entrabi devo i miei ringraziamenti più sinceri.

Un ringraziamento speciale va anche a Serena Dipierro, che è stata molto presente nel mio dottorato, come collaboratrice e quasi come un terzo relatore, si è impegnata a coinvolgermi e promuovermi nella comunità matematica.
Ringrazio moltissimo anche Henri Berestycki, per la grande ispirazione che mi ha dato, per i suoi preziosi consigli e per aver aiutato me e i miei relatori a formare l'accordo di cotutela.

I would like to thank the two anonymous referees for their precious time and their nice comments on the report. I am also happy to thank the members of the defence commission, Luis Almeida, Sepideh Mirrhaimi, Fabiana Leoni and again Henri Berestycki, for accepting the task and devoting their valuable time to me.

Ringrazio anche per l'accoglienza il Dipartimento di Matematica dell'Università degli Studi di Milano, in particolare nelle persone di Vieri Mastropietro, coordinatore del corso di dottorato, e di Stefania Leonardi. Sono molto riconoscente anche a Daniela Lipari per avermi aitato a stringere l'accordo di cotutela. Un grande ringraziamento va anche ai miei colleghi dottorandi, per avermi condiviso con me gioie e dolori del dottorato e per aver reso molto più divertente il tempo a Milano.
Dedico un pensiero particolare agli altri dottorandi (ormai dottori!) e postdoc di Enrico e Serena, che mi hanno accompagnato in numerose conferenze in giro per il mondo e nei due mesi in Australia, facendomi sentire sempre a casa: Pietro, Luca, Claudia, Giorgio, Matteo, Matteo, Julien.

Je suis reconnaissante également à Sorbonne Université et au laboratoire CAMS pour m'avoir accueilli, déjà depuis mon stage de M2.
Un grand merci à Sandrine Nadal, Nathalie Brusseaux, Jean-François Venuti, Corentin Lacombe et Patricia Zizzo pour leur travail administratif, et à tous les personnes du labo  pour la belle ambiance, les discussions passionnantes, et pour m'avoir appris beaucoup sur la culture et la langue française.
Ici aussi un grand groupe de doctorants et postdoc m'ont aidé avec leurs conseils et leur amitié, pendant ma thèse et le stage: merci à Romain, Samuel, Charles, Alessandro, Benedetta,  Federico, Julien, François, Noemi, Imke, Jérémie, Milim, José, Elisa.

Vorrei ringraziare tutti gli amici anche al di fuori dell'università che mi hanno sostenuto in questi anni di viaggi, conferenze e traslochi sfrenati tra Padova, Milano, Parigi e Stoccolma, e tutti gli amici che c'erano da molto prima che il mio dottorato iniziasse.
Sono divisa tra la felicità di aver conosciuto così tante persone meravigliose e il rammarico di non aver passato abbastanza tempo con ciascuno.

Per la mia famiglia, il mondo dell'università e della matematica sono sempre stati estranei, ma questo non ha impedito loro di sostenermi e di avere fiducia in me. Grazie a mio fratello Andrea per avermi insegnato le sottrazioni e aver sopportato le mie domande sugli strani simboli che apparivano nei suoi libri di matematica. Grazie anche a Emanuela, Nicolò e Matteo per aver arricchito la nostra famiglia con tanta gioia e affetto. Grazie a mamma e papà per avermi dato tutto, senza mai chiedere niente.

Ad Andrea potrei dedicare intere pagine di ringraziamenti, ma credo sia meglio farglieli di persona.

CHAPTER: INTRODUCTION

The main motivation behind research is to enhance mankind ability to predict and keep under control natural and artificial processes.
To this purpose, mathematical models have revealed to be a very compelling instrument. A mathematical model is a simplified representation of a phenomenon through several meaningful, quantitative parameters evolving with analytical laws. Once some faithful evolution equations are established, the role of mathematics is to provide as much information as possible on the solutions, even if often only qualitative properties can be derived. That is, mathematics does not study the reality, but the language in which we read it.

On the other side, given a model, people often find the mathematical challenges interesting in themselves. It is natural that some
questions on the mathematical tools arise, or that variations of the model are proposed and discussed.
This way, knowledge of mathematics is expanded, and more equipment is available to write new models.

At present, the problem of climate change and environment anthropization is a great concern for humankind. In order to activate effective countermeasures against biodiversity loss, it is important to understand as deeply as possible what conditions would entail such event. These conditions depend on quantitative and qualitative properties of the environment where the species lives, on a population's resilience to changes, but also on its interaction  with other species sharing the same habitat. We still know too little about the effects that these elements and their alteration have on the survival chances of species.

This thesis is far from giving a solution to these dreadful problems but aims to give a contribution to the field of evolution equations and systems with possible application to population dynamics.

§.§.§ Topics and aims of the thesis

The thesis consists of three parts, each treating a different problem.

In the first part, corresponding to Chapter <ref>, we start from a reaction-diffusion model in a periodic environment with a fast diffusion line. The aim is to find conditions entailing survival or extinction of the population and to understand the influence of the line and the environment on the dynamics. Our analysis permits a comparison with the scenario where the fast diffusion line is not present for the general case of a medium with heterogeneity in one direction.
The content of Chapter <ref> is reflects the content of the paper [2] by the author of this thesis.

The second part, contained in Chapter <ref>, is consecrated to a model of aggressive, asymmetric competition between two populations, derived from a Lotka-Volterra system.
The presence of the aggression term naturally leads to a control problem, where a population tries to prevail on the other using an appropriate strategy.
Hence, once the dynamics of the system is understood, we investigate conditions for the victory of the aggressive population, which quite surprisingly is not always possible.
Moreover it is found that, depending on the initial condition, either a bang-bang or a constant strategy leads to the desired scenario.
Chapter <ref> corresponds to the paper [3] by Serena Dipierro, Luca Rossi, Enrico Valdinoci and the author of this thesis.

The last part of this thesis deals with a more abstract and general problem; we investigate asymptotic behaviour for a class of evolution equations with both fractional and classical time derivatives.
Our setting consists of an homogeneous evolution equation working on a bounded set.
The framework comprehends both real and complex, local and nonlocal diffusion operators, and allow us to evaluate the impact of time derivatives on the decay of solutions.
Depending on the type of time derivative, polynomial or exponential decays are entailed.
The results of Chapter <ref> are presented in the paper [5] in collaboration with Enrico Valdinoci and the note [4] in collaboration with Serena Dipierro and Enrico Valdinoci.

§.§.§ Organisation of the manuscript

In this introductory chapter, we make the reader familiar with the problems we investigate and the framework they are enclosed in.
Following the historical path,
we start by a general introduction that then branches in three sections corresponding to the precise research niches of our problems. In each section, after an overview of the state of the art of the topic, we introduce the corresponding problem in details and provide precise statements of our results.

As mentioned before, the rest of the manuscript consists of three chapters, corresponding respectively and in the same order to the topics we introduce in this introduction.
Each chapter is meant to be a self-standing script.

§ GENERAL HISTORIC BACKGROUND

For apparent reasons of population control and resource organisation, one of the first themes for which modelisation has been used is population dynamics.
The first example in this sense was written by Leonardo Fibonacci in Liber Abaci and treats the size of a population of rabbits.
Fibonacci supposed that each couple of rabbits that are older than one month gives birth to another couple of rabbits; calling $u_n$ the size of the population at the $n-$th month, under the previous hypothesis one deduces that

\begin{equation*}
u_{n+2}=u_{n+1}+ u_n.
\end{equation*}

Staring with $u_0=1$, it can be deduced that $u_n$ has an exponential behaviour [8].
This deduction corresponds to the reality only as long as the food is abundant for all the individuals; moreover, the relation is involved and not easy to treat.

Another discrete model was proposed by Euler in the treatise Introduction to the Analysis of the Infinite, published in 1748 [8]. He assumed the annual growth rate to be a fixed quantity $\alpha>0$. Then, calling $P_n$ the size of the population at the year $n$, one has that

\begin{equation*}
\end{equation*}

so one derives, calling $P_0$ the population at the initial time,

\begin{equation*}
P_n=(1+\alpha)^{n} P_0.
\end{equation*}

The sequence $\{P_n\}_{n\in\N}$ is called a geometric sequence, and its behaviour is again exponential.
Thanks to these formulae, Euler treated some problems linked to the growth of urban population and he investigated the reliability of the biblical story of the Float. However, his model involve many computations that were hard to perform before the introduction of computers.

Thomas Malthus, in his work Essay on the Principle of Population [81], used a simpler relation to represent the evolution of a population size; he supposed the growth of a population to be proportional to its size, that is, the growth rate to be a fixed constant, $a>0$.
Moreover, as simplification, he assumed the size of the population to evolve in a continuous fashion with respect to time.
With these hypothesis, the evolution of $u$ follows the law

\begin{equation}\label{eq:malthus}
u'(t)= a u(t) \quad \text{for} \ t\geq 0.
\end{equation}

The solutions to equation (<ref>) are exponentials, in accordance with the result of Fibonacci.

Again in [81], Malthus pointed out that the growth of a population is limited by the quantity of resources. This idea was taken into the equation by Verhulst [120]. He considered the number $k>0$ of individuals that the environment can support indefinitely with the available resources; this is called carrying capacity of the environment. Then, he corrected Malthus's equation (<ref>) with the following:

\begin{equation}\label{logistic}
u'(t)= a u(t)\left( 1-\frac{u(t)}{k}  \right) \quad \text{for} \ t\geq 0.
\end{equation}

Equation (<ref>) presents two equilibria: $u=0$, that is repulsive, and $u=k$, which is attractive.
In fact, for all $u(t)<k$, one has $u'(t)>0$, while for $u(t)>k$, it holds $u'(t)<0$; in both cases, the solution tends to get closer to the value $k$.
This means that, independently of the starting condition, as long as the initial datum is positive, the population size evolves approaching the value $k$, which is the maximum number of individuals that the environment can sustain.
The logistic model is much more realistic that the previous estimates. It is considered the precursor of interesting mathematical branches, including Lotka-Volterra systems and reaction-diffusion equations.

§ THE ROAD-FIELD MODEL IN A PERIODIC MEDIUM

§.§ Reaction diffusion equations in the literature

One important feature that is not taken into account in the logistic equation is dependence on space.
The first effect to take into account for a space structured model is the fact that a population is subject to dispersion.
This is a result of the free movement for animals and of the dispersion of seeds for plants.
The first hypothesis in the literature was to consider the individuals to move with random brownian walk, as particles of a gas.

Without taking account reproduction, calling $u(t, x)$ the size of a population and considering it in continuous dependence on time,
the dispersal would follow the well-known heat equation

\begin{equation}\label{eq1}
\partial_t u - \Delta u=0.
\end{equation}

Note that when speaking of population denisties and sizes, we only consider nonnegative solutions.

The first mathematicians who added a reaction term to equation (<ref>) were
Fisher [50] and Kolmogorov, Petrovsky and Piskunov [72].
They considered a function $u(t,x)$ representing the concentration of an advantageous gene in a population; it was supposed that the population lives in a one-dimensional environment and that the individuals move randomly. Taking these hypothesis, once the gene was introduced, it spreads according to the equation

\begin{equation}\label{eq:KPP}
\partial_t u - \partial_{xx}^2 u = f(u)
\end{equation}

where $f$ is a function such that

\begin{equation}\label{0225}
\end{equation}

moreover it is monostable, that is,

\begin{equation*}
\quad f(u)>0 \ \text{for} \ u\in(0,1),
\end{equation*}

and respects the condition called KPP hypothesis

\begin{equation}\label{0024}
f(u) < f'(0)u.
\end{equation}

The function $f$ represents the birth-death rate of individuals carrying the gene. The fact that $f(0)=0$ is a very natural assumption: if no individuals are present, no new individual is generated. On the other hand, the choice $f(1)=0$ suggests a saturation at the size $u=1$.
The hypothesis (<ref>) reflects the fact that the growth rate decreases as the size of the population grows, as it is the case for the logistic equation (<ref>).
Actually, Fisher supposed $f(u)=au(1-u)$ for $a>0$, which is exactly the nonlinearity proposed by Verhulst, while Kolmogorov, Petrovsky and Piskunov selected $f(u)=a u(1-u)^2$.

For a large class of initial data, among which the Heaviside functions, the solutions to (<ref>) asymptotically converge to a function of the shape

\begin{equation}\label{sol}
u(t,x )=U(z) \quad \text{for} \ z=x+ct.
\end{equation}

Solutions of the form (<ref>) are called travelling waves and the quantity $c$ is called speed of propagation of the travelling wave. The travelling wave found in [50] and [72] has speed corresponding to $c_{KPP}=2\sqrt{ f'(0)}$; actually, a travelling wave exists for all $c\geq c_{KPP}$ and $c_{KPP}$ corresponds to the minimal speed.

The main questions addressed in [50] and [72] have been later asked for larger and larger class of nonlinearites. These questions concerns the existence of stationary solutions, the existence of travelling fronts and the asymptotic speed of propagation for the Cauchy problem.

For the sake of completeness, we must here name other two important settings.
In [48] and in [7] for the multidimensional case, Fife and McLeod and Aronson and Weinberger treated equation (<ref>) in the case of a function $f$ satisfying the hypothesis (<ref>) and such that there exists a value $\theta$ for which

\begin{equation}\label{0044}
f(u)<0 \quad \text{if} \ u\in(0,\theta), \qquad f(u)>0 \quad \text{if} \ u\in(\theta, 1).
\end{equation}

A function satisfying (<ref>) is called bistable, from the fact that the related equation has two attractive states, $0$ and $1$. This type of nonlinearity is particularly interesting because it embodies an important phenomenon in population dynamics, called the Allee effect from the name of the scientist who discover it in the '30s.  It happens that in social animals, aggregation increases the survival rate of individuals; therefore, when the size of a population is under a certain threshold, the growth rate is negative; when the group size passes the threshold, the growth rate becomes positive.

A third important setting is the combustion case, in which there exists a quantity $\theta\in(0,1)$ such that

\begin{equation*}
f(u)=0 \quad \text{if} \ u\in[0,\theta], \qquad f(u)>0 \quad \text{if} \ u\in(\theta, 1).
\end{equation*}

This type of nonlinearity is used for ignition models, where to activate the combustion process the  temperature must pass a threshold.

