# The effect of “fear” on two species competition

###### Abstract.

Non-consumptive effects such as fear of depredation, can strongly influence
predator-prey dynamics. These effects have not been as well studied in the
case of purely competitive systems, despite ecological and social motivations
for the same. In this work we consider the classic two species ODE and PDE
Lokta-Volterra competition models, where _one_ of the competitors is “fearful”
of the other. We find that the presence of fear can have several interesting
dynamical effects on the classical scenarios of weak and strong competition,
and competitive exclusion. Notably, for fear levels in certain regimes, we
show bi-stability between interior equilibrium and boundary equilibrium is
possible - contrary to the classical strong competition situation where bi-
stability is only possible between boundary equilibrium. Furthermore, in the
spatially explicit setting, the effects of several spatially heterogeneous
fear functions are investigated. In particular, we show that under certain
$\mathbb{L}^{1}$ restrictions on the fear function, a weak competition type
situation can change to competitive exclusion. Applications of these results
to ecological as well as sociopolitical settings are discussed, that connect
to the “landscape of fear” (LOF) concept in ecology.

Vaibhava Srivastava 1, Eric M. Takyi 2 and Rana D. Parshad 1

1)Department of Mathematics,

Iowa State University,

Ames, IA 50011, USA.

2)Department of Mathematics and Computer Science,

Ursinus College,

Collegeville, PA 19426, USA.

##  1\. Introduction

Fear, is defined as,

> _An unpleasant emotion caused by the belief that someone or something is
> dangerous_ [77].

It is a complex emotion, that is critical as a safety measure, and can trigger
the “fight or flight” response [2] \- in particular it can change the way one
acts, even when there is no threat present [24]. In predator-prey systems,
this is most naturally observed among prey, due to their perceived threat of
depredation [10]. This perception can lead to non-consumptive effects or
trait-mediated interactions, which are behavioral, morphological or
physiological changes in prey phenotype, due to this threat [23, 10]. Such
effects are known to strongly influence predator-prey dynamics [21]. From a
mathematical viewpoint, the effects of fear in predator-prey systems has been
intensely investigated since the seminal work of Brown et. al. [28], where
optimal foraging theory is extended to consider a game theoretic setup, played
out by predator and prey, exhibiting stealth and fear, in which an animal
follows a map or a “landscape of fear” (LOF), which describes its predation
risk while it navigates the physical landscape. In recent work, Wang et. al.
[50], model fear of depredation, as a (predator) density dependent effect,
that negatively effects the prey population. In essence, the prey’s growth
rate is modeled as a monotonically decreasing function of predator density.
Dynamically, a key finding in [50] is that under the parametric restrictions
of a Hopf bifurcation, an increase in the fear parameter (and prey’s birth
rate parameter) can alter the direction of a Hopf bifurcation from
supercritical to subcritical. Thus, fear enables both supercritical and
subcritical Hopf bifurcations, contrary to only the supercritical bifurcations
found in classical predator-prey systems. In essence, the fear effect can
change the fundamental cylical patterns of predator-prey dynamics, leading to
large scale ecological consequences [12].

These results have since initiated a host of activities in diverse ecological
scenarios such as when refuges are present [59, 51], when the prey has
tendencies to avoid predators [52], or when the predators responses are
influenced by interference pressures, for instance, via a Beddington-DeAngelis
functional response [60]. Various works have considered the fear effect in
case of group defense by the prey [65, 64]. It has been investigated in the
context of cooperative and competitive systems within the larger predator-prey
context. These include the fear effect when predators are cooperating [55] in
the hunting process, or when they are hunting for competing prey [14]. These
effects have been investigated in the three and multi-species settings as well
[53, 68] where fear can damp population explosions [71]. Various authors have
considered the fear effect in a stochastic setting [70] as well as a spatially
explicit setting, in the context of taxis type movements, as well as pattern
formation [56, 62]. It can also lead to chaotic dynamics [67]. However, the
effect of fear has been far less investigated in classical monotone systems,
such as purely cooperative or competitive two species systems - that are
outside the predator-prey setting.

Competition among two species, typically modeled via the Lotka–Volterra
competition model and its variants have been intensely investigated in the
last few decades. These models take into account growth and
inter/intraspecific competition [41], and predict well-observed states in
biology of co-existence, competitive exclusion of one competitor, and bi-
stability, and find diverse applications in ecology and invasion science [13,
15, 3, 6, 1]. There are several ecological motivations for competitors being
fearful of each other. This is perhaps most naturally seen to occur with
intraguild predation - a widespread phenomenon in many food webs, where
competitors will kill and consume each other [7]. Recent evidence of non-
consumptive effects exerted by intraguild predator mites (_Blattisocius
dentriticus_) on their competitor (_Neoseiulus cucumeris_) show this can be an
important factor in determining food web dynamics in biological control [73,
4]. However, there is strong evidence for fear in purely competitive two
species systems _without_ predatory effects. Barred owls (_Strix varia_) are a
species of owl, native to eastern North America. They have expanded their
range westward over the last century and are considered invasive in western
North America. Currently, their range overlaps with the spotted owl (_Strix
occidentalis_), which is native to the north west and western North America.
This has resulted in intense competition between the two species [72]. Barred
owls exert a strong negative influence on spotted owls, threatening their
possible competitive exclusion [75]. Field observations report frequent barred
owl attacks on spotted owls, and even on surveyors imitating spotted owl calls
[74]. There is also evidence of barred owls aggressively chasing spotted owls
out of shared habitat - but _not_ the opposite [76]. Such evidence clearly
motivates considering fear type dynamics into a purely competitive two species
model where one of the competitors is fearful of the other.