As a matter of fact, Aronson and Weinberger investigated the equation

\begin{equation}\label{aw}
\partial_t u- \Delta u= f(u) \quad \text{for} \ x \in\R^n
\end{equation}

and asked under which conditions on the function $f$, other than (<ref>), and on the initial datum $u_0$ one has invasion or spreading, that is,

\begin{equation*}
u(t,x) \overset{t\to+\infty}{\longrightarrow} 1 \quad \text{locally uniformly in} \ x.
\end{equation*}

The opposite behaviour is called extinction, and it occurs when

\begin{equation*}
u(t,x) \overset{t\to+\infty}{\rightarrow} 0 \quad \text{uniformly in} \ x.
\end{equation*}

We point out that for extinction a uniform convergence is required, otherwise, in some scenarios, one could have a positive mass escaping further and further in space as $t$ goes to infinity.
The authors found that for a compactly supported initial datum which is “sufficiently large” (depending on the nonlinearity), invasion occurs if and only if

\begin{equation*}
\int_0^1 f(x)dx>0.
\end{equation*}

Let us give more details on the minimal requirements for the initial datum.
In the monostable case, it is sufficient for $u_0$ to be greater than a positive constant in $(0,1)$ in a large enough ball. Moreover, if $f'(0)>0$, then  all solutions issued from a non zero, non negative initial datum  converges to 1 as $t$ goes to infinity; this is called hair trigger effect.
In the bistable and monostable cases, the positive constant is necessarily greater than the threshold $\theta$.

Equation (<ref>) was the first example of a whole class of PDEs, the reaction-diffusion equations. From the initial works [50, 72, 48, 7], the literature on reaction-diffusion equations and the study on travelling waves have flourished. What is present here is a circumscribed niche, which is handy to provide context to our work.

§.§.§ Reaction-diffusion equations in periodic media

One of the other natural applications of equations (<ref>) and (<ref>) is of course population dynamics. Skellam [109] was one of the firsts to study the effects of random dispersion on a population subject to  the malthusian law, after noticing that the framework given by [50] and [72] could be adapted to this problem.

In the optic of studying the survival and the distribution of a population in space, a homogeneous environment is not satisfying and one expects the growth of the population to vary according to the habitat conditions.
On the other hand, from a mathematical point of view, heterogeneity in the nonlinearity creates great difficulties. Many new techniques were required to overcome these obstacles.

A first analysis was carried out by Shigesada, Kawasaki and Teramoto
in [108, 107].

The authors observed that natural environments are a mosaic of different habitats, such as forests, meadows, brush, cultivated fields and villages.
This led them to consider an environment which consists of two periodically alternating homogeneous habitats, one favourable, $E^+$, and one unfavourable, $E^-$, for the considered species.

The heterogeneity of the living conditions is reflected by the birth-death rate, which they chose to be

\begin{equation*}
f(x, u)= \left\{
\begin{array}{ll}
u(\mu^+ -u), & \text{in} \ E^+, \\
u(\mu^- -u), & \text{in} \ E^-,
\end{array}
\right.
\end{equation*}

for some $\mu^+>\mu^-$.
Moreover, they also consider possibly varying diffusivity, hence they took

\begin{equation*}
A(x)= \left\{
\begin{array}{ll}
A^+, & \text{in} \ E^+, \\
A^-, & \text{in} \ E^-.
\end{array}
\right.
\end{equation*}

This is due to the observation of increased speed in unfavourable environments; hence we expect $A^+<A^-$ for a real population.
Then, the authors studied in [108] the equation

\begin{equation}\label{0325}
\partial_t u - \nabla \cdot (A(x) \nabla u) = f(x,u) \quad \text{for} \ x\in\R^n.
\end{equation}

This is known as the patch model; they investigated long time behaviour, convergence to travelling fronts and propagation speeds.
Actually, since $u=1$ is no longer an equilibrium for equation (<ref>), we have to modify our definition for species survival; from now on, we intend that  persistence occurs if $u(x,t)$ approaches a non null stationary solution locally uniformly as $t$ tends to infinity.

By making use of numerical simulations, it was found that the stability of the trivial solution $u=0$ plays a key role in determining if the population survives or not.
It was already known (see [35]) that a negative or positive sign of the principal eigenvalue resulting from the linearisation around $u=0$ entails respectively stability or instability of the $0$ solution.
In [108], it was shown numerically that the stability of the trivial solution entails extinction, while its instability causes persistence of the population.
The authors also studied the sign of the eigenvalue depending on the values of $L$, the measures of $E^+$ and $E^-$ and the values of the parameters; this was possible because of the simplicity of the framework.

Equation (<ref>) was later considered in [70] and [18] for general $A(x)$ and $f(x,u)$ depending on $x$ in a continuous fashion and perdiodically of period $L$ for some $L\in\R^n$.
In this second article, Berestycki, Hamel and Roques comprehended that the extinction or persistence of the population depends on the sign of a periodic eigenvalue $\lambda_p(-\mathcal{L}', \R^n)$, that is the unique real number such that the problem

\begin{equation}\label{sys:L_RN_p}
\left\{
\begin{array}{ll}
\mathcal{L'}(\psi) + \lambda \psi = 0, & x\in\R^n, \\
\psi> 0, &  x\in\R^n, \\
|| \psi ||_{\infty}=1, \\
\psi \ \text{is periodic in $x$ of periods $L$},
\end{array}
\right.
\end{equation}

where $\mathcal{L'}$ is given by

\begin{equation*}\label{def:mathcal_L'}
\mathcal{L'}(\psi):=  \nabla \cdot(A(x) \nabla  \psi)  + f_u(x,0)\psi,
\end{equation*}

has a solution $\psi_p\in W_{loc}^{2, 3}(\R^n)$. It was proved that when
$\lambda_p(-\mathcal{L}', \R^n)\geq 0$ extinction occurs. On the other hand,
when $\lambda_p(-\mathcal{L}', \R^n)<0$ there is persistence; moreover, there exists a unique stationary solution to (<ref>), that is periodic of period $L$, and attracts all the solutions starting from a non negative, non zero bounded initial datum.

The studies on the patch model [108, 107] and the ones on periodic media [70, 18] evidenced also the effect of fragmentation on the survival chances of a population. It was found that $\lambda_p(-\mathcal{L}', \R^n)$ decreases as the homogeneity increases, that is, a species has better survival chances when the environment is less fragmented.

§.§.§ The case of a changing climate

A new aspect that one may consider while studying ecological problems is a changing climate. If the environment changes in time, so does the fitness of a population. In this paragraph, we are going to analyse the difficulties produced by the new type on nonlinearity and how it has been overcome.

A 1-dimensional model for population persistence under climate change was first proposed by Berestycki, Diekmann, Nagelkerke and Zegeling in
The authors first imagined that a population lives in a favourable region enclosed into disadvantageous environment.
Assuming that a global warming is in place, and that the population lives in the Boreal Emisphere,
the authors imagined that the favourable region moves to the north, so that for every favourable area lost in the South, an equivalent favourable area is gained in the North.
The resulting equation is

\begin{equation*}
\partial_t u -  \partial_{xx}^2 u=f(x-ct,u) \quad \text{for} \ x\in\R.
\end{equation*}

Later, in [21], Berestycki and Rossi presented a model for climate change in $\R^n$ and for a larger class of nonlinearites; they dealt with equation

\begin{equation}\label{1709}
\partial_t u -  \Delta u=f(x-ct e,u) \quad \text{for} \ x\in\R^n,
\end{equation}

with $e$ a direction in $\mathbb{S}^{n-1}$ and $f: \R^n\times \R^+ \to \R$.
The authors focused on solutions
in the form of a travelling waves $u(x,t)=U(x-cte)$ which solve the equation

\begin{equation}\label{eq:cc}
\partial_t U -  \Delta U- c\,e\cdot \nabla U=f(x,U) \quad \text{for} \ x\in\R^n.
\end{equation}

This second equation is more treatable: in fact, the dependence in time of the nonlinearity, which poses a lot of problems, is transformed into a transport term; now, the equation has a nonlinearity depending only on space, and techniques for this type of heterogeneity are more familiar.

The main question is if the population keeps pace with the shifting climate, that is, if a large enough group is able to migrate with the same speed of the climate. The answer to this question is positive if a solution to (<ref>) exists; as happened for the periodic equation (<ref>), this depends on the sign of the principal eigenvalue coming from the linearisation in $0$.

§.§.§ The road-field model

Spatial heterogeneity in natural environments may be the consequence not only of the diversity of the habitats, but also of the presence of obstacles or fast diffusion channels that affects the fitness and the mobility of individuals.

In recent years, humans activity has caused drastic changes in the environment, causing different species to become invasive in areas they were not present [107]. In the case of the Processionary pine tree caterpillar, the diffusion in France has been even faster than anticipated. It has been observed that the insect was incidentally transported by humans from town to town, and from these settlements it spread in the surroundings [103].
This in not the only example of ecological diffusion acceleration by fast diffusion lines. In Western Canadian Forest, GPS observations on wolves proved  that the animals exploit seismic lines, that are straight roads used by the oil companies to test reservoirs, to move faster and therefore to increase their probability of meeting a prey [83].

Roads play a strong role also in the spreading of epidemics. The “black death” plague in the 14th century was one of the most devastating epidemics known in Europe. It is known that the plague was transported by animals and humans along the commercial trade line of the silk road, and from that  spread all over Europe. More recently, a similar effect has been conjectured for the COVID-19 infection. By tracing the spreading in Northen Italy in early March 2020, it was found that the diffusion occurred first along highways and then spread in the surrounding territory [52].

Inspired by this behaviour, Berestycki, Roquejoffre and Rossi proposed in [20] a model of spreading in an environment presenting a fast diffusion channel. As a simplification, they considered the channel to be a straight line in $\R^2$, the $x$ axis $\R\times \{ y=0 \}$.
Their idea was to split the population into two groups; the first one, of density $u$, occupies the one dimensional environment $\R\times \{ y=0 \}$ representing the road, and the second one, of density $v$, occupies the surrounding territory; by symmetry, they considered just one half of the plan, thus $\Omega:=\{(x, y) \in\R^2 : y>0 \}$, which they called “the field”. These two groups continuously exchange along the road: a fraction $\nu>0$ of the population in $\Omega$ at $y=0$ passes in the road, and a fraction $\mu>0$ of the population in the road passes in the field.
The diffusivity is different in the two environments; its values are $D$ on the road and $d$ on the field, both positive.
Moreover, it is supposed that population reproduces only in the field and that the environment is homogeneous; the corresponding function $f$ is required to satisfy (<ref>), $f'(0)>0$ and a stronger version of the KPP hypothesis,  that is

\begin{equation*}
v \mapsto \frac{f(v)}{v} \quad  \text{is decreasing}.
\end{equation*}

The resulting system, called road-field model, is

\begin{equation}\label{sys:rf}
\left\{
\begin{array}{ll}
\partial_t u(x,t) - D \partial_{xx}^2 u (x,t) = \nu v (x,0,t) - \mu u(x,t), &  x\in \R, \ t > 0, \\
\partial_t v(x,y,t) - d \Delta v (x,y,t)= f(v), & (x,y) \in \Omega, \ t>0, \\
-d \partial_y v(x,0,t) = -\nu v(x,0,t) + \mu u(x,t), & x \in \R, \ t>0.
\end{array} \right.
\end{equation}

The authors of [20] found that invasion occurs for any non negative, non zero initial datum, so the hair trigger effect holds; solutions converge to the unique steady state $\left(\frac{\nu}{\mu},1  \right)$. Moreover, they studied spreading speeds and found that it is enhanced by the presence of the road.
In a second paper [19], the same authors investigated system (<ref>) with a transport term and a reaction term on the line.

Many variations of the road-field model were proposed.
In [94, 95], the system was modified by introducing nonlocal exchanges between the road and the field.
The case of a general nonlocal diffusion has been treated in [14, 13].
Different geometric settings have also been considered; in  [105], the model was extended in higher dimensions.
For a complete list, we refer to the chapter in [112] by Tellini.

Treating system (<ref>) poses some difficulties because of the interaction between functions living in different dimensions and the unusual boundary condition. Adding some heterogeneity in space increases the difficulties. This is why very few studies of this type have carried on so far, a part from an article by Giletti, Monsaingeon and Zhou [58], where the authors considered the case of exchanges terms depending periodically on $x$.

Recently, Berestycki, Ducasse and Rossi introduced in [16] a new generalised principal eigenvalue fitting road-field models for a possibly heterogeneous reaction term.
Hence, they considered the system

\begin{equation*}
\left\{
\begin{array}{ll}
\partial_t u(x,t) - D \partial_{xx}^2 u (x,t) -c \partial_x u(t,x)= \nu v (x,0,t) - \mu u(x,t), &  x\in \R, \ t > 0, \\
\partial_t v(x,y,t) - d \Delta v (x,y,t)-c \partial_x u(t,x)= f(x,y,v), & (x,y) \in \Omega, \ t>0, \\
-d \partial_y v(x,0,t) = -\nu v(x,0,t) + \mu u(x,t), & x \in \R, \ t>0.
\end{array} \right.
\end{equation*}

\begin{equation*}\label{sys:operators}
\left\{
\begin{array}{l}
\mathcal{R}(\phi, \psi):=D \phi''+c \phi'+\nu {\psi}|_{y=0}-\mu \phi, \\
\mathcal{L}(\psi):= d\Delta \psi +c \partial_x \psi -f_v(x,y,0)\psi, \\
B(\phi, \psi):=d \partial_y {\psi}|_{y=0}+\mu \phi- \nu {\psi}|_{y=0},
\end{array}
\right.
\end{equation*}

this eigenvalue is defined as

\begin{equation}\label{def:lambda1_S_Omega}
\begin{split}
\lambda_1( \Omega)=\sup \{ \lambda \in \R \ : \ \exists (\phi, \psi)\geq (0,0), \ (\phi, \psi) \not\equiv(0,0), \ \text{such that} \\ \mathcal{L}(\psi) + \lambda \psi \leq 0 \ \text{in} \ \Omega, \ \mathcal{R}(\phi, \psi) +\lambda \phi \leq 0
\ \text{and} \ B(\phi, \psi)\leq 0 \ \text{in} \ \R \},
\end{split}
\end{equation}

with $(\phi, \psi)$ belonging to $W_{loc}^{2,3}(\R)\times W_{loc}^{2,3}(\overline{\Omega})$. Together with the definition, many interesting properties and bounds were studied.

Thanks to that, the same authors were able to investigate the case of
a favourable ecological niche, possibly facing climate change, in [17]. It was proven that the sign of $\lambda_1( \Omega)$ characterises
the extinction or the persistence of the population; moreover, comparing the results with  the ones found for the model without the road, in the absence of climate change a deleterious effect of the road on the survival chances was found.
On the other hand, if the ecological niche shifts, the road has in some cases a positive effect on the persistence.

§.§ A KPP model with a fast diffusion line in a periodic medium

We are now ready to introduce in details the first problem dealt with in this thesis.
We are going to investigate a road-field model in a periodic medium.
This problem combines the interests of studying the effect of a fast diffusion line with the one of treating a heterogeneous nonlinearity, that, as we pointed out before, reflects a natural territory in a more realistic way than a homogeneous term. From a technical point of view, it also combines the difficulties of the two settings.

§.§.§ The model

We have already presented the road-field model.
In our problem, we treat a road-field system with possible climate change and with a reaction term depending on the spatial variable $x$; in particular, we will focus on the case of periodic dependence.
There is no dependence in the variable $y$, the heterogeneity in that direction is only due to the presence of the road.

Keeping the notation used so far, the system we investigate reads

\begin{equation}\label{sys:fieldroad}
\left\{
\begin{array}{lr}
\partial_t u-D  u '' -c u' - \nu  v|_{y=0} + \mu u= 0,   & x\in \R,  \\
\partial_t v -d \Delta v-c\partial_x v  =f(x,v),  & (x, y)\in \Omega, \\
-d  \partial_y{v}|_{y=0} + \nu v|_{y=0} -\mu u=0, & x\in\R.
\end{array}
\right.
\end{equation}

Recall that $D$, $d$, $\nu$, $\mu$ are positive constants and $c\geq 0$.
The function  $f:\R\times \R_{\geq 0}\to \R $
is always supposed to be $\mathcal{C}^{1}$ in $x$, locally in $v$, and Lipschitz in $v$, uniformly in $x$; moreover we suppose that the value $v=0$ is an equilibrium, that is

\begin{equation}\label{hyp:0}
f(x,0)=0, \quad \text{for all} \ x\in \R,
\end{equation}

and that

\begin{equation}\label{hyp:M}
\exists M>0 \ \text{such that} \ f(x, v)<0 \quad \text{for all} \ v>M \ \text{and all} \ x\in \R,
\end{equation}

which indicates that there is a saturation level.
We will derive some inequalities on the generalised principal eigenvalue of (<ref>) for the general case of $f$ respecting these hypothesis and
$c$ possibly nonzero.

The characterisation of extinction or persistence of the species is addressed in the case of $c=0$ and $f$ a periodic function, reflecting the periodicity of the environment in which the population diffuses, as we require with the forthcoming hypothesis.
We will analyse the  case of a KPP nonlinearity, that is, we require that

\begin{equation}\label{hyp:KPP}
\frac{f(x,s_2)}{s_2}< \frac{f(x,s_1)}{s_1} \quad  \text{for all} \ s_2>s_1>0 \  \text{and all} \ x\in\R.
\end{equation}

Then, we suppose that there exists $\ell> 0$ such that

\begin{equation}\label{hyp:per}
f(x+\ell, s)=f(x,s) \quad \text{for all} \ s >0 \  \text{and all} \ x\in\R.
\end{equation}

To study the effect of the line of fast diffusion, we will compare the behaviour of (<ref>) to the one of the system

\begin{equation}\label{sys:symmetric}
\left\{
\begin{array}{ll}
v_t-d\Delta v - c\partial_x v= f(x,v),  & (x,y)\in\Omega,\\
-\partial_y v|_{y=0} =0, & x\in\R,
\end{array}
\right.
\end{equation}

whose solution is a function $v(x,y)$ that can be extended by symmetry to the whole plane.
It is natural to consider system (<ref>) as the counterpart of system (<ref>) in the case without the road, since it presents the same geometry, including the same boundary condition, exception made for the exchange terms that are in place for the case of a fast diffusion channel.

§.§ Our results

We are now ready to present the main results of this part of the thesis.

The case of a periodic $f(x,v)$.

Here, we consider the case of a nonlinearity that respects the KPP hypothesis and is periodic in the direction of the road. Moreover, we consider $c=0$.

We begin by the following result on the long time behaviour for solutions of system (<ref>). As already seen for similar problems, the key point lies in the stability of the $0$ solution. This is linked to the sign of the generalised principal eigenvalue for the road-field model, that we have defined in (<ref>). With this notation, we have the following:

Let $f$ satisfy (<ref>)-(<ref>) and $c=0$.
Then the following holds:

* if $\lambda_1( \Omega)\geq 0$, then  extinction occurs.

* if $\lambda_1(\Omega)<0$, then persistence occurs and the  positive stationary solution $(u_{\infty}, v_{\infty})$ is unique and periodic in $x$.

Next, we compare the behaviour of solutions of the system (<ref>) with the ones of (<ref>), or, equivalently, after  extension by symmetry to the whole plane, of

\begin{equation}\label{eq:bhroques}
\partial_t v - d \Delta v = f(x,v), \quad \text{for} \ (x,y)\in\R^2.
\end{equation}

Recalling the results of [18], we know that the persistence or extinction of a population for a periodic equation in the whole $\R^2$ depends on the sign of the periodic eigenvalue $\lambda_p(-\mathcal{L}, \R^2)$, that was defined in (<ref>) for a general case.
We obtain the following:

Assume $f$ fulfils hypotheses (<ref>)-(<ref>) and let $c=0$. Then:

* if  $\lambda_p(-\mathcal{L}, \R^2)<0$, then $\lambda_1( \Omega)<0$, that is, if persistence occurs  for the system “without the road” (<ref>), then it occurs also for system “with the road” (<ref>).

* if $\lambda_p(-\mathcal{L}, \R^2)\geq 0$, then $\lambda_1( \Omega)\geq 0$, that is, if extinction occurs for the system “without the road” (<ref>), then it occurs also for system “with the road” (<ref>).

Theorem <ref> asserts that the road has no negative impact on the survival chances of the population in the case of a medium depending periodically on with respect to variable in the direction of the road.

We recall the fact that fragmentation lowers the survival possibilities of a species (see [18, 108]); also, even if we are not in the framework of an ecological niche, we remember from [17] the fact that a road has a negative impact in the setting without climate change. For those reasons, the result in Theorem <ref> may be somehow unexpected.
However, despite the fact that no reproduction takes place on the road, in the case of periodic media the presence of the fast diffusion channel does not interfere with the long time behaviour of the population, which depends only on the  environment of a periodicity cell.
As seen in [18], where the dependence of persistence on the amplitude of fragmentation was studied, if the favourable zones are sufficiently large, the population will eventually spread in all of them; the presence of the road does not cause loss of favourable environment and consequently of persistence chances.
However, we expect the spreading speed to be influenced by the presence of the road, as it has been already proven in the case of homogeneous environment.

We point out that Theorem (<ref>) completes and is in accordance with the results on long time behaviour found in [20] for a homogeneous reaction function, which we can consider as a particular case of periodicity, satisfying a positive KPP request (thanks to the hypothesis $f'(0)>0$). In [20], Theorem 4.1 states the convergence of any positive solution to the unique positive stationary solution of the system.  Since it is well known that for the homogeneous case it holds $\lambda_1(-\mathcal{L}, \R^2)=- f'(0)$, the hypothesis gives that $\lambda_1(-\mathcal{L}, \R^2)<0$ and, as a consequence of Theorem <ref>, that persistence occurs.
Instead if $f'(0)\leq0$, then we would be in the first case of Theorem <ref>, yielding extinction of the population.

Effects of amplitude of heterogeneity

One may ask if the presence of a road may alter the complex interaction between more favourable and less favourable zones; in particular, one could wonder if this could penalise the persistence, since it was shown that populations prefer a less fragmented environment. Nevertheless, owing from Theorem <ref> that the road has no effect on the survival chances of the species, we can recover all the results on the effect of fragmentation.

Take a parameter $\alpha>0$ and consider system (<ref>) with nonlinearity

\begin{equation}\label{1421}
\tilde{f}(x,v)=\alpha f(x,v).
\end{equation}

To highlight the dependence on $\alpha$, we will call $\lambda_1(\Omega, \alpha)$ the  generalised principal eigenvalue defined in (<ref>) with nonlinearity $\tilde{f}$.
As a direct consequence of our Theorem <ref> and of Theorem 2.12 in [18], we have the following result on the amplitude of heterogeneity:

Assume $\tilde{f}$ is defined as in (<ref>), $f$ satisfies (<ref>)-(<ref>), and $c=0$. Then:

* if $ \int_{0}^{\ell} f_v(x,0)>0$, or if $ \int_{0}^{\ell} f_v(x,0)=0$ and $f\neq 0$, then for all $\alpha >0$ we have $\lambda_1(\Omega, \alpha  )<0$.

* if $ \int_{0}^{\ell} f_v(x,0)<0$, then  $\lambda_1(\Omega, \alpha )>0$ for $\alpha$ small enough; if moreover there exists $x_0\in[0,\ell]$ such that $f_v(x_0,0)>0$, then for all $\alpha$ large enough $\lambda_1(\Omega, \alpha )<0$.

A climate change setting for a general $f(x,v)$.

We consider now a general nonlinearity that depends on the variable in the direction of the road. We stress the fact that we do not suppose any periodicity, but the case of a periodic $f$ is a particular case of this setting. Moreover, the following results are done in the general framework of a possible climate change, so the parameter $c$ may be different from $0$.

Comparison between the systems with and without the road, in the general case, are done through comparison between $\lambda_1(\Omega)$ and the generalised principal eigenvalue of system (<ref>), given by

\begin{equation}\label{lambda:L_Omega}
\begin{split}
\lambda_1(-\mathcal{L}, \Omega)=\sup \{ \lambda \in \R \ : \ \exists \psi \geq 0, \psi \not\equiv 0 \ \text{such that} \\
\mathcal{L}(\psi) + \lambda \psi \leq 0 \ \text{on} \ \Omega, \ -\partial_y \psi|_{y=0}\leq 0 \ \text{on} \ \R \}
\end{split}
\end{equation}

for $\psi\in W_{loc}^{2,3}(\Omega)$. With this notation, we have the following:

Assume $\lambda_1(-\mathcal{L}, \R^2)$ as in (<ref>) and $\lambda_1(\Omega)$ as in (<ref>); then $\lambda_1(-\mathcal{L}, \R^2) \geq \lambda_1(\Omega)$.

In the special case $c=0$, some information on the relations between $\lambda_1(-\mathcal{L}, \R^2)$ and $\lambda_1(\Omega)$  was already available in [17]: Proposition 3.1 gives that if $\lambda_1(-\mathcal{L}, \R^2)\geq 0$ then $\lambda_1(\Omega)\geq 0$. Thanks to that and Theorem <ref>, the following result holds:

If $c=0$, we have $\lambda_1(-\mathcal{L}, \R^2)<0$ if and only if $\lambda_1(\Omega)<0$.

As already pointed out in [16], even for $c=0$ it is not true that $\lambda_1(-\mathcal{L}, \R^2) =\lambda_1(\Omega)$. In fact, it has been found that $\lambda_1(\Omega) \leq \mu$, while playing with $f$ one can have $\lambda_1(-\mathcal{L}, \R^2)$ as large as desired. However, the fact that they have the same sign reveals that they are profoundly linked.

§.§.§ Perspectives

The next problem to tackle for system (<ref>) in a periodic medium regards  the existence of travelling fronts and the study of their speed in all the direction of the plane.
We point out that, with respect to the classical case, there are great difficulties linked to the anisotropy of the space, due both to the road and to the periodicity of the medium.
An acceleration effect due to the presence of the road is expected to be found when $D>d$; however, the repercussions of the periodicity of the medium on the spreading speed in a general direction is hard to predict.

We also mention that it would be nice to extend the current results to the case of heterogeneous exchange terms, periodic in $x$, as already treated in [58]. The key point for attacking that problem is in the generalisation of the definition of $\lambda_1(\Omega)$ for non homogeneous coefficients.

§ A NEW MODEL FOR AGGRESSIVE COMPETITION

§.§ Lotka-Volterra models: a literature overview

Another issue that is overlooked in the logistic equation is the interaction of a species with the other ones living in the same environment. In the '20s, Lotka [78] and Volterra [121] observed independently some curios transitory oscillations in the concentration of chemicals during a reaction and in the population sizes of fishes.

They formulated the following model; let $u$ be the quantity of a species of plants present in the environment and $v$ the size of a population of herbivores. It is supposed that the plants have a constant growth rate at all times, $a>0$. The herbivorous feed exclusively on the observed plant and have limitless appetite.
The consumption of plants eaten by the animals is supposed to depend on the probability of meeting of the two, represented by $uv$; the actual loss of the plants is $-buv$ and the gain for the herbivores is $duv$ with $b>d>0$, owning the fact that some plants could be torn but not consumed.
Moreover, as in the malthusian equation, the increase of a population is suppose to depend on its size.
It is also supposed that the environment conditions are stable and than no mutation in the behaviour of the two species is possible.

Then, the model reads

\begin{equation}\label{model:lv}
\left\{
\begin{array}{llr}
\dot{u}&= au-buv, & {\mbox{ for }}t>0,\\
\dot{v}&= -cv+duv, & {\mbox{ for }}t>0.
\end{array}
\right.
\end{equation}

This system has two equilibria, $(0,0)$ and $\left(  \frac{c}{d},\frac{a}{b}  \right)$. If the initial datum is any point of positive coordinates distinct from the equilibrium, the population sizes oscillate in time, running on a closed curve on the phase portrait.