There are also several socio-economic-political settings, where pure
competitors may be fearful of each other. Small/new businesses may be fearful
of large businesses, due to their already large market share [43]. But large
business may also be fearful of small local businesses, due to their
familiarity with local nuances, that may yield competitive advantage at a
small local scale [42]. Fear is also conceivable among two competing political
parties, where the weaker party on a national scale, may have a stronger voter
bank at a regional scale [18]. Or perhaps two warring drug cartels, where the
weaker cartel has certain local/territorial strongholds [19, 44] \- within
which they might be able to induce fear among the stronger cartel [19]. Such
phenomenon becomes even more interesting in the spatially explicit case where
this fear could be heterogeneous in the spatial domain of interest. This
connects back to the LOF concept, where the fear function is essentially the
map that describes how the fear levels change as a species disperses over a
physical landscape.

Motivated by all of the affore mentioned sociopolitical, economic as well as
ecological settings, the current manuscript considers the effect of fear in a
competitive two species system. We restrict our analysis to the case where
only _one_ of the competitors is fearful of the other. Our investigations show
that:

  * •

Sufficiently large fear can change a situation of competitive exclusion, to a
strong competition type scenario, where there is bi-stability between boundary
equilibrium. See Fig. 5 (C). Dynamically, this occurs via a transcritical
bifurcation. This is shown via Lemma 2.19, see Fig. 9.

  * •

Fear in a certain parametric regime can change a situation of competitive
exclusion to bi-stability between boundary equilibrium and interior
equilibrium, see Fig. 4 and Fig. 5 (B). Dynamically, this occurs via a saddle-
node bifurcation. This is shown via Lemma 2.18, see Fig. 8. This is in sharp
contrast with classical competition theory, where bi-stability occurs only
between boundary equilibriums.

  * •

Sufficiently large fear can change a situation of weak competition to a
competitive exclusion type scenario. This is shown via Lemma 2.5, see Fig. 2.

  * •

Fear cannot qualitatively change a strong competition type scenario. This is
shown via Lemma 2.8, see Fig. 3. Also, fear cannot produce periodic orbits.
This is demonstrated via Lemma 2.10.

  * •

In the spatially explicit setting, comparison theory is used to determine
point-wise restrictions on the fear functions such that competitive exclusion
or strong competition type dynamics abounds. These are shown via Theorem 3.8,
Theorem 3.10 and Theorem 3.15, see Figs. [10,11,13].

  * •

In the spatially explicit setting, fear can change a situation of weak
competition to a competitive exclusion type scenario, for fear functions with
certain $\mathbb{L}^{1}$ restrictions. This is shown via Theorem 3.13 and
Lemma 3.14, see Figs. [15,16,17]. In particular the fear functions need not
lie uniformly above the critical fear levels derived in the ODE case via Lemma
2.5.

  * •

Various heterogeneous fear functions are constructed to demonstrate these
results numerically, see Fig. 18(b). Applications of these to ecological as
well as socio-political settings are discussed in section 4.

##  2\. The ODE case

###  2.1. Model formulation

Consider the classical two species Lotka-Volterra ODE competition model,

(1)
$\left\\{\begin{array}[]{ll}\dfrac{du}{dt}&~{}=u(a_{1}-b_{1}u-c_{1}v),\\\\[8.61108pt]
\dfrac{dv}{dt}&~{}=v(a_{2}-b_{2}v-c_{2}u),\end{array}\right.$

where $u$ and $v$ are the population densities of two competing species,
$a_{1}$ and $a_{2}$ are the intrinsic (per capita) growth rates, $b_{1}$ and
$b_{2}$ are the intraspecific competition rates, $c_{1}$ and $c_{2}$ are the
interspecific competition rates. All parameters considered are positive. The
dynamics of this system are well studied [20]. We recap these briefly,

  * •

$E_{0}=(0,0)$ is always unstable.

  * •

$E_{u}=(\frac{a_{1}}{b_{1}},0)$ is globally asymptotically stable if
$\dfrac{a_{1}}{a_{2}}>\max\left\\{\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\right\\}$.
Herein $u$ is said to competitively exclude $v$.

  * •

$E_{v}=(0,\frac{a_{2}}{b_{2}})$ is globally asymptotically stable if
$\dfrac{a_{1}}{a_{2}}<\min\left\\{\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\right\\}$.
Herein $v$ is said to competitively exclude $u$.

  * •

$E^{*}=\Big{(}\frac{a_{1}b_{2}-a_{2}c_{1}}{b_{1}b_{2}-c_{1}c_{2}},\frac{a_{2}b_{1}-a_{1}c_{2}}{b_{1}b_{2}-c_{1}c_{2}}\Big{)}$
exists when $b_{1}b_{2}-c_{1}c_{2}\neq 0$. The positivity of the equilibrium
holds if $\frac{c_{2}}{b_{1}}<\frac{a_{2}}{a_{1}}<\frac{b_{2}}{c_{1}}$ and is
globally asymptotically stable if $b_{1}b_{2}-c_{1}c_{2}>0$. This is said to
be the case of weak competition.

  * •

If $b_{1}b_{2}-c_{1}c_{2}<0$, then
$E^{*}=\Big{(}\frac{a_{1}b_{2}-a_{2}c_{1}}{b_{1}b_{2}-c_{1}c_{2}},\frac{a_{2}b_{1}-a_{1}c_{2}}{b_{1}b_{2}-c_{1}c_{2}}\Big{)}$
is unstable as a saddle. In this setting, one has initial condition dependent
attraction to either $E_{u}(\frac{a_{1}}{b_{1}},0)$ or
$E_{v}(0,\frac{a_{2}}{b_{2}})$. This is the case of strong competition.

We proceed by considering the effects of fear on the classical model (1), when
_one_ of the competitors is fearful of the other.

###  2.2. The case of $v$ fearing $u$

We consider the case of the competitor $v$ being fearful of $u$. Thus in the
classical model (1), we model the fear effect as in [50], where the growth
rate of the fearful competitor $v$, is not constant but rather density
dependent. Essentially, the growth rate is decreased by a factor
$\approx\frac{1}{1+ku}$, where $k\geq 0$ is a fear coefficient. Thus a higher
density of the competitor $u$ increases the fear in $v$. When $k=0$, the
assumption is there is no fear and one recovers the classical model (1). If
fear is present, we obtain the following ODE model for two competing species
$u$ and $v$, where $v$ is fearful of $u$.