§.§.§ Competitive Lotka-Volterra models

Since the pioneer works, many studies on the interaction between populations were carried out. In particular, after the studies of Gause [54], another model has been employed to investigate the dynamics between two populations in competition, that is, exploiting at least partly the same resources.
We propose here its construction using the example of two population of squirrels, the grey one and the red one, following the work in [90].
These two species, one of the two recently introduced in Britain, both inhabit hardwood forests and rely on the same resources to live. Keeping in mind the derivation of the logistic equation, we realize that the resource term in this scenario depends on the size of both population.
Moreover, we take into consideration the fact that, due to the social organisation and sometimes the segregation between competing species, the presence of individuals of the rival group may obstruct food collection; if this is the case, there is an additional decrease of the available resources for both population.
Adding these corrections to the logistic of both groups,  the Lotka-Volterra competitive system reads

\begin{equation}\label{lv}
\begin{cases}
\dot{u}=a_u  u\left(1-\displaystyle
\frac{u+\alpha_{uv} v}{k_u}  \right), & t>0,\\
\dot{v}
=a_v v\left(1- \displaystyle\frac{v+\alpha_{vu} u}{k_v}  \right), & t>0,
\end{cases}
\end{equation}

where $a_u$, $a_v$, $\alpha_{uv}$, $\alpha_{vu}$, $k_u$ and $k_v$ are nonnegative real numbers.
The coefficients $a_u$ and $a_v$ are the intrinsic growth rates of the two population; $k_u$ and $k_v$ represent the carrying capacities of the environment for the two groups.
The coefficients $\alpha_{uv}$ and $\alpha_{vu}$ represent the competition between individuals of different species, and indeed they appear multiplied by the term $uv$, which represents a probability of meeting.
Taking the example of the squirrels, we expect that $\alpha_{uv}, \alpha_{vu} >1$.
However, for other couple of populations relying on only partially overlapping food sets, one could have also $\alpha_{uv}, \alpha_{vu} \leq 1$. If finally the first population feeds on a subset of the resources of the second one, it would be $\alpha_{uv} \geq1$ and $\alpha_{vu} <1$.

For the sake of completeness, we recall that in the case of species mutually benefiting from the presence of the other, which is not part of the competitive framework, the dynamics prescribes negative values for $\alpha_{uv}$ and $\alpha_{vu}$.

The dynamics of system (<ref>) depends indeed on the values of the interspecific competition terms:
if $\alpha_{uv}<1<\alpha_{vu}$, then the first species $u$
has an advantage over the second one $v$
and will eventually prevail; if $\alpha_{uv}, \ \alpha_{vu} >1$, then the first population that penetrates the environment (that is, the one that has a greater size at the initial time) will persist while the other will extinguish;  if $\alpha_{uv}, \ \alpha_{vu} <1$, there exists an attractive coexistence steady state.
The fact that, if two populations' ecological niches completely overlap, then one of the two species gets extinct, is exactly the statement of the Gause principle, a well-established law in ecology.

The Lotka-Volterra models of ODEs have been extended in many ways and its applications range from  technology substitution to business competition. In the stochastic analysis community, system (<ref>) with additioned noise terms has been largely studied [62].
Another branch were these systems have been of huge influence is, of course, reaction-diffusion equations.
In the next paragraph we are spending some words on the results for diffusive Lotka-Volterra competitive systems.

§.§.§ Competitive Lotka-Volterra model with diffusion

In the interaction between different population, as already happens in the dynamic of a single species, spatial organisation plays an important role.
A great literature has been devoted to the competitive Lotka-Volterra system with diffusion, that is, up to a rescaling,

\begin{equation}\label{model:diffusion}
\begin{cases}
\partial u - \Delta u=  u\left(1-u-\alpha_{uv} v  \right), & x\in\R^n, \  t>0,\\
\partial v - d\Delta v
=a v\left(1- v- \alpha_{vu} u \right), & x\in\R^n, \  t>0,
\end{cases}
\end{equation}

for some $d$, $a$, $\alpha_{uv}$, $\alpha_{vu}$  positive constants.
Richer dynamics and more questions naturally arise for system (<ref>) but, unsurprisingly, the study of these  involves many more difficulties.
Just to give some flavour of those, we provide some details on the study of the speed of propagation of travelling waves connecting the steady states  $\left(1, 0\right)$ and $\left( 0, 1\right)$, to which many studies have been consecrated. From the works of Lewis, Li and Weinberger [75, 76] the minimal speed of propagation of a monotonic wave is called $c_{LLW}$. Even it the simplest case stated in (<ref>), the exact value of $c_{LLW}$ is still not known. Calling $c_{KPP}$ the minimal speed of diffusion for the second equation under the assumption $u\equiv 0$, it holds that $c_{LLW}\geq c_{KPP}$, but the inequality may be strict depending on the parameters [64, 65].

Nevertheless, system (<ref>) is one of the simplest among many possibilities; in the literature one finds systems considering
nonlocal diffusion [91],
free boundary [44],
cross diffusion  [79],
and many other variations.

§.§ A model of Lotka-Volterra type for aggressive competition and analysis of the strategies

Among the several models dealing with the
dynamics of biological systems, the case of populations in open hostility
seems to be rather unexplored.
Our model considers the case of
two populations competing for the same resource with one aggressive population
which attacks the other:
concretely, one may think of
a situation in which
two populations live together in the same territory and share the same
environmental resources,
till one population wants to prevail and try to overwhelm the other.
We consider this situation as a “civil war”, since the two populations share
land and resources.

§.§.§ The model

We now describe in further detail our model of conflict between the
two populations and the attack strategies pursued by the aggressive population.
Given the lack of reliable data related to civil wars,
the equations were derived by deduction
from the principles of population dynamics.
Our idea is to modify the Lotka-Volterra competitive system for two populations with
density $u$ and $v$,
adding to the usual competition for resources the fact that both populations suffer some losses as an outcome of the attacks.
The key point in our analysis
is that the clashes do not depend on the chance of meeting of the two populations, given by the quantity $uv$, as it happens in many other works in the literature, but they are sought by the first population and
depend only on the size $u$ of the first population and on its level of aggressiveness $a$.
The resulting model is

\begin{equation}\label{model}
\left\{
\begin{array}{llr}
\dot{u}&= u(1-u-v) - acu, & {\mbox{ for }}t>0,\\
\dot{v}&= \rho v(1-u-v) -au, & {\mbox{ for }}t>0,
\end{array}
\right.
\end{equation}

where $a$, $c$ and $\rho$ are nonnegative real numbers. Here, the coefficient $\rho$ models the fitness of the second population with respect to the first one. The parameter $c$ here stands for the quotient of endured per inflicted  damages for the first population.

§.§.§ Behaviour of solutions

We denote by $(u(t), v(t))$ a solution of (<ref>) starting from a point $(u(0),v(0))\in [0,1] \times [0,1]$.
We will also refer to the orbit of $(u(0), v(0))$  as the collection of
points $(u(t), v(t))$ for $t\in \R$, thus both positive and negative times, while the trajectory is the collection of points $(u(t), v(t))$ for $t\geq0$.

From the equations in (<ref>), one promptly sees that $v=0$ is not an equilibrium, hence,
$v$ can reach the value $0$ and even negative values in finite time.
From a modelling point of view, negative values of $v$ are not acceptable, being it a population density.
However, we will suppose that the dynamics stops when the value $v=0$ is reached for the first time.
At this point, the conflict ends with the victory of the first population $u$, that can continue its evolution with a classical Lotka-Volterra equation of the form

\begin{equation*}
\dot{u}= u (1- u)
\end{equation*}

and that would certainly fall into the attractive equilibrium $u=1$.

In order to state our first result on the dynamics of the system (<ref>),
we first observe that, in a real-world situation, the value of $a$ would probably be non-constant and discontinuous, so we allow this coefficient to take values in the class $\mathcal{A}$
defined as follows:

\begin{equation}\begin{split}\label{DEFA}
\mathcal{A}&\; :=
\big\{a: [0, +\infty) \to [0, +\infty) {\mbox{ s.t.~$a$ is continuous}}\\
&\qquad \qquad {\mbox{except at most at a finite number of points}}\big\}.\end{split}\end{equation}

A solution related to a strategy $a(t)\in \mathcal{A}$ is a pair $(u(t), v(t)) \in C_0 (0,+\infty)\times C_0 (0,+\infty)$, which is $C^1$ outside the discontinuity points of $a(t)$ and
solves system (<ref>).
Moreover, once the initial datum is imposed, the solution is assumed to be
continuous at $t=0$.

In this setting, we establish the existence of the solutions to problem (<ref>)
and we classify their behavior with respect to the possible exit from the domain $[0,1]\times[0,1]$. Given $(u(0), v(0))\in [0,1] \times [0,1]$ and $a(t)\in\mathcal{A}$, two scenarios are possible for a solution $(u(t),v(t))$ with $a=a(t)$
of system (<ref>) starting at $(u(0), v(0))$:

(1) The solution $(u(t), v(t))$ issued from $(u(0), v(0))$ belongs to $ [0,1]\times (0,1]$ for all $t\geq 0$.

(2) There exists $T\geq0$ such that the solution $(u(t), v(t))$ issued from $(u(0), v(0))$ exists unique for all $t\leq T$, and $v(T)=0$ and $u(T)>0$.

As a consequence, we can define the
the stopping time of the solution $(u(t), v(t))$ as

\begin{equation}\label{def:T_s}
T_s (u(0), v(0)) =
\left\{
\begin{array}{ll}
+\infty & \text{if situation (1) occurs}, \\
T & \text{if situation (2) occurs}.
\end{array}
\right.
\end{equation}

From now on, we will implicitly  consider solutions $(u(t),v(t))$ only for $t\leq T_s(u(0), v(0))$.

We call victory of the first population the scenario where $T_s < +\infty$, that corresponds to the case where $v(T_s)=0$ and $u(T_s)>0$.
On the other hand,  we call victory of the second population the scenario where $(u,v)$ tends to $(0,1)$ as $t$ tends to infinity.

Now we are going to analyze the dynamics of (<ref>) with a particular focus on possible strategies. To do this, we now define the basins of attraction.
The first one is the basin of attraction of the point $(0,1)$, that is

\begin{equation}\label{DEFB}\begin{split}
\mathcal{B}&\;:= \Big\{ (u(0),v(0))\in [0,1]\times[0,1] \;{\mbox{ s.t. }}\;\\
&\qquad\qquad T_s (u(0), v(0)) = +\infty, \ (u(t),v(t)) \overset{t\to\infty}{\longrightarrow} (0,1)  \Big\},
\end{split}\end{equation}

namely the set of the initial points for which the first population gets extinct (in infinite time) and the second one survives.
The other one is

\begin{equation}\label{DEFE}
\mathcal{E}:= \left\{ (u(0),v(0))\in ([0,1]\times[0,1])\setminus(0,0) \;{\mbox{ s.t. }}\;  T_s(u(0),v(0))< + \infty \right\},
\end{equation}

the set of initial points for which we have the victory of the first population and the extinction of the second one.

§.§.§ A control problem

In a rational, or at least well-organised population, one may expect that the parameter $a$, representing aggressiveness, is subject to control; we are suggesting that a population, by performing premeditated attacks, may control its strategy in the conflict and  would be able to choose the most appropriate one.

From now on, we may refer to the parameter $a$ as the strategy, that may also depend on time, and we will say that it is winning if it leads to victory of the first population.
We also notice that, with this choice, (<ref>) is a control-affine system.

The main problems that
we deal with are:

* The characterization of the initial conditions for which there exists a winning strategy.

* The success of the constant strategies, compared to all  possible strategies.

* The construction of a winning strategy for a given initial datum.

* The existence of a single winning strategy independently of the initial datum.

* The existence of a winning strategy minimizing duration of the war.

The first question is a problem of target reachability for a control-affine system.
The second point regards the choice of a suitable functional space where to choose the strategy.
We also construct an actual winning strategy when victory is possible, answering the third and fourth question.
The last question results to be an optimisation problem.

§.§ Our results

§.§.§ Dynamics for a constant strategy

The first step towards the understanding of the dynamics of the system
in (<ref>) is
is to analyze the behavior of the system for constant coefficients.

To this end, we introduce some notation.
Following the terminology on pages 9-10 in [123],
we say that an equilibrium point (or fixed point) of the dynamics
is a (hyperbolic) sink
if all the eigenvalues of the linearized map have strictly
negative real parts, a (hyperbolic) source
if all the eigenvalues of the linearized map have strictly
positive real parts, and a (hyperbolic) saddle
if some of the eigenvalues of the linearized map have strictly
positive real parts
and some have negative real parts
(since in this problem we work in dimension $2$,
saddles correspond to linearized maps with one
eigenvalue with
strictly positive real part
and one eigenvalue with
strictly negative real part).
We also recall that
sinks are asymptotically stable (and sources are
asymptotically stable for the reversed-time dynamics), see e.g. Theorem 1.1.1
in [123].

With this terminology, we state the following theorem:

For $a > 0$, $c>0$ and ${\rho}> 0$ the system (<ref>) has the following features:

(i) When $0<ac<1$, the system has 3 equilibria: $(0,0)$ is a source, $(0,1)$ is
a sink, and

\begin{equation}\label{usvs}
(u_s, v_s):= \left( \frac{1-ac}{1+{\rho}c} {\rho}c, \frac{1-ac}{1+{\rho}c} \right) \in (0,1)\times (0,1)
\end{equation}

is a saddle.

(ii) When $ac>1$, the system has 2 equilibria: $(0,1)$ is a sink and $(0,0)$ is a saddle.

(iii) When $ac=1$, the system has 2 equilibria: $(0,1)$ is a sink and $(0,0)$
corresponds to a strictly positive eigenvalue and a null one.

(iv) We have

\begin{equation} \label{fml:division}
[0,1]\times [0,1] = \mathcal{B} \cup \mathcal{E} \cup \mathcal{M}
\end{equation}

where $\mathcal{B}~$ and $\mathcal{E}$ are defined in (<ref>)
and (<ref>), respectively, and $\mathcal{M}$ is a smooth curve.

(v) The trajectories starting in $\mathcal{M}$ tend to $(u_s,v_s)$ if $0<ac<1$,
and to $(0,0)$ if $ac\ge1$ as $t$ goes to $+\infty$.