(2) $\displaystyle\begin{split}\dfrac{du}{dt}&=a_{1}u-b_{1}u^{2}-c_{1}uv,\\\
\dfrac{dv}{dt}&=\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv.\end{split}$

####  2.2.1. Existence

The nullclines associated with the problem (2) are

$u(a_{1}-b_{1}u-c_{1}v)=0\quad\text{and}\quad
v\Big{(}\dfrac{a_{2}}{1+ku}-b_{2}v-c_{2}u\Big{)}.$

Hence, the boundary equilibrium points are obtained by substituting $u=0$ and
$v=0$ in the above equations of the nullclines, respectively. Denote the
boundary equilibrium points as $\widehat{E}_{1}=(0,0)$,
$\widehat{E}_{2}=(\frac{a_{1}}{b_{1}},0)$ and
$\widehat{E}_{3}=(0,\frac{a_{2}}{b_{2}})$.

For the interior equilibrium, substitute
$u^{*}=\frac{a_{1}}{b_{1}}-\frac{c_{1}}{b_{1}}v^{*}$ in the second nullcline
equation, i.e.,

$\displaystyle\begin{split}\dfrac{a_{2}}{1+k\Big{(}\frac{a_{1}}{b_{1}}-\frac{c_{1}}{b_{1}}v^{*}\Big{)}}-b_{2}v^{*}-c_{2}\Big{(}\frac{a_{1}}{b_{1}}-\frac{c_{1}}{b_{1}}v^{*}\Big{)}=0.\end{split}$

On simplification, we have that $v^{*}$ solves a quadratic equation of the
form $A(v^{*})^{2}+Bv^{*}+C=0$, where

(3) $\begin{split}A&=c_{1}k(b_{2}b_{1}-c_{1}c_{2}),\\\
B&=b_{1}(c_{1}c_{2}-b_{2}b_{1})-a_{1}k(b_{2}b_{1}-2c_{1}c_{2}),\\\
C&=b_{1}(a_{2}b_{1}-a_{1}c_{2})-a_{1}^{2}c_{2}k.\end{split}$

Let

(4) $v_{1,2}^{*}=\frac{-B\pm\sqrt{B^{2}-4AC}}{2A}$

be the two roots of above qudratic equation. WLOG assume
$v_{1}^{*}<v_{2}^{*}.$ Moreover, consider the following parametric restriction

(5) $\displaystyle
b_{1}(a_{1}c_{2}-a_{2}b_{1})+a_{1}^{2}c_{2}k<\Big{[}(b_{2}b_{1}-c_{1}c_{2})(2a_{1}k-b_{1})-a_{1}kc_{1}c_{2}\Big{]}\dfrac{a_{1}}{c_{1}}.$

We can prove the existence of a positive equilibrium point $\widehat{E}_{4}$
with the choice of specific parameters. Let us use Descartes’s rule of sign to
establish some sufficient conditions for the existence of one or two positive
equilibrium points.

Two positive equilibrium points: Under the assumption $A>0,B<0$ and $C>0$,
i.e., $b_{2}b_{1}>2c_{1}c_{2}>c_{1}c_{2}$ and
$k<\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)},$ we
have two positive roots. In order to claim that these two roots correspond to
two positive interior equilibria, we need some extra assumption given by:

$v_{1}^{*}<v_{2}^{*}:=\dfrac{-B+\sqrt{B^{2}-4AC}}{2A}<\dfrac{a_{1}}{c_{1}}\implies-C<(A\dfrac{a_{1}}{c_{1}}+B)\dfrac{a_{1}}{c_{1}}.$

Hence, if
$b_{1}(a_{1}c_{2}-a_{2}b_{1})+a_{1}^{2}c_{2}k<[(b_{2}b_{1}-c_{1}c_{2})(2a_{1}k-b_{1})-a_{1}kc_{1}c_{2}]\frac{a_{1}}{c_{1}},$
we have two positive interior equilibrium points
$\widehat{E}_{4}=(u_{i}^{*},v_{i}^{*})$ for $i=1,2$.

One positive equilibrium point: Under the assumption $A>0,B<0$, i.e.,
$b_{2}b_{1}>2c_{1}c_{2}>c_{1}c_{2}$, we have at least one positive root of the
quadratic equation. If $C<0$, which is
$k>\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$
along with $\eqref{pos_u_null}$ gives existence of one positive equilibrium
point $\widehat{E}_{4}=(u^{*},v^{*})$. Moreover, if $C>0$, which is
$k<\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$, and
$v_{1}^{*}<\frac{a_{1}}{c_{1}}$ and $v_{2}^{*}>\frac{a_{1}}{c_{1}}$, then we
have existence of one positive equilibrium point
$\widehat{E}_{4}=(u^{*},v^{*})$.

We formulate all these restrictions as an existence theorem:

######  Theorem 2.1.

For the given ODE system (2), we always have three boundary equilibrium
points, namely
$\widehat{E}_{1}:=(0,0),\widehat{E}_{2}:=(\frac{a_{1}}{b_{1}},0)$ and
$\widehat{E}_{3}:=(0,\frac{a_{2}}{b_{2}})$. For the case of two positive
interior equilibrium points
$\widehat{E}_{4}=\Big{(}u_{i}^{*},v_{i}^{*}\Big{)}_{i=1,2}$, we have the
following parametric restrictions:

$\displaystyle b_{2}b_{1}>$ $\displaystyle 2c_{1}c_{2}>c_{1}c_{2},\quad
k<\frac{1}{a_{1}^{2}c_{2}}\Big{[}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{]}\quad\&\quad\eqref{pos_u_null}\hskip
7.22743pt\text{holds true}.$

Lastly, for the case of one positive interior equilibrium point
$\widehat{E}_{4}=(u^{*},v^{*})$, we have either one of the following choices
of parametric restrictions:

  1. (1)

$b_{2}b_{1}>2c_{1}c_{2}>c_{1}c_{2},\quad
k>\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$ and
$\eqref{pos_u_null}$ holds true.

  2. (2)

$b_{2}b_{1}>2c_{1}c_{2}>c_{1}c_{2},\quad
k<\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$, and
$v_{1}^{*}<\frac{a_{1}}{c_{1}}$ and $v_{2}^{*}>\frac{a_{1}}{c_{1}}$.

  3. (3)

$b_{2}b_{1}<c_{1}c_{2}<2c_{1}c_{2},\quad
k>\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$, and
$v_{1}^{*}<\frac{a_{1}}{c_{1}}$ and $v_{2}^{*}>\frac{a_{1}}{c_{1}}$,

where $v_{i}^{*}$ are the roots of the quadratic equation defined as
$\eqref{quad_roots}.$

We now provide several lemmas, so that we can compare the effect of fear to
the classical competition case.

######  Lemma 2.2.

Consider the given ODE system (2), with $k=0$, s.t we are in the weak
competition setting with $b_{2}b_{1}>2c_{1}c_{2}$. Then for a fear coefficient
$k$ s.t.
$k>k_{c}=\dfrac{1}{a_{1}^{2}c_{2}}\Big{(}b_{1}^{2}a_{2}-a_{1}c_{2}b_{1}\Big{)}$,
$(\frac{a_{1}}{b_{1}},0)$ is globally asymptotically stable. That is, $u$ will
competitively exclude $v$.

######  Remark 1.

For the existence of one positive interior equilibrium point for the given
system (2), we have either one of the following choices of parametric
restrictions: For weak competition,

(6) $\displaystyle
b_{2}b_{1}>2c_{1}c_{2}>c_{1}c_{2}\quad\&\quad\dfrac{c_{2}}{b_{1}}<\dfrac{a_{2}}{a_{1}}<\dfrac{b_{2}}{c_{1}},$

and for strong competition,

(7) $\displaystyle
b_{2}b_{1}<c_{1}c_{2}<2c_{1}c_{2}\quad\&\quad\dfrac{c_{2}}{b_{1}}>\dfrac{a_{2}}{a_{1}}>\dfrac{b_{2}}{c_{1}}.$

####  2.2.2. Linear Stability Analysis

We next perform stability analysis on the equilibrium points of system (2).
The Jacobian matrix of system (2) is given by

(8)
$\widehat{J}^{*}(u^{*},v^{*})=\left(\begin{array}[]{cc}a_{1}-2b_{1}u^{*}-c_{1}v^{*}&-c_{1}u^{*}\\\
-\dfrac{a_{2}kv^{*}}{(ku^{*}+1)^{2}}-c_{2}v^{*}&\dfrac{a_{2}}{ku^{*}+1}-2b_{2}v^{*}-c_{2}u^{*}\\\
\end{array}\right).$

We state the following lemmas.

######  Lemma 2.3.

$\widehat{E}_{1}$ is locally unstable.

###### Proof.

On evaluating Eq.$(\ref{jacob_second})$ at $\widehat{E}_{1}$, we have

$\widehat{J}^{*}(\widehat{E}_{1})=\left(\begin{array}[]{cc}a_{1}&0\\\
0&a_{2}\\\ \end{array}\right).$

Being a triangular matrix, we know that the above matrix has two positive
eigenvalues $a_{1}$ and $a_{2}$. Hence, the equilibrium point
$\widehat{E}_{1}$ is locally unstable. ∎

######  Lemma 2.4.

$\widehat{E}_{2}$ is locally stable iff
$k>\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}$.

###### Proof.

We again evaluate Eq.$(\ref{jacob_second})$ at $\widehat{E}_{2}$ and obtain

$\widehat{J}^{*}(\widehat{E}_{2})=\left(\begin{array}[]{cc}-a_{1}&-\dfrac{c_{1}a_{1}}{b_{1}}\\\
0&\dfrac{a_{2}b_{1}}{b_{1}+ka_{1}}-\dfrac{c_{2}a_{1}}{b_{1}}\\\
\end{array}\right).$

Being a triangular matrix, the above matrix has two eigenvalues,
$\lambda_{1}=-a_{1}$ and
$\lambda_{2}=\frac{a_{2}b_{1}}{b_{1}+ka_{1}}-\frac{c_{2}a_{1}}{b_{1}}$. As
$\lambda_{1}$ is always negative, if we can show that $\lambda_{2}$ is
negative, we are done. We make the assumption that,

$k>\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}\iff\lambda_{2}=\dfrac{a_{2}b_{1}}{b_{1}+ka_{1}}-\dfrac{c_{2}a_{1}}{b_{1}}<0.$

Therefore, the boundary equilibrium point $\widehat{E}_{2}$ is locally stable.
∎

Local stability of $\widehat{E}_{2}$ actually implies global stability, we can
see this via a simple geometric argument.

######  Lemma 2.5.

$\widehat{E}_{2}$ is globally stable if
$k>k_{c}=\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}$.

###### Proof.

Consider the nullclines of $u$ and $v$, where,

$v=\dfrac{1}{c_{1}}(a_{1}-b_{1}u)\quad\&\quad
v=\dfrac{1}{b_{2}}\Big{(}\dfrac{a_{2}}{1+ku}-c_{2}u\Big{)}.$

In order to establish the global stability of $\widehat{E}_{2}$, via the
geometry of the nullclines, it suffices to show that

$\Big{[}\dfrac{a_{1}-b_{1}u}{c_{1}}\Big{]}>\Big{[}\dfrac{1}{b_{2}}\Big{(}\dfrac{a_{2}}{1+ku}-c_{2}u\Big{)}\Big{]}$

when $u=\frac{a_{1}}{b_{1}}$,

i.e., when $k>\frac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}$. Herein,
the $v$-nullcline lies completely below the $u$-nullcline, and via the
convexity of the $v$-nullcline, it lies completely below the straight line
connecting its $v$ and $u$ intercepts - which lies completely below the
$u$-nullcline. Now, via the standard theory of competition and a comparison
argument, where $v$ is compared to the $\tilde{v}$ that is a solution to the
straight line nullcline connecting the $v$ and $u$ intercepts of the
$v$-nullcline, we have the global stability of $\widehat{E}_{2}$. ∎

(a) $k=0$

(b) $k=10$

Figure 1. Phase plots showing dynamics under competition exclusion parametric
restriction with $f=0$. The orange curve is the u-nullcline and blue curve is
the v-nullcline. Here, $(u^{*},0)$ always wins. Parameters used are
$a_{1}=3,a_{2}=1,b_{1}=1,b_{2}=1,c_{1}=0.5,c_{2}=0.5$.