More precisely, one can say that
the curve $\mathcal{M}$ in Theorem <ref> is the stable manifold of the saddle
point $(u_s,v_s)$ when $0<ac<1$, and of the
saddle point $(0,0)$ when $ac>1$. The case $ac=1$ needs a special treatment,
due to the degeneracy of one eigenvalue, and in this case the curve $\mathcal{M}$
corresponds to the center manifold of $(0,0)$, and an ad-hoc argument will
be exploited
to show that also in this degenerate case orbits that start in $\mathcal{M}$
are asymptotic in the future to $(0,0)$.
As a matter of fact, $\mathcal{M}$
acts as a dividing wall between the two basins of attraction $\mathcal{B}$ and $\mathcal{E}$, as described in (iv)
of Theorem <ref>.

From a modelling point of view, Theorem <ref> shows that, also for our model, the Gause principle of exclusion is respected; that is, in general, two competing populations cannot coexist in the same territory, see e.g. [47].

One peculiar feature of our system is that, if the aggressiveness is too strong, the equilibrium $(0,0)$ changes its “stability” properties, passing from a source (as in (i) of
Theorem <ref>)
to a saddle point (as in (ii) of
Theorem <ref>). This shows that the war may have self-destructive outcomes, therefore it is important for the first population to analyze the situation in order to choose a proper level of aggressiveness.
Figure <ref> shows one example of dynamics for each case.

The dedicated chapter contains further results on the dependence of $\mathcal{B}$ and $\mathcal{E}$ on the parameters $\rho$ and $c$. The parameter $a$, a part from having a more intricate influence on the system, may be interpreted not as a biological constant but rather as a choice of the first population.
Therefore, we perform a deeper analysis, whose result are presented in the next paragraph.

$a=0.8$, $c=0.5$, $\rho=2$

$a=0.8$, $c=3$, $\rho=2$

The figures show a phase portrait for the indicated values of the coefficients. In blue, the orbits of the points. The red dots represent the equilibria. The images are realised with Python.

§.§.§ Dynamics for variable strategies and optimisation results

We now introduce some terminology.
Recalling (<ref>),
for any $\mathcal{T}\subseteq \mathcal{A}$, we set

\begin{equation}\label{DEFNU}
\mathcal{V}_{\mathcal{T}}:= \underset{a(\cdot)\in \mathcal{T}}{\bigcup} \mathcal{E}(a(\cdot)),
\end{equation}

where $\mathcal{E}(a(\cdot))$ denotes the set of initial data $(u_0,v_0)$
such that $T_s(u_0,v_0)< +\infty$, when the coefficient $a$ in (<ref>) is replaced by the function $a(t)$.
Namely, $\mathcal{V}_{\mathcal{T}}$ represents the set of initial conditions for which $u$ is able to win by choosing a suitable strategy in $\mathcal{T}$; we call $\mathcal{V}_{\mathcal{T}}$ the victory set with admissible strategies in $\mathcal{T}$.
We also say that $a(\cdot)$ is a winning strategy for the point $(u_0,v_0)$
if $(u_0,v_0)\in \mathcal{E}(a(\cdot) )$.

Moreover, we will call

\begin{equation}\label{u0v0}
(u_s^0, v_s^0):= \left(\frac{\rho c}{1+\rho c}, \frac{1}{1+\rho c}\right).
\end{equation}

Notice that $(u_s^0, v_s^0)$ is the limit point as $a$ tends to $0$ of the sequence of saddle points $\{(u_s^a, v_s^a)\}_{a>0}$
defined in (<ref>).

With this notation,
the first question that we address is for which initial configurations it is possible for the population $u$
to have a winning strategy, that is, to characterize the victory set. For this, we allow the strategy to take all the values in $[0, +\infty)$.
In this setting, we have the following result:

(i) For $\rho=1$, we have that

\begin{equation}\label{Vbound1}\begin{split}
\mathcal{V}_{\mathcal{A}} = \,&\Big\{ (u,v)\in[0,1] \times [0,1] \;
{\mbox{ s.t. }}\;  v-\frac{u}{c}<0 \; {\mbox{ if }} u\in[0,c]\\
&\qquad\qquad\qquad {\mbox{ and }}\; v\le1 \; {\mbox{ if }} u\in(c,1]\Big\},
\end{split}\end{equation}

with the convention that the last line in (<ref>) is not present if $c\ge1$.

For $\rho<1$, we have that

\begin{equation}\label{bound:rho<1}
\begin{split}
\mathcal{V}_{\mathcal{A}} &\;= \Bigg\{ (u,v)\in[0,1] \times [0,1] \;{\mbox{ s.t. }}\;
v< \gamma_0(u) \ \text{if} \ u\in [0, u_s^0], \\
v< \frac{u}{c} + \frac{1-\rho}{1+\rho c} \ \text{if} \ u\in \left[u_s^0,
\frac{\rho c(c+1)}{1+\rho c}\right]\\
{\mbox{and }}\; v\le1\ \text{if} \ u\in \left(
\frac{\rho c(c+1)}{1+\rho c},1\right]
\Bigg\},
\end{split}
\end{equation}

\begin{equation*}
\gamma_0(u):= \frac{u^{\rho}}{\rho c(u_s^0)^{\rho-1}},
\end{equation*}

and we use the convention that the last line
in (<ref>) is not present if $ \frac{\rho c(c+1)}{1+\rho c}\ge1$.

(iii) For $\rho>1$, we have that

\begin{equation}\label{bound:rho>1}
\begin{split}
\mathcal{V}_{\mathcal{A}} &\;= \Bigg\{ (u,v)\in[0,1] \times [0,1]\;
{\mbox{ s.t. }}\; v< \frac{u}{c} \ \text{if} \ u\in [0, u_{\infty}],\\&\qquad
\qquad\qquad\qquad
v< \zeta(u)  \ \text{if} \ u\in\left(u_{\infty}, \frac{c}{(c+1)^{\frac{\rho-1}\rho}}\right] \\&\qquad
\qquad\qquad\qquad
{\mbox{and }}\; v\le 1
\ \text{if} \ u\in\left(\frac{c}{(c+1)^{\frac{\rho-1}\rho}},1\right]
\Bigg\},
\end{split}
\end{equation}

\begin{equation}\label{ZETADEF}
u_{\infty}:= \frac{c}{c+1}
\quad {\mbox{ and }}\quad \zeta (u):= \frac{u^{\rho}}{c \, u_{\infty}^{\rho-1}} .
%, \quad z:=\min \left\{1, \frac{c}{(c+1)^{1-\frac{1}{\rho}}}\right\}.
\end{equation}

and we use the convention that the last line
in (<ref>) is not present if $  \frac{c}{(c+1)^{\frac{\rho-1}\rho}}\ge1$.

Theorem <ref> implies that the problem is not controllable, that is, for some initial conditions the first population is not able to reach its target.

In practice,
constant strategies could be certainly easier to implement and
it is therefore natural to investigate whether or not
it suffices to restrict the control to constant strategies
without altering the possibility of victory.
The next result addresses this problem by showing that when $\rho=1$
constant strategies are as good as all strategies,
but instead when $\rho\ne 1$ victory cannot be achieved by only
exploiting constant strategies:

Let $\mathcal{K}\subset \mathcal{A}$ be the set of constant functions. Then the following holds:

(i) For $\rho= 1$, we have that $ \mathcal{V}_{\mathcal{A}}=\mathcal{V}_{\mathcal{K}}=\mathcal{E}(a)$ for all $a>0$;

(ii) For $\rho\neq 1$, we have that $\mathcal{V}_{\mathcal{K}} \subsetneq \mathcal{V}_{\mathcal{A}}$.

The result of Theorem <ref>, part (i),
reveals a special rigidity of the case $\rho=1$
in which the victory depends only on the initial conditions, but it is independent of the strategy $a(t)$.
Instead, as stated in
Theorem <ref>, part (ii),
for $\rho \neq 1$ the choice of $a(t)$ plays a crucial role in determining which population is going to win and constant strategies do not exhaust all the
possible winning scenarios.
We stress that $\rho=1$ plays also
a special role in the biological interpretation of the model, since in this case the two
populations have the same fitness to the environmental resource, and hence, in a sense,
they are indistinguishable, up to the possible aggressive behavior of the first population.

Next, we show that for all points in the set $\mathcal{V}_{\mathcal{A}}$ we can choose an appropriate piecewise constant strategy with at most one discontinuity; functions with these properties are called Heaviside functions.

There holds that $\mathcal{V}_{\mathcal{A}} = \mathcal{V}_{\mathcal{H}}$, where $\mathcal{H}$ is the set of Heaviside functions.

The proof of Theorem <ref> solves also the third question mentioned in the introduction. As a matter of fact, it proves that for each point we either have a constant winning strategy or
a winning strategy of type

\begin{equation*}
a(t) = \left\{
\begin{array}{lr}
a_1  &{\mbox{ if }} t<T ,\\
a_2  &{\mbox{ if }} t\geq T,
\end{array}
\right.
\end{equation*}

for some $T\in(0,T_s)$, and
for suitable values $a_1$, $a_2 \in (0,+\infty)$ such that one is very small and the other one very large, the order depending on $\rho$.
The construction that
we give also puts in light the fact that the choice of the strategy depends on the initial datum, answering also our fourth question.

It is interesting to observe that the winning strategy that switches abruptly from a small to a large value
could be considered, in the optimal control terminology, as a “bang-bang” strategy.
Even in a target reachability problem, the structure predicted by Pontryagin's Maximum Principle is brought in light: the bounds of the set $\mathcal{V}_{\mathcal{A}}$, as
given in Theorem <ref>, depend on the bounds that
we impose on the strategy, that are, $a \in[0,+\infty)$.

It is natural to consider also the case
in which the level of aggressiveness
is constrained between a minimal and maximal threshold,
which corresponds to the setting $a\in[m,M]$ for suitable $M\geq m\geq 0$, with the hypothesis that $M>0$.
In this setting, we denote by $\mathcal{A}_{m,M}$ the class of piecewise continuous strategies $a(\cdot)$
in ${\mathcal{A}}$ such that $
m\leq a(t)\leq M$ for all $t>0$ and we let

\begin{equation}\label{SPE}
\mathcal{V}_{m,M}:=\mathcal{V}_{\mathcal{A}_{m,M}}=\underset{{a(\cdot)\in \mathcal{A}}\atop{m\leq a(t)\leq M}
}{\bigcup} \mathcal{E}(a(\cdot))=
\underset{{a(\cdot)\in \mathcal{A}}_{m,M}
}{\bigcup} \mathcal{E}(a(\cdot)).\end{equation}

Then we have the following:

Let $M$ and $m$ be two real numbers such that $M\geq m\geq 0$ and $M>0$. Then, for $\rho\neq 1$ we have the strict inclusion  $\mathcal{V}_{{m,M}}\subsetneq \mathcal{V}_{\mathcal{A}}$.

Notice that for $\rho=1$, Theorem <ref> gives instead that $\mathcal{V}_{{m,M}}= \mathcal{V}_{\mathcal{A}}$
and we think that this is a nice feature, outlining a special role played by the parameter $\rho$
(roughly speaking, when $\rho=1$ constant strategies suffice
to detect all possible winning configurations, thanks to
Theorem <ref>, while when $\rho\ne1$ non-constant strategies are necessary to detect
all winning configurations).

Time minimizing strategy.
Once established that it is possible to win starting in a certain initial condition, we are interested in knowing which of the possible strategies is best to choose. One condition that may be taken into account is the duration of the war. Now, this question can be written as a minimization problem with a proper functional to minimize and therefore the classical Pontryagin theory applies.

To state our next result,
we recall the setting in (<ref>) and define

\begin{equation*}
\mathcal{S}(u_0, v_0) := \Big\{ a(\cdot)\in \mathcal{A}_{m,M}
\;\mbox{ s.t. }\; (u_0, v_0) \in \mathcal{E}(a(\cdot))  \Big\},
\end{equation*}

that is the set of all bounded strategies for which the trajectory starting at $(u_0, v_0)$ leads to the victory of the first population.
To each $a(\cdot)\in\mathcal{S}(u_0, v_0)$ we associate the stopping time defined in (<ref>), and we express its dependence on $a(\cdot)$ by writing $T_s(a(\cdot))$.
In this setting, we provide the following statement concerning the strategy leading
to the quickest possible victory for the first population:

Given a point $(u_0, v_0)\in \mathcal{V}_{m,M}$, there exists a winning strategy $\tilde{a}(t)\in
\mathcal{S}(u_0, v_0)$, and a trajectory $(\tilde{u}(t), \tilde{v}(t) )$ associated with $\tilde{a}(t)$,
for $t\in[0,T]$,
with $(\tilde{u}(0), \tilde{v}(0) )=(u_0,v_0)$, where $T$ is given by

\begin{equation*}
T = \underset{a(\cdot)\in\mathcal{S}}{\min} T_s(a(\cdot)).
\end{equation*}

\begin{equation*}
\tilde{a}(t)\in \left\{m, \ M, \    a_s(t) \right\},
\end{equation*}

\begin{equation}\label{KSM94rt3rjjjdfe}
{a}_s(t) := \dfrac{(1-\tilde{u}(t)-\tilde{v}(t))[\tilde{u}(t) \, (2c+1-\rho c)+\rho c]}{\tilde{u}(t) \, 2c(c+1)}.
\end{equation}

The surprising fact given by Theorem <ref>
is that the
minimizing strategy is not only of bang-bang type, but it may assume some values along a singular arc, given by $a_s(t)$.
This possibility is realized in some concrete cases, as we verified by running some numerical simulations, whose results can be visualized in Figure <ref>.

The figure shows the result of a numerical simulation searching a minimizing time strategy $\tilde{a}(t)$ for the problem starting in $(0.5, 0.1875)$ for the
parameters $\rho=0.5$, $c=4.0$, $m=0$ and $M=10$. In blue, the value
found for $\tilde{a}(t)$; in red, the value of $a_s(t)$ for the corresponding trajectory $(u(t), v(t))$. As one can observe, $\tilde{a}(t)\equiv a_s(t)$ in a long trait.
The simulation was done using AMPL-Ipopt on the server NEOS and pictures have been made with Python.

§.§.§ Perspectives

The system of ODEs is the cornerstone for the study of the reaction-diffusion system

\begin{equation}\label{model2}
\left\{
\begin{array}{lr}
\partial_t u- \partial_{xx}^2 u = u(1-u-v) - acu, & {\mbox{ for }}  x\in\R, \ t>0,\\
\partial_t v- d\partial_{xx}^2 v = \rho v(1-u-v) -a \int_{\R} u, & {\mbox{ for }} x\in\R, \ t>0,
\end{array}
\right.
\end{equation}

for some $d>0$.
We expect solutions to this system to have very interesting behaviours. It is possible that the second population reaches the value $0$ in only some points of the domain, giving an example of the interesting phenomenon known as dead-core, see e.g. [61].