######  Lemma 2.6.

The equilibrium point $\widehat{E}_{3}$ is locally stable iff
$a_{1}b_{2}<c_{1}a_{2}$.

###### Proof.

Similar evaluation of Eq.$(\ref{jacob_second})$ at $\widehat{E}_{3}$ yields

$\widehat{J}^{*}(\widehat{E}_{3})=\left(\begin{array}[]{cc}a_{1}-\dfrac{c_{1}a_{2}}{b_{2}}&0\\\
-\dfrac{ka_{2}^{2}}{b_{2}}-\dfrac{c_{2}a_{2}}{b_{2}}&-a_{1}\\\
\end{array}\right).$

Being a triangular matrix, the above matrix has two eigenvalues,
$\lambda_{1}=-a_{1}$ and $\lambda_{2}=a_{1}-\dfrac{c_{1}a_{2}}{b_{1}}$. Under
the assumed parametric restriction,

$a_{1}b_{2}<c_{1}a_{2}\iff\lambda_{2}<0.$

Hence, the equilibrium point $\widehat{E}_{3}$ is locally stable.

∎

######  Lemma 2.7.

The interior equilibrium $\widehat{E}_{4}$ exists and is locally stable if
$k<\frac{1}{a_{2}}\left(\frac{b_{1}b_{2}}{c_{1}}-c_{2}\right).$

###### Proof.

On evaluating Eq.$(\ref{jacob_second})$ again at $\widehat{E}_{4}$, we have

$\widehat{J}^{*}(\widehat{E}_{4})=\left(\begin{array}[]{cc}-b_{1}u^{*}&-c_{1}u^{*}\\\
-\dfrac{ka_{2}v^{*}}{(1+ku^{*})^{2}}-c_{2}v^{*}&-b_{2}v^{*}\\\
\end{array}\right).$

For the local stability of $\widehat{E}_{4}$, it is enough to show that
$Trace(\widehat{J}^{*}(\widehat{E}_{4}))<0$ and
$Det(\widehat{J}^{*}(\widehat{E}_{4}))>0.$ Simple computations yield

$Trace(\widehat{J}^{*}(\widehat{E}_{4}))=-b_{1}u^{*}-b_{2}v^{*}<0,$

and

$Det(\widehat{J}^{*}(\widehat{E}_{4})=b_{1}b_{2}u^{*}v^{*}-c_{1}u^{*}\Big{(}\dfrac{ka_{2}v^{*}}{(1+ku^{*})^{2}}+c_{2}v^{*}\Big{)}=u^{*}v^{*}\Big{\\{}b_{1}b_{2}-c_{1}\Big{(}\dfrac{ka_{2}}{(1+ku^{*})^{2}}+c_{2}\Big{)}\Big{\\}}.$

Note that

$ka_{2}>\frac{ka_{2}}{(1+ku^{*})^{2}}.$

Therefore, if $k$ is chosen s.t,
$k<\frac{1}{a_{2}}\left(\frac{b_{1}b_{2}}{c_{1}}-c_{2}\right)$, then,

$b_{1}b_{2}>c_{1}(ka_{2}+c_{2})>c_{1}\Big{[}\dfrac{ka_{2}}{(1+ku^{*})^{2}}+c_{2}\Big{]}\implies
Det(\widehat{J}^{*}(\widehat{E}_{4}))>0,$

and the result follows.

∎

(a) $k=0$

(b) $k=10^{-3}$

(c) $k=10^{-4}$

(d) $k=0.3$

Figure 2. Phase plots showing various dynamics under weak competition
parametric restriction with $f=0$. Here $(u^{*},v^{*})$ is a sink and both
$(0,v^{*})$ and $(u^{*},0)$ are saddles in $(A),(B)$ and $(C)$. In $(D)$ there
is competition exclusion and $(u^{*},0)$ wins. The orange curve is the
u-nullcline and blue curve is the v-nullcline. Parameters used are
$a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=2,c_{1}=0.3,c_{2}=1.8$.

######  Lemma 2.8.

The interior equilibrium $\widehat{E}_{4}$ exists and is a saddle if

$\Big{(}\dfrac{b_{2}b_{1}}{c_{1}}-c_{2}\Big{)}<\frac{ka_{2}b_{1}^{2}}{(b_{1}+ka_{1})^{2}}.$

###### Proof.

In order to claim that the interior equilibrium $\widehat{E}_{4}$ is a saddle,
it is enough to show that $Trace(\widehat{J}^{*}(\widehat{E}_{4}))<0$ and
$Det(\widehat{J}^{*}(\widehat{E}_{4}))<0.$ We have that

$Trace(\widehat{J}^{*}(\widehat{E}_{4}))=-b_{1}u^{*}-b_{2}v^{*}<0,$

and

$Det(\widehat{J}^{*}(\widehat{E}_{4})=b_{1}b_{2}u^{*}v^{*}-c_{1}u^{*}\Big{(}\dfrac{ka_{2}v^{*}}{(1+ku^{*})^{2}}+c_{2}v^{*}\Big{)}=u^{*}v^{*}\Big{\\{}b_{1}b_{2}-c_{1}\Big{(}\dfrac{ka_{2}}{(1+ku^{*})^{2}}+c_{2}\Big{)}\Big{\\}}.$

Under the assumption and density of reals, we have

$\displaystyle\Big{(}\dfrac{b_{2}b_{1}}{c_{1}}-c_{2}\Big{)}$
$\displaystyle<\frac{ka_{2}}{(1+\frac{ka_{1}}{b_{1}})^{2}}<\frac{ka_{2}}{(1+ku^{*})^{2}}\implies
Det(\widehat{J}^{*}(\widehat{E}_{4})<0.$

Hence, $\widehat{E}_{4}$ is a saddle. ∎

(a) $k=0$

(b) $k=11$

Figure 3. Phase diagrams showing dynamics under strong competition parametric
restriction with $f=0$. The orange curve is the u-nullcline, blue curve is the
v-nullcline and separatrix/stable manifold is in green. Here, $(u^{*},v^{*})$
is saddle. Parameters used are
$a_{1}=0.5,a_{2}=2,b_{1}=0.5,b_{2}=0.8,c_{1}=4,c_{2}=4$.

######  Lemma 2.9.

Consider the system (2). For any given $k>0$ s.t., there exist two positive
interior equilibria, a sink and a saddle.

###### Proof.

Theorem 2.1 provides conditions under which two positive interior equilibria
exists. Since stability has to alternate by standard theory [9] for planar
systems, one of the equilibrium is stable while the other is unstable.
Instability as a source is impossible due to the lack of periodic dynamics in
the system via Lemma 2.10. Thus the unstable equilibrium must be a saddle.
This proves the lemma. ∎

######  Lemma 2.10.

Consider the ODE system (2). There do not exist any periodic orbits for the
system, for any values of the fear parameter $k$.

###### Proof.

Consider the function $\phi(u,v)=\frac{1}{uv}$ where $u$ and $v$ are both non-
zero. Let,

$\begin{split}F_{1}(u,v)&=a_{1}u-b_{1}u^{2}-c_{1}uv,\\\
F_{2}(u,v)&=\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv.\end{split}$

Then we have

$\displaystyle\dfrac{\partial(F_{1}\phi)}{\partial
u}+\dfrac{\partial(F_{2}\phi)}{\partial v}$ $\displaystyle=$
$\displaystyle\dfrac{\partial}{\partial
u}\left(\dfrac{1}{uv}(a_{1}u-b_{1}u^{2}-c_{1}uv)\right)+\dfrac{\partial}{\partial
v}\left(\dfrac{1}{uv}\left(\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv\right)\right),$
$\displaystyle=$ $\displaystyle-\frac{b_{1}}{v}-\frac{b_{2}}{u}<0.$

The result follows by application of the Dulac criterion [9]. ∎

(a) $k=0$

(b) $k=15$

Figure 4. Phase plot showing competition exclusion in $(A)$ when fear is
absent in both competing species where $(0,v^{*})$ wins. In $(B)$, we observe
the occurrence of two positive interior equilibria when $f=0$ and $k=15$. The
orange curve is the u-nullcline, blue curve is the v-nullcline and
separatrix/stable manifold is in green. Parameters used are
$a_{1}=1,a_{2}=2,b_{1}=2,b_{2}=0.3,c_{1}=0.3,c_{2}=0.05$.

######  Remark 2.

Some observations:

  1. (1)

From Fig. 4, we observe that when there is no fear, then $(0,v^{*})$ is
globally stable. For a sufficiently large level of fear in species $v$, a bi-
stability situation is created. That is, for a certain initial data, species
$u$ is completely excluded by $v$ and initial data is attracted to the co-
existence state (See Fig. 4).

  2. (2)

If $(0,v^{*})$ is globally attracting, a much higher level of fear in species
$v$ ($\approx 200$) can change the dynamics to a strong competition type case.
However most data in this setting is attracted to $(u^{*},0)$. For initial
data $(u_{0},v_{0})$ to be attracted to $(0,v^{*})$, we would need
$v_{0}>>1,u_{0}<<1$.

###  2.3. The case of $u$ fearing $v$

In this subsection, we shall consider the case when the competitor $u$ is
being fearful of $v$. For the modeling construct, we will follow the modeling
approach of fear effect as in the model in [50], where the growth rate of the
fearful competitor $u$, is not constant but rather density dependent.
Essentially, the growth rate is decreased by a factor $\approx\frac{1}{1+fv}$,
where $f\geq 0$ is a fear coefficient. Thus a higher density of the competitor
$v$ increases the fear in $u$. When $f=0$, the assumption is there is no fear,
and one recovers the classical model (1). If fear is present, we obtain the
following ODE model for two competing species $u$ and $v$, where $u$ is
fearful of $v$.

(10)
$\displaystyle\begin{split}\dfrac{du}{dt}&=\dfrac{a_{1}u}{1+fv}-b_{1}u^{2}-c_{1}uv,\\\
\dfrac{dv}{dt}&=a_{2}v-b_{2}v^{2}-c_{2}uv.\end{split}$

The system (10) possesses the following biologically feasible non-negative
equilibria. These are

  * •

$E_{1}=(0,0)$,

  * •

$E_{2}=(\frac{a_{1}}{b_{1}},0)$,

  * •

$E_{3}=(0,\frac{a_{2}}{b_{2}})$,

  * •

$E_{4}=(u^{*},v^{*})$,

where $u^{*}$ is given by $(\ref{u_star})$ and $v^{*}$ is a positive root of
the following third order polynomial in Eq.$(\ref{three_post1})$.

####  2.3.1. Existence

The system (10) can have one or two positive interior equilibria, but not
three. This is established via the following lemma,

######  Lemma 2.11.

Consider the ODE system (10), with $f=0$. If we are in the competitive
exclusion, weak-competition, or strong-competition setting, then for any
$f>0$, under the parametric restrictions in $(\ref{three_post})$, it is
impossible to find three positive interior equilibria.