§ EVOLUTION EQUATIONS WITH CLASSICAL AND FRACTIONAL DERIVATIVES

§.§ Fractional derivatives in evolution equations

The idea of fractional calculus first appears in the discussions between Leibniz and  De l'Hospital (see [104]); namely, given the classical derivative $\frac{d^n f(x)}{dx^n}$ for $n\in\N$, it is quite natural to ask if it is possible to define a generalisation of this operator but with non entire order, thus $\frac{d^{\alpha} f(x)}{dx^{\alpha}}$ with $\alpha\in\R$.

However, it is only in the last few decades that a good number of mathematicians started to work on fractional calculus. One of the reasons of this interest is the fact that fractional derivative can help in the modelization of processes with memory or of diffusion phenomena with spreading behaviour different from the one prescribed by Brownian motion. This has applications in charge carrier transport in semicondutors, nuclear magnetic resonance diffusometry, porous systems, dynamics in polymeric systems (see [84] and the references therein).

Here, we introduce some fractional derivatives and operators and justify their meaning and relations with the previous material.
Providing an overview on the state of art over existence, regularity and behaviour of evolution equations dealing with fractional operators is far from our purposes, due to the complexity of the topic. For that, we refer to [6, 33, 34, 98, 106, 124] and the reference therein.

§.§.§ The Caputo derivative

There are several way of defining a fractional derivative. We choose to work with Caputo derivative,
that was first proposed in a geology model by Caputo in [30].
The Caputo derivative of order $\alpha\in(0,1)$ is defined by

\begin{equation*}
D_t^{\alpha} f(t):= \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t} \frac{\dot{f}(\tau)}{(t-\tau)^{\alpha}} d\tau
\end{equation*}

where $\Gamma$ is Euler's Gamma-function and $\dot{f}$ is the classical derivative of $f$.
For simplicity, we will omit the constant and work with

\begin{equation*} \label{def:caputo}
\partial_t^{\alpha} f(t):= \Gamma(1-\alpha)	D_t^{\alpha} f(t).
\end{equation*}

Notice that $\partial_t^{\alpha} f(t)$ is defined for all $f\in\mathcal{C}^1([0, t])$ such that $\dot{f}\in L^1(0,t)$. It is also possible to define Caputo derivatives of higher order, thus with $m-1<\alpha<m$ for $m\in\N$, by the following:

\begin{equation*}
D_t^{\alpha} f(t):= \frac{1}{\Gamma(m -\alpha)} \int_{0}^{t} \frac{{f}^{(m)}(\tau)}{(t-\tau)^{1+\alpha-m}} d\tau.
\end{equation*}

The Caputo derivative describes a process “with memory”, in the sense that the history of the process is encoded in the derivative, though old events “count less” than recent ones, since their value is counted with a smaller weight. Due to this memory property, this operator has
found several applications in models of hydrogeology, heat transmission, percolation.

The Caputo derivative is considered to be an operator of “fractional” order, as opposed to the entire order, that is proper to the classical derivatives.
The value $\alpha$ corresponds to the order of the derivative:
indeed, for a power of order $r>0$, it holds $\partial_t^{\alpha} t^{r}=C t^{r-\alpha}$ for some constant $C>0$.

Among the other types of fractional derivatives, it is worth mentioning the Riemann-Liouville derivative because of its large diffusion. The Riemann-Liouville derivative is defined by

\begin{equation}
\mathcal{D}_t^{\alpha} f(t):= \frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int_{0}^{t} \frac{f(\tau)}{(t-\tau)^{\alpha}} d\tau,
\end{equation}

and making the calculations one can show that it differs from the Caputo derivative by the term $\frac{f(0)}{t^{\alpha}}$. One of the reasons of the popularity of the Riemann-Liouville derivative is that its limit for $\alpha\to1$ coincides with the classical derivative.

Evolution equations with Caputo time derivative.

Classical partial differential equations are often divided in three groups depending on the order of the time derivative: elliptic, parabolic, hyperbolic. Nevertheless, even in the Preface of Partial Differential Equations by Evans [45], the author states that this subdivision is fictive and “creates the false impression that there is some kind of general and useful classification scheme available”.
This subdivision is supposed to put together object with similar behaviours and classic theory results are usually meant for one of these clusters.

An evolution equation with the Caputo time derivative, for example

\begin{equation}\label{cap1}
\partial_t^{\alpha} u - \Delta u =0,
\end{equation}

is not part of any of this groups.
Thus, many results that one may want to use are not available and must be recovered.

However, relations with the classical objects are present.
Because of some behaviour similarities, evolution equations with Caputo time derivative have often been compared to parabolic ones [84].
Recently, in [42], Dipierro and Valdinoci inspected a model of transmission in neural structures and derive equation (<ref>) from basic principles.
Doing so, they realized that it can be seen as a superposition of several hyperbolic equations acting with delay.
Despite this, the behaviour of solutions of (<ref>) is not similar to the one of wave equations: in fact, in opposition to hyperbolic equations, (<ref>) has a regularising effect on initial data.

§.§.§ The fractional Laplacian

An operator that has been very popular in recent years is the fractional Laplacian, which is considered in some sense the fractional counterpart of the classic homonym operator.
Given $s\in(0,1)$, we define the fractional Laplacian as

\begin{equation}\label{0302}
- \left( \Delta\right)^s u(x) := \text{P.V.} \int_{\R^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} dy,
\end{equation}

where “P.V.” stands for Principal Value.
Curiously, many equivalent definitions of the fractional Laplacian are possible: a part the one in (<ref>), in [1, 73] one can find a very exhaustive list together with the most important properties of the operator known so far.

One of the reason for the popularity of the fractional Laplacian is its connection with Lévy processes. We now give the idea behind the derivation of an evolution equation containing the fractional Laplacian from a discrete Lévy process, which is similar to the derivation of the heat equation from a Brownian movement. Consider  an infinite grid $h \Z^n$, for some step $h>0$, and a discrete evolution of time step $\tau=h^{2s}$. Imagine to put a particle at the origin. At each time step $t\in \tau \N$, the particle can jump to any vertex of the grill different from its actual position, with a probability that depends on the length of the jump;  namely, if the particle is at the position $hk$, with $k\in\Z^n$, the probability to jump into the position $hj$, if $j\neq k$, is

\begin{equation*}
\end{equation*}

with $C$ a normalisation constant. Then, we call $u(t,x)$ the probability of finding the particle in $x\in h\Z^n$ at the time $t\in\tau\N$. The function $u(t,x)$ evolves according to the probability of the jumps; for example, the probability of finding the particle in the origin at some time $t+\tau$ is

\begin{equation*}
u(t+\tau, 0)= \sum_{{j\in\Z^n\setminus 0}} P_h(0,j) u(t, j)= \sum_{{j\in\Z^n\setminus 0}} \frac{C }{|j|^{n+2s}}u(t,j)
\end{equation*}

By taking the limit as $h$ tends to $0$,
and by performing suitable manipulations,
the last equality becomes the evolution equation

\begin{equation*}
\partial_t u - (\Delta)^s u=0.
\end{equation*}

For all the details of the proof, we refer to [68].

Satellite-based measures of animal movement performed in the last years have shown that Lévy process are a better approximation of animal movement than Brownian motion. Some examples are provided by honey bees displacements and by movement of marine predators when prey is scarce [10, 66, 101]. In general, it appears that Lévy-flights are more fitting hunt strategies than Brownian walks [77].
For this reason, the fractional Laplacian has been introduced in population dynamics model, see [11, 36, 55] and the reference therein. However, the technical difficulties of dealing with such delicate operators have not been totally overcome.

§.§ Decay estimates for evolution equations with classical and fractional derivatives

Among the many open questions for fractional operators, we choose to study decay estimates of a class of evolution equations with possibly nonlocal or nonlinear diffusion operators.
In particular, we are going to study the decay in time of the Lebesgue norm of solutions to a Cauchy problem in a bounded domain.
We present some general results that apply to a wide class of evolution equations, namely all the ones involving a diffusion operator that is satisfying a certain ellipticity property, involving an “energy functional” that suits for both local and non local operators, possibly complex. The time derivative may be of two types: purely classical or a linear combination of a classical derivative and a Caputo derivative.

§.§.§ The problem

We now set the problem.
Let $\lambda_1, \lambda_2 \geq 0$ be fixed positive numbers.
We suppose, for concreteness,
$$\lambda_1 + \lambda_2=1,$$

but up to a rescaling of the operator we can take $\lambda_1, \lambda_2$
any nonnegative numbers with positive sum.
Let $\Omega \subset \R^n$ be a
bounded open set and let $u_0\in L^{\infty}(\R^n)$ such that $\text{supp} \,u_0 \subset \Omega$.
Consider the Cauchy problem

\begin{equation} \label{sys:generalform}
\left\{ \begin{array}{lr}
(\lambda_1 \partial_t^{\alpha} + \lambda_2 \partial_t) [u] + \mathcal{N}[u]=0, & {\mbox{for all }}x\in \Omega, \ t>0, \\
u(x,t)=0, & {\mbox{for all }}x\in \R^n \setminus \Omega , \ t>0, \\
u(x,0)=u_0(x), & {\mbox{for all }}x\in \R^n ,
\end{array} \right.
\end{equation}

where $\mathcal{N}$ is an operator, possibly involving fractional derivatives.

We underline that we consider smooth ($C^1$ often, $C^2$ if also the second derivative appears) and bounded solutions of the problem (<ref>). In fact, we want to avoid convergence problems with the integrals that appear in the statements and in the proofs. However, for certain operators, weaker hypothesis may be taken.

Let us recall that for a complex valued function $v:\Omega\to\C$ the Lebesgue norm is

\begin{equation*}
\Vert v \Vert_{L^s(\Omega)} = \left( \int_{\Omega} |v(x)|^s \; dx \right)^{\frac{1}{s}}
\end{equation*}

for any $s\in[1, +\infty)$. Also, we call $\Re \{ z\}$ the real part of $z\in\C$.
The main assumption we take is the following: there exist $\gamma \in (0,+\infty) $ and $C\in (0,+\infty)$ such that

\begin{equation} \label{cond:complexstr}
\Vert u(\cdot,t) \Vert_{L^{s}(\Omega) }^{s-1+\gamma} \leq C \int_{\Omega} |u(x,t)|^{s-2} \Re \{ \bar{u}(x,t)\mathcal{N} [u](x,t)\} \; dx.
\end{equation}

The constants $\gamma$ and $C$ and their dependence from the parameters of the problem may vary from case to case. The righthandside of the equation may be seen as an energy functional linked to the diffusion operator.
This inequality implies, essentially, that the operator $\mathcal{N}$ is not too degenerate and the energy of the solution should control a certain power of the solution itself; here $\gamma$ plays the role of the degree of ellipticity.
The inequality (<ref>) strongly depends  on the validity of a Sobolev inequality for the solutions of the evolution equation.
To get an intuition of the roles of the factors, take the case of the Laplacian with $s=2$; integrating by parts on the righthandside one obtains $\Vert \nabla u (\cdot, t) \Vert_{L^{2}(\Omega) }^2$, thus the energy, which controls the $L^2$ norm of the solution by the Gagliardo-Nirenberg-Sobolev inequality.
In our setting, the structural inequality in (<ref>)
will be the cornerstone to obtain general energy estimates,
which, combined with appropriate barriers, in turn
produce time-decay estimates.

§.§ Our Results

Extending the method of [43], we obtain a power-law decay in time
of the $L^s$ norm with $s\geq 1$.
Also, for the case of classical time-derivatives,
we obtain exponential decays in time. The difference between
polynomial and exponential decays in time is thus related to
the possible presence of a fractional derivative in the operator involving the time variable.

§.§.§ Decay estimate theorems

First, we present this result for the more general setting, hence for a linear combination of classical and Caputo time derivative. We have the following:

Let $u$ be a solution of the Cauchy problem (<ref>), with $\mathcal{N}$ possibly complex
valued. Suppose that there exist $s\in[1, +\infty)$, $\gamma\in(0,+\infty)$ and $C\in(0,+\infty)$ such that $u$ satisfies (<ref>).

\begin{equation} \label{claim1gen}
(\lambda_1\partial_t^{\alpha} + \lambda_2\partial_t) \Vert u(\cdot,t) \Vert_{L^{s}(\Omega) } \leq -\dfrac{\Vert u(\cdot,t) \Vert_{L^{s}(\Omega) }^{\gamma}}{C},
\qquad{\mbox{ for all }}t>0,\end{equation}

where $C$ and $\gamma$ are the constants appearing in (<ref>).

\begin{equation} \label{claim2gen}
\Vert u(\cdot,t) \Vert_{L^{s}(\Omega) } \leq
\dfrac{C_*}{1+t^{\frac{\alpha}{\gamma}}},\qquad{\mbox{ for all }}t>0,
\end{equation}

for some $C_*>0$, depending only on $C$, $\gamma$, $\alpha$
and $\Vert u_0(\cdot) \Vert_{L^{s}(\R^n)}$.

A polynomial decay is a nice piece of information on the solution and we can expect this to be the best decay we can get for some fractional evolution equations [56, 25]. However, there is also evidence that for classical settings better decays can be achieved. In fact, the following theorem holds:

Let $u$ be a solution of the Cauchy problem (<ref>) with only classical derivative (that is, $\lambda_1=0$) and $\mathcal{N}$ possibly complex
valued. Suppose that there exist $s\in[1, +\infty)$, $\gamma\in(0,+\infty)$ and $C\in(0,+\infty)$ such that $u$ satisfies (<ref>).
Then, for some $C_*>0$, depending only on the constants $C$ and $\gamma$
in (<ref>),
and on $\Vert u_0(\cdot) \Vert_{L^{s}(\R^n)}$, we have that:

1.	if $0<\gamma \leq 1$ the solution $u$ satisfies

\begin{equation} \label{claim3}
\Vert u(\cdot,t) \Vert_{L^{s}(\Omega) } \leq
C_* \, e^{-\frac{t}{C}},\qquad{\mbox{for all }}t>0;
\end{equation}

2. if $ \gamma>1$, the solution $u$ satisfies

\begin{equation} \label{claim4}
\Vert u(\cdot,t) \Vert_{L^{s}(\Omega) } \leq
\dfrac{C_*}{1+t^{\frac{1}{\gamma-1}}},\qquad{\mbox{for all }}t>0.
\end{equation}

As we will see in the proofs
of these two theorems, the idea is to find a supersolution of (<ref>) and use a comparison principle in order to estimate the decay of the solution $u$. For the case of mixed derivatives, Vergara and Zacher [119] find both a supersolution and a subsolution decaying as $t^{-\frac{\alpha}{\gamma}}$. But, when $\alpha \to 1$, the subsolution tends to 0.
On the other hand, the classical equation $\partial_t e =- e^{\gamma}$ has some exponential supersolutions.
This allows possibly better decays, which are in fact proven.