###### Proof.

If $k=0$ and $f>0$, Eq.(37) reduces to the cubic equation
$B(v^{*})^{3}+C(v^{*})^{2}+D(v^{*})+E=0$, where

(11) $\begin{split}B&=b_{1}f^{2}(c_{1}c_{2}-b_{1}b_{2}),\\\
C&=a_{2}b_{1}^{2}f^{2}+2b_{1}f(c_{1}c_{2}-b_{2}b_{1}),\\\
D&=b_{1}f(-a_{1}c_{2}+2a_{2}b_{1})+b_{1}(c_{1}c_{2}-b_{2}b_{1}),\\\
E&=b_{1}(-a_{1}c_{2}+a_{2}b_{1}).\end{split}$

We will require the following conditions to satisfy Descartes’s rule of signs,
so as to obtain three positive roots to the cubic equation. These are:

(12) $\displaystyle\begin{split}c_{1}c_{2}-b_{1}b_{2}&<0,\\\
a_{2}b_{1}f+2(c_{1}c_{2}-b_{2}b_{1})&>0,\\\
f(-a_{1}c_{2}+2a_{2}b_{1})&<-(c_{1}c_{2}-b_{2}b_{1}),\\\
-a_{1}c_{2}+a_{2}b_{1}&>0.\end{split}$

Let us prove this is an impossible claim by contradiction: First, assume the
parameter set satisfies both competitive exclusion-state parametric
restriction and $(\ref{three_post})$. We know that competitive exclusion-state
is asymptotically stable if
$\frac{a_{1}}{a_{2}}>\max\Big{\\{}\frac{b_{1}}{c_{2}},\frac{c_{1}}{b_{2}}\Big{\\}}$.
On using these parametric restriction, $-a_{1}c_{2}+a_{2}b_{1}<0$, which is a
contradiction to the last inequality in $(\ref{three_post})$.

If the parameter set satisfies both the strong competition state parametric
restriction and $(\ref{three_post})$, then we have a contradiction because of
the first inequality in $(\ref{three_post})$, as under the strong competition
state parametric restriction that inequality should be positive.

For the weak competition, recall the parametric restrictions:

$\frac{c_{2}}{b_{1}}<\frac{a_{2}}{a_{1}}<\frac{b_{2}}{c_{1}}.$

Let’s re-write the third inequality in $(\ref{three_post})$,

$-(c_{1}c_{2}-b_{2}b_{1})-f(-a_{1}c_{2}+2a_{2}b_{1})>0.$

On adding the second and third inequality in $(\ref{three_post})$, we have

$(c_{1}c_{2}-b_{1}b_{2})+f(a_{1}c_{2}-a_{2}b_{1})>0,$

which is a contradiction as the added inequality should be negative by the
parametric restrictions of weak-competition. This proves the lemma.

∎

####  2.3.2. Linear Stability Analysis

The Jacobian matrix of system (10) is given by

(13)
$J^{*}=\left(\begin{array}[]{cc}\dfrac{a_{1}}{fv^{*}+1}-2b_{1}u^{*}-c_{1}v^{*}&-\dfrac{a_{1}fu^{*}}{(fv^{*}+1)^{2}}-c_{1}u^{*}\\\
-c_{2}v^{*}&a_{2}-2b_{2}v^{*}-c_{2}u^{*}\\\ \end{array}\right).$

######  Lemma 2.12.

The trivial steady state $E_{1}$ is locally unstable.

###### Proof.

For proof details refer to (6.1) ∎

(a) $f=0$

(b) $f=0.42$

Figure 5. Phase plots showing dynamics under competition exclusion parametric
restriction with $k=0$ in each plot. In $(A)$, $(0,v^{*})$ is saddle point. In
$(B)$ we observe multiple coexistence points and $(0,v^{*})$ and $(u^{*},0)$
are saddle points. In $(C)$ a further increase in $f$ still leads to bi-
stability but multiple coexistence states are lost. The parameters
$a_{1}=3.6,a_{2}=3,b_{1}=1.8,b_{2}=1,c_{1}=0.5,c_{2}=1.8$ are used for each
simulation. In each plot, the orange curve is the u-nullcline, blue curve is
the v-nullcline and the green curve is the separatrix/stable manifold.

######  Lemma 2.13.

The boundary equilibrium point $E_{2}$ is locally stable iff
$a_{2}b_{1}<c_{2}a_{1}$.

###### Proof.

For proof details refer to (6.2) ∎

######  Lemma 2.14.

The boundary equilibrium point $E_{3}$ is locally stable iff
$f>\dfrac{a_{1}b_{2}^{2}-a_{2}b_{2}c_{1}}{a_{2}^{2}c_{1}}$.

###### Proof.

For proof details refer to (6.3) ∎

######  Lemma 2.15.

The interior equilibrium $E_{4}$ exists and is locally stable if

$f<\frac{1}{a_{1}}\Big{(}\frac{b_{1}b_{2}}{c_{2}}-c_{1}\Big{)}.$

###### Proof.

For proof details refer to (6.4) ∎

######  Lemma 2.16.

The interior equilibrium $E_{4}$ exists and is a saddle if

$\Big{(}\dfrac{b_{1}b_{2}}{c_{2}}-c_{1}\Big{)}<\dfrac{a_{1}c_{2}^{2}f}{(fa_{2}+c_{2})^{2}}.$

###### Proof.

For proof details refer to (6.5) ∎

######  Lemma 2.17.

Consider the ODE system (10). There do not exist any periodic orbits for the
system, for any values of the fear parameter $f$.

###### Proof.

The proof follows as in the proof of Lemma 2.10. ∎

(a) $f=0$

(b) $f=0.1$

(c) $f=0.3$

(d) $f=0.5$

Figure 6.  Dynamics under weak competition parametric restriction with $k=0$
in each plot. Here $(u^{*},v^{*})$ is a sink and both $(0,v^{*})$ and
$(u^{*},0)$ are saddles in $(A),(B)$ and $(C)$. $(0,0)$ is a source. The
orange curve is the u-nullcline and blue curve is the v-nullcline. In $(D)$,
$(0,v^{*})$ changes from a saddle to a sink. Parameters used are
$a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=1,c_{1}=0.3,c_{2}=1.8$.

(a) $f=0$

(b) $f=20$

Figure 7. Phase plots showing dynamics under strong competition parametric
restriction with $k=0$. Parameters used are
$a_{1}=1,a_{2}=1,b_{1}=1,b_{2}=1,c_{1}=2,c_{2}=2$.

###  2.4. Bifurcation Analysis

A bifurcation is said to occur in a dynamical system when the behavior of
solutions changes when a parameter is varied. Bifurcation analysis is useful
in understanding and measuring these qualitative changes as the system
switches from stable to unstable and vice-versa.

####  2.4.1. Saddle-node bifurcation

The following theorem is connected to the existence of a saddle-node
bifurcation for the growth rate $a_{1}$ when there is no fear effect in
competitor $u$.

######  Theorem 2.18.

The ODE system (2) undergoes a saddle-node bifurcation around $E_{4}^{*}$ at
$a_{1}=a_{1}^{*}$ when the conditions $det(J^{*})=0$ and $tr(J^{*})<0$ hold
for the system parameters.

###### Proof.

We shall use the Sotomayor’s Theorem [9] to show the occurrence of a saddle-
node bifurcation at $a_{1}=a_{1}^{*}$. At $a_{1}=a_{1}^{*}$, we can have
$det(J^{*})=0$ and $tr(J^{*})<0$. This implies that, $det(J^{*})$ admits a
zero eigenvalue. Now let $P=(p_{1},p_{2})^{T}$ and $Q=(q_{1},q_{2})^{T}$ be
the eigenvectors of $J^{*}$ and $J^{*T}$ corresponding to the zero eigenvalue
respectively.

We have that, $P=\left(\dfrac{-B}{A},1\right)^{T}$ and
$Q=\left(\dfrac{-A}{F},1\right)^{T}$
where $A=a_{1}-2b_{1}u^{*}-c_{1}v^{*}$, $B=-c_{1}u^{*}$,
$F=-\dfrac{a_{2}kv^{*}}{(1+ku^{*})^{2}}-c_{2}v^{*}$.

Now, let $X=(X_{1},X_{2})^{T}$ where

$\begin{split}X_{1}&=a_{1}u-b_{1}u^{2}-c_{1}uv,\\\
X_{2}&=\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv.\end{split}$

Furthermore,

$Q^{T}X_{a_{1}}(E_{3}^{*},a_{1})=\left(\dfrac{-A}{F},1\right)\left(u^{*},0\right)^{T}=-\dfrac{Au^{*}}{F}\neq
0$

and

$\displaystyle Q^{T}[D^{2}X(E_{3}^{*},a_{1})(P,P)]$
$\displaystyle=\left(\dfrac{-A}{F},1\right)\left(\dfrac{2B}{A}\left(b_{1}+c_{1}\right),2\left(\dfrac{a_{2}kB}{A\left(1+ku^{*}\right)^{2}}\left[1-\dfrac{kv^{*}}{1+ku^{*}}\right]-b_{2}\right)\right)^{T}$
$\displaystyle\neq 0.$

Therefore by the Sotomayor’s theorem system (2) undergoes a saddle-node
bifurcation at $a_{1}=a_{1}^{*}$ around $E_{4}^{*}$. ∎

######  Remark 3.

Consider the case of two interior equilibria, such as in Fig. 4. Then
decreasing the fear coefficient $k$ results in a saddle-node bifurcation,
where the interior equilibrium (the saddle $E_{4}$ and the node $E_{5}$)
collide and disappear, resulting in the boundary equilibrium $E_{2}=(u^{*},0)$
becoming globally asymptotically stable. Thus in this setting, a certain
critical level of “fear” can maintain a co-existence state, but fear less than
this level takes the system back to a competitive exclusion type scenario.
This transition occurs via a saddle-node bifurcation. This can be rigorously
proven by adopting the methods of Theorem 2.18, to the parameter $k$ instead
of using $a_{1}$.

####  2.4.2. Transcritical bifurcation

######  Theorem 2.19.

The ODE system (10) experiences a transcritical bifurcation around $E_{3}^{*}$
at $f=f^{*}=\dfrac{b_{2}\left(a_{1}b_{2}-a_{2}c_{1}\right)}{a_{2}^{2}c_{1}}$
and when
$\left(\dfrac{a_{1}f}{\left(1+fv^{*}\right)^{2}}+c_{1}-\dfrac{b_{1}b_{2}}{c_{2}}\right)\neq
0$.

###### Proof.

An evaluation of the Jacobian matrix for system (10) at $E_{3}$ with
$f^{*}=\dfrac{b_{2}\left(a_{1}b_{2}-a_{2}c_{1}\right)}{a_{2}^{2}c_{1}}$ yields

(14) $J_{f}^{*}=\left(\begin{array}[]{cc}0&0\\\
-\dfrac{a_{2}c_{2}}{b_{2}}&-a_{2}\\\ \end{array}\right).$

A calculation of the eigenvalues of the Jacobian matrix in Eq. (14) are
$\lambda_{1}=0$ and $\lambda_{2}=-a_{2}$. Next, we let $G=(g_{1},g_{2})^{T}$
and $H=(h_{1},h_{2})^{T}$ denote the eigenvectors corresponding to the zero
eigenvalue of the matrices $J_{f}^{*}$ and $J_{f}^{*T}$ respectively.

We have $G=\left(-\dfrac{b_{2}}{c_{2}},1\right)^{T}$ and
$H=\left(1,0\right)^{T}$. Now, let $S=(S_{1},S_{2})^{T}$ where

$\begin{split}S_{1}&=\dfrac{a_{1}u}{1+fv}-b_{1}u^{2}-c_{1}uv