We point out that the case of an evolution equation with only Caputo time derivative, i.e. of $\lambda_2=0$, was treated in [43]. The authors find in this case that the supersolution is still asymptotic to $t^{-\frac{\alpha}{\gamma}}$ and the decay is of polynomial type.

It is interesting to notice the presence of a “decoupling” effect: for evolution equations with classical time derivative and fractional space derivative (take for example the fractional Laplacian, $-(\Delta)^{\sigma}u$, $\sigma\in(0,1)$, see [43]), the space derivative does not asymptotically interfere with the time derivative;
thus the polynomial decay, typical of fractional derivatives, does not appear, leaving place for the exponential decay given by the classical time derivative.
An example of this behaviour is found in [93], where a model inspired to atoms dislocation was studied.

§.§.§ Applications

What makes Theorems <ref> and <ref> interesting is the fact that they may be applied to a wide range of equations. Indeed, the only hypothesis required in order to apply the theorems is the validity of the inequality (<ref>) for suitable parameters $C$ and $\gamma$. In [43] and in our work, (<ref>) was verified for many operators, that we are listing here together with some references on the origins of these operators:

* the classic and fractional Laplacian, [27],

* the classic and fractional $p$-Laplacian, [32],

* the doubly nonlinear equation, [100]

* the classic and fractional porous medium equations, [118, 29] and [39],

* the classic and fractional mean curvature equation, [28]

* the classic and fractional Kirchhoff equation, [23] and [49],

* the classic and fractional magnetic operator, [67] and [38].

The list is not supposed to be exhaustive; in fact, the aim is only to provide some example of operators satisfying (<ref>) and to encourage other mathematicians looking for some decay estimates to attempt with operators they are struggling with.

CHAPTER:  A FISHER-KPP MODEL WITH A FAST DIFFUSION LINE IN PERIODIC MEDIA

In this chapter, we treat a model of population dynamics in a periodic environment presenting a fast diffusion line. The
“road-field” model, introduced in [20], is a system of coupled reaction-diffusion equations set in domains of different dimensions. Here, we consider for the first time the case of a reaction term depending on a spatial variable in a periodic fashion, which is of great interest for both its mathematical difficult and for its applications. We derive necessary and sufficient conditions for the survival of the species in terms of the sign of a suitable generalised principal eigenvalue, defined recently in [16]. Moreover, we compare the long time behaviour of a population in the same environment without the fast diffusion line, finding  that this element has no impact on the survival chances. This chapter corresponds to the paper [2].

§ SETTING AND MAIN RESULTS

This chapter investigates some effects of a fast diffusion line in an ecological dynamics problem.
Various examples in the literature showed that, in the presence of roads or trade lines, some species or infections spread faster along these lines, and then diffuse in the surroundings. This was observed in the case of the Processionary caterpillar, whose spreading in France and Europe has been accelerated by accidental human transport [103]. Another striking proof was given in  [52], where the authors point out that the COVID-19 epidemics in Northern Italy at the beginning of 2020 diffused faster along the highways.

A model for biological diffusion in a homogeneous medium presenting a fast diffusion line was proposed by Berestycki, Roquejoffre and Rossi in [20], and since then is called the road-field model. The authors proved an acceleration effect due to the road on the spreading speed of an invading species.
Since then, a growing number of articles treated variations of the same system, investigating in particular the effect of different type of diffusion or different geometries [13, 14, 105].

However, natural environments are usually far from being homogeneous and, more often than not, territories are a composition of different habitats. Living conditions and heterogeneity play a strong impact on the survival chances of a species and on the equilibria at which the population can settle.

Road-field models on heterogeneous environments have been  little studied so far, being more complex to treat. One of the few example is the paper [58] for periodic exchange terms between the population on the road and the one in the field. Recently, Berestycki, Ducasse and Rossi introduced a notion of generalised principal eigenvalue for the road-field system in [16] and, thanks to it, they were able to treat the case of an ecological niche facing climate change in [17].

Here, we propose an analysis of the asymptotic behaviour of an invasive population under the assumption of spatial periodicity of the reaction term. Of course, under this hypothesis we can investigate deeper the dependence of the population on a natural-like environment and the effects of the road in this balance.
Under which conditions does the population survive in a periodic medium? And does the road play some role on the survival chances of a species, perturbing the environment and scattering the individuals, or rather permitting them to reach advantageous zones more easily?
These are the questions we are going to tackle.

§.§ The model

In this chapter, we study the reaction-diffusion model regulating the dynamics of a population living in a periodic environment with a fast diffusion channel. The equivalent of this model for homogeneous media was first introduced by Berestycki, Roquejoffre and Rossi in [20]. Consider the half plane $\Omega:=\R\times \R^+$, where we mean $\R^+=(0, +\infty)$.
The proposed model imposes the diffusion of a species in $\Omega$ and prescribes that on $\partial \Omega=\R\times \{ y=0\}$ the population diffuses at a different speed.
We call $v(x,t)$ the density of population for $(x,y)\in\Omega$, hence on the “field”, and $u(x)$ the density of population for $x\in\R$, i.e. on the “road”; moreover, we take $D$, $d$, $\nu$, $\mu$ positive constants and $c\geq 0$. Then, the system we analyse reads

\begin{equation}\label{ch1sys:fieldroad}
\left\{
\begin{array}{lr}
\partial_t u-D  u '' -c u' - \nu  v|_{y=0} + \mu u= 0,   & x\in \R,  \\
\partial_t v -d \Delta v-c\partial_x v  =f(x,v),  & (x, y)\in \Omega, \\
-d  \partial_y{v}|_{y=0} + \nu v|_{y=0} -\mu u=0, & x\in\R.
\end{array}
\right.
\end{equation}

In $\Omega$, the population evolves with a net birth-death rate represented by $f$, that depends on the variable $x$.
This embodies the heterogeneity of the media: in fact, environments are typically not uniform and some zone are more favourable than others.
There is no dependence in the variable $y$, since the presence of the road itself creates enough heterogeneity in that direction.
The function  $f:\R\times \R_{\geq 0}\to \R $
is always supposed to be $\mathcal{C}^{1}$ in $x$, locally in $v$, and Lipschitz in $v$, uniformly in $x$; moreover, we suppose that the value $v=0$ is an equilibrium, that is

\begin{equation}\label{ch1hyp:0}
f(x,0)=0, \quad \text{for all} \ x\in \R,
\end{equation}

and that

\begin{equation}\label{ch1hyp:M}
\exists M>0 \ \text{such that} \ f(x, v)<0 \quad \text{for all} \ v>M \ \text{and all} \ x\in \R.
\end{equation}

We will derive some inequalities on the generalised principal eigenvalue of (<ref>) for the general case of $f$ respecting these hypothesis and
$c$ possibly nonzero.

The characterisation of extinction or persistence of the species is performed for the case of $c=0$ and $f$ a periodic function, reflecting the periodicity of the environment in which the population diffuses.
We will analyse the  case of a KPP nonlinearity, that is, we require that

\begin{equation}\label{ch1hyp:KPP}
\frac{f(x,s_2)}{s_2}< \frac{f(x,s_1)}{s_1} \quad  \text{for all} \ s_2>s_1>0 \  \text{and all} \ x\in\R.
\end{equation}

Then, we suppose that there exists $\ell> 0$ such that

\begin{equation}\label{ch1hyp:per}
f(x+\ell, s)=f(x,s) \quad \text{for all} \ s >0 \  \text{and all} \ x\in\R.
\end{equation}

To study the effect of the line of fast diffusion, we will compare the behaviour of (<ref>) to the one of the system

\begin{equation}\label{ch1sys:symmetric}
\left\{
\begin{array}{ll}
v_t-d\Delta v - c\partial_x v= f(x,v),  & (x,y)\in\Omega,\\
-\partial_y v|_{y=0} =0, & x\in\R,
\end{array}
\right.
\end{equation}

whose solution is a function $v(x,y)$ that can be extended by symmetry to the whole plane, thanks to the Neumann border condition.
It is natural to consider system (<ref>) as the counterpart of system (<ref>) in the case without the road, since it presents the same geometry, including the same boundary condition exception made for the exchange terms that are in place for the case of a fast diffusion channel.

§.§ State of the art

We present here the background that led us consider system (<ref>) and some useful results that are known in the community.

The study of reaction-diffusion equations started with the works by Fisher [50] and by Kolmogorov, Petrowskii and Piskunov [72], who modelled the spacial diffusion of an advantageous gene in a population living in a one-dimensional environment through the equation

\begin{equation}\label{ch1eq:KPP}
\partial_t v -d \, \partial_{xx}^2 v = f(v)
\end{equation}

for $x\in\R$ and $t\geq 0$. For (<ref>), it is supposed that $d>0$ and $f\geq 0$ is a $\mathcal{C}^1$ function satisfying $f(0)=f(1)=0$ and the KPP hypothesis $f(v)\leq f'(0)v$ for $v\in[0,1]$. The first example was a nonlinearity of logistic type, so $f(v)= av(1-v)$ for some $a>0$.
It was shown any solution $v$ issued from a nonnegative initial datum $v_0$ converges to 1 as $t$ goes to infinity, locally uniformly in space; this long time behaviour is called invasion.
The generalisation in higher dimension of equation (<ref>) was then used to study the spatial diffusion of animals, plants, bacteria and epidemics [109, 89].

A vast literature has been originated from the pioneer works, studying various aspects of the homogeneous equation (<ref>), in particular concerning the travelling fronts. These are solutions of the form $v(t,x)= V(x \cdot e +ct)$ with $V:\R\to[0,1]$, for $e$ a direction, the direction of propagation,  and $c$ the speed of propagation of the travelling front. Other than this, researchers have investigated the asymptotic speed of propagation at which level sets of a solution starting from $v_0$ expands. These topics arose already in [50] and [72], and their investigation was continued in many interesting articles, among which [48] and [7].

The correspondence of the theoretical results with actual data as seen in [109] was encouraging, however it was clear that natural environments, even at macroscopic levels, were not well represented by a homogeneous medium, due to the alternation of forests, cultivated fields, plains, scrubs and many other habitats, as well as roads, rivers and other barriers [70]. It was necessary to look at more sophisticated features, as the effects of inhomogeneity, fragmentation, barriers and fast diffusion channels, and on the top of that, climate change.

A first analysis was carried out
in [108, 107] and
in [70] for the so-called the patch model. The authors considered a periodic mosaic of two different homogeneous habitats, one favorable and one unfavorable for the invading species.
In [70], the authors studied the long time behaviour of the population starting from any nonnegative initial datum. For further convenience, let us give the following definition:

For the equation of (<ref>) or the system (<ref>), we say that

* extinction occurs if any solution starting from a non negative bounded initial datum converges to $0$ or to $(0,0)$  uniformly as $t$ goes to infinity.

* persistence occurs if any solution starting from a non negative, non zero, bounded initial datum converges to a positive  stationary solution locally uniformly as $t$ goes to infinity.

In [70], it was empirically shown that the stability of the trivial solution $v=0$ determines the long time behaviour of the solutions. A solid mathematical framework for a general periodic environment was given in [18]. There, the authors considered the equation

\begin{equation}\label{ch1eq:bhroques}
\partial_t v - \nabla \cdot (A(x)\cdot \nabla v) = f(x, v)
\end{equation}

for $x\in \R^N$ and $t\geq 0$.
The diffusion matrix $A(x)$ is supposed to be $\mathcal{C}^{1, \alpha}$, uniformly elliptic and periodic; however, for our interest we can suppose $A(x)=d\, I_{N}$, where $I_N$ is the identity matrix. The nonlinearity $f: \R^N \times \R_{\geq0} \to \R$ is supposed to be $\mathcal{C}^{1}$ in $x$, locally in $v$, and Lipshitz in $v$, uniformly in $x$, respecting hypothesis (<ref>)-(<ref>) and such that for some $L=(L_1, \dots, L_N)$, with $L_i\geq 0$, it holds

\begin{equation}\label{ch1hyp:per'}
f(x+L,s)=f(x,s) \quad \text{for all} \  s\geq 0 \ \text{and all} \ x\in\R^N.
\end{equation}

The criterion for persistence or extinction is given via a notion of periodic eigenvalue, that is the unique number $\lambda_p(-\mathcal{L}, \R^N)$
such that there exists a solution $\psi\in W_{loc}^{2, p}(\R^N)$ to the system

\begin{equation}\label{ch1sys:L_RN_p}
\left\{
\begin{array}{ll}
\mathcal{L'}(\psi) + \lambda \psi = 0, & x\in\R^N, \\
\psi> 0, &  x\in\R^N, \\
|| \psi ||_{\infty}=1, \\
\psi \ \text{is periodic in $x$ of periods $L$},
\end{array}
\right.
\end{equation}

where $\mathcal{L'}$ is given by

\begin{equation}\label{ch1def:mathcal_L'}
\mathcal{L'}(\psi):=  d \Delta \psi  + f_v(x,0)\psi.
\end{equation}

We point out that the existence and uniqueness of $\lambda_p(-\mathcal{L}, \R^N)$ is guaranteed by Krein-Rutman theory. The long time behaviour result in [18] is the following:

Assume $f$ satisfies (<ref>)-(<ref>) and (<ref>). Then:

* If $\lambda_p(-\mathcal{L'}, \R^N)<0$, persistence occurs for (<ref>).

* If $\lambda_p(-\mathcal{L'}, \R^N)\geq 0$, extinction occurs for (<ref>).

To prove Theorem <ref>, the authors performed an analysis of $\lambda_p(-\mathcal{L}, \R^N)$, proving that it coincide with the limit of eigenvalues for a sequence of domains invading $\R^N$, so that it coincides with the generalised principal eigenvalue of the system “without the road” (<ref>). Nowadays, that and many other properties of this eigenvalue can be found as part of a broader framework in [22]. In Section <ref>, we will provide further comments on it.

Another important fact highlighted both in the series in [108, 107, 70] and in [18] is that the presence of multiple small unfavourable zones gives less chances of survival than one large one, the surface being equal.

A new difficulty that one may consider while studying ecological problems is, sadly, the issue of a changing climate. A 1-dimensional model in this sense was first proposed in
[15] and [97], and was later treated in higher dimension in [21].
The authors first imagined that a population lives in a favourable region enclosed into a disadvantageous environment; due to the climate change, the favourable zone starts to move in one direction, but keeps the same surface. The resulting equation is

\begin{equation}\label{ch11709}
\partial_t v -  \Delta v=f(x-ct e,v) \quad \text{for} \ x\in\R^N,
\end{equation}

with $e$ a direction in $\mathbb{S}^{N-1}$ and $f: \R^N\times \R_{\geq 0} \to \R$. It was observed that a solution to (<ref>) in the form of a travelling wave $v(x,t)=V(x-cte)$ solves the equation

\begin{equation}\label{ch1eq:cc}
\partial_t V -  \Delta V- c\,e\cdot \nabla V=f(x,V) \quad \text{for} \ x\in\R^N,
\end{equation}

which is more treatable. The main question is if the population keeps pace with the shifting climate, that is, if the species is able to migrate with the same speed of the climate. The answer to this question is positive if a solution to (<ref>) exists; this depends on the value of $c$. We point out that already in [21] the authors considered the general case of a possible periodic $f(x,v)$.

As mentioned before, another feature worth investigation is the effect of fast diffusion channels on the survival and the spreading of species. In fact, the propagation of invasive species as well as epidemics is influenced by the presence of roads [103, 52].
This observations led Berestycki, Roquejoffre and Rossi to propose a model for ecological diffusion in the presence of a fast diffusion channel in [20], the so-called road-field model. The field is modelled with the halfplane $\Omega=\R \times \R_{+}$ and the line with the $x$ axis; the main idea is to use two different variables for modelling the density of population along the line, $u$, and on the half plane, $v$. The system reads

\begin{equation*}
\left\{
\begin{array}{ll}
\partial_t u(x,t) - D \partial_{xx}^2 u (x,t) = \nu v (x,0,t) - \mu u(x,t), &  x\in \R, t > 0, \\
\partial_t v(x,y,t) - d \Delta v (x,y,t)= f(v), & (x,y) \in \Omega, t>0, \\
-d \partial_y v(x,0,t) = -\nu v(x,0,t) + \mu u(x,t), & x \in \R, t>0,
\end{array} \right.
\end{equation*}

for $D$, $d$, $\nu$, $\mu$ positive constants; moreover, $f\in \mathcal{C}^1$ was supposed to satisfy

\begin{equation*}
f(0)=f(1)=0, \quad 0< f(s) < f'(0)s \ \text{for} \ s \in (0,1), \quad f(s)<0 \ \text{for}  \ s>1.
\end{equation*}

The three equations describe, respectively, the dynamic on the line, the dynamic on the half plane and the exchanges of population between the line and the half plane. On the line, the diffusion is faster than in  $\Omega $ if $D>d$.
In [20], the authors identify the unique positive stationary solution $\left(\frac{1}{\mu}, 1 \right)$ and prove persistence of the population.
Moreover, they show that the presence of the line increases the spreading speed. Another version of the model with a reaction term for the line was presented by the same authors in [19], while many variation of the models were proposed by other authors: with nonlocal exchanges in the direction of the road [94, 95], with nonlocal diffusion [14, 13], and with different geometric settings [105]. For a complete list, we refer to [112].

The case of heterogeneous media for systems of road-field type has been so far not much treated, due to its difficulties. A first road-field model with exchange terms that are periodic in the direction of the road was proposed in [58]. There, the authors
recovered the results of persistence and of acceleration on the propagation speed due to the road known in the homogeneous case;
they also studied the spreading of solution with exponentially decaying initial data and calculated their speeds.

Recently, Berestycki, Ducasse and Rossi introduced in [16] a new generalised principal eigenvalue fitting road-field system for possibly heterogeneous reaction term; here, we give its definition directly for the system (<ref>).

\begin{equation}\label{ch1sys:operators}
\left\{
\begin{array}{l}
\mathcal{R}(\phi, \psi):=D \phi''+c \phi'+\nu {\psi}|_{y=0}-\mu \phi, \\
\mathcal{L}(\psi):= d\Delta \psi +c \partial_x \psi -f_v(x,0)\psi, \\
B(\phi, \psi):=d \partial_y {\psi}|_{y=0}+\mu \phi- \nu {\psi}|_{y=0},
\end{array}
\right.
\end{equation}

this eigenvalue is defined as

\begin{equation}\label{ch1def:lambda1_S_Omega}
\begin{split}
\lambda_1( \Omega)=\sup \{ \lambda \in \R \ : \ \exists (\phi, \psi)\geq (0,0), \ (\phi, \psi) \not\equiv(0,0), \ \text{such that} \\ \mathcal{L}(\psi) + \lambda \psi \leq 0 \ \text{in} \ \Omega, \ \mathcal{R}(\phi, \psi) +\lambda \phi \leq 0
\ \text{and} \ B(\phi, \psi)\leq 0 \ \text{in} \ \R \},
\end{split}
\end{equation}

with $(\phi, \psi)$ belonging to $W_{loc}^{2,3}(\R)\times W_{loc}^{2,3}(\overline{\Omega})$. Together with the definition, many interesting properties and bounds were studied; we will recall some of them later.

Thanks to that, the same authors were able to investigate the case of
a favourable ecological niche, possibly facing climate change, in [17]. It was proven that the sign of $\lambda_1( \Omega)$ characterises
the extinction or the persistence of the population; moreover, comparing the results with  the ones found for the model without the road, a deleterious effect of the road on the survival chances is always found when there is no climate change. On the other hand, if the ecological niche shifts, the road has in some cases a positive effect on the persistence.

§.§ Main results

We are now ready to present the main results of this chapter.

§.§.§ The case of a periodic $f(x,v)$

Here, we consider the case of a nonlinearity that respects the KPP hypothesis and is periodic in the direction of the road. Moreover, here we always consider $c=0$.

We begin by the following result on long time behaviour for solutions of system (<ref>):

Assume $f$ satisfy (<ref>)-(<ref>), $c=0$ and let $\lambda_1(\Omega)$ be as in (<ref>).
Then the following holds:

* if $\lambda_1( \Omega)\geq 0$, then  extinction occurs.

* if $\lambda_1(\Omega)<0$, then persistence occurs and the  positive stationary solution $(u_{\infty}, v_{\infty})$ is unique and periodic in $x$.

Now, we compare the behaviour of solutions to the system (<ref>) with the ones of system (<ref>).
This allows us to highlight the effects of the fast diffusion channel on the survival chances of the population.
Actually, since solutions of (<ref>) can be extended by refection to the whole plane, we can make the comparison with equation (<ref>) for $A(x)=d I_2$ and $L=(\ell, 0)$.
The comparison is performed thanks to the generalised principal eigenvalue $\lambda_1(\Omega)$ for system (<ref>) and the periodic eigenvalue $\lambda_p(-\mathcal{L}, \R^2)$, as defined in (<ref>), for the operator $\mathcal{L}$ in dimension 2.
We obtain the following:

Assume $f$ respects hypothesis (<ref>)-(<ref>), $c=0$. Then:

* if  $\lambda_p(-\mathcal{L}, \R^2)<0$, then $\lambda_1( \Omega)<0$, that is, if persistence occurs  for the system “without the road” (<ref>), then it occurs also for system “with the road” (<ref>).

* if $\lambda_p(-\mathcal{L}, \R^2)\geq 0$, then $\lambda_1( \Omega)\geq 0$, that is, if extinction occurs for the system “without the road”  (<ref>), then it occurs also for system “with the road” (<ref>).

Theorem <ref> says that the road has no negative impact on the survival chances of the population in the case of a periodic medium depending only on the variable in the direction of the road.
This is surprising if compared to the results obtained in [17] (precisely Theorem 1.5, part (ii)), where the authors find that the existence of the road is deleterious in presence of an ecological niche, and even more counter-intuitive owing the fact that fragmentation of the environment lessens the survival chances of a population, as shown in [18]. This means that, in the case of periodic media, the presence of the fast diffusion channel does not interfere with the persistence of the population, which depends only on the  environment of a periodicity cell.
As seen in [18], where the dependence of persistence on the amplitude of fragmentation was studied, if the favourable zones are sufficiently large, the population will eventually spread in all of them; the presence of the road does not cause loss of favourable environment and consequently of persistence chances.
However, we expect the spreading speed to be influenced by the presence of the road, as it has been already proven in the case of homogeneous environment.

We point out that Theorem (<ref>) completes and is in accordance with the results on long time behaviour found in [20] for a homogeneous reaction term, which we can see as a particular case of periodicity, which respects positive KPP hypothesis (where the positivity is requested through $f'(0)>0$). In [20], Theorem 4.1 states the convergence of any positive solution to the unique positive stationary solution of the system.  Since it is well known that for the homogeneous case it holds $\lambda_1(-\mathcal{L}, \R^2)=- f'(0)$, the positivity hypothesis gives that $\lambda_1(-\mathcal{L}, \R^2)<0$ and, as a consequence of Theorem <ref>, that the second case in our Theorem <ref> occurs.
If instead we asked for $f'(0)\leq0$, then we would be in the first case of Theorem <ref>, yielding extinction of the population.

Effects of amplitude of heterogeneity.
One may expect that the presence of a road may alter the complex interaction between more favourable and less favourable zones, in particular penalising the persistence, since it was shown that populations prefer a less fragmented environment. However, the road does not interfere with that; as a consequence, also for environments presenting fast diffusion channels, some results of the analysis on the effect of fragmentation performed in [18] holds.

Take a parameter $\alpha>0$ and consider system (<ref>) with nonlinearity

\begin{equation}\label{ch11421}
\tilde{f}(x,v)=\alpha f(x,v).
\end{equation}

To highlight the dependence on $\alpha$, we will call $\lambda_1(\Omega, \alpha)$ the  generalised principal eigenvalue defined in (<ref>) with nonlinearity $\tilde{f}$.
As a direct consequence of Theorem (<ref>) and Theorem 2.12 in [18], we have the following result on the amplitude of heterogeneity:

Assume $\tilde{f}$ is defined as in (<ref>), $f$ satisfies (<ref>)-(<ref>), and $c=0$. Then:

* if $ \int_{0}^{\ell} f_v(x,0)>0$, or if $ \int_{0}^{\ell} f_v(x,0)=0$ and $f\not\equiv 0$, then for all $\alpha >0$ we have $\lambda_1(\Omega, \alpha  )<0$.

* if $ \int_{0}^{\ell} f_v(x,0)<0$, then  $\lambda_1(\Omega, \alpha )>0$ for $\alpha$ small enough; if moreover there exists $x_0\in[0,\ell]$ such that $f_v(x_0,0)>0$, then for all $\alpha$ large enough $\lambda_1(\Omega, \alpha )<0$.

This result describes with precision the fact that, to persist, a species must have a sufficiently large favourable zone available. If the territory is more advantageous than not, then the population persist. If however there environment is generally unfavourable, the population persists only if there are some contiguous advantageous zones large enough; if instead the advantageous zones are fragmented, even if there is unlimited favourable territory, the population will encounter extinction.

§.§.§ A climate change setting for a general $f(x,v)$

We consider now a general nonlinearity that depends on the spatial variable in the direction of the road. We stress the fact that we do not suppose any periodicity, but the case of a periodic $f$ is a particular case of this setting. Moreover, the following result is done in the general framework of a possible climate change, so the parameter $c$ may be different from $0$.

Comparison between the systems with and without the road, in the general case, are done through comparison between $\lambda_1(\Omega)$ and the generalised principal eigenvalue of system (<ref>), given by

\begin{equation}\label{ch1lambda:L_Omega}
\begin{split}
\lambda_1(-\mathcal{L}, \Omega)=\sup \{ \lambda \in \R \ : \ \exists \psi \geq 0, \psi \not\equiv 0 \ \text{such that} \\
\mathcal{L}(\psi) + \lambda \psi \leq 0 \ \text{on} \ \Omega, \ -\partial_y \psi|_{y=0}\leq 0 \ \text{on} \ \R \}
\end{split}
\end{equation}

for $\psi\in W_{loc}^{2,3}(\Omega)$. With this notation, we have the following:

Assume $\lambda_1(-\mathcal{L}, \R^2)$ as in (<ref>) and $\lambda_1(\Omega)$ as in (<ref>); then $\lambda_1(-\mathcal{L}, \R^2) \geq \lambda_1(\Omega)$.

In the special case $c=0$, some information on the relations between $\lambda_1(-\mathcal{L}, \R^2)$ and $\lambda_1(\Omega)$  was already available in [17]: Proposition 3.1 yields that $\lambda_1(-\mathcal{L}, \R^2)\geq 0$ implies $\lambda_1(\Omega)\geq 0$. Thanks to that and Theorem <ref>, the following result holds:

If $c=0$, we have $\lambda_1(-\mathcal{L}, \R^2)<0$ if and only if $\lambda_1(\Omega)<0$.

As already pointed out in [16], even for $c=0$ it is not true that $\lambda_1(-\mathcal{L}, \R^2) =\lambda_1(\Omega)$. In fact, it has been found that $\lambda_1(\Omega) \leq \mu$, while playing with $f$ one can have $\lambda_1(-\mathcal{L}, \R^2)$ as large as desired. However, the fact that the two eigenvalues have the same sign reveals that they are profoundly linked.

§.§ Organisation of the chapter

In Section <ref>, we recall and discuss the properties of the eigenvalues $\lambda_1(\Omega)$, $\lambda_1(-\mathcal{L}, \R^2)$ and $\lambda_p(-\mathcal{L}, \R^2)$ already known in the literature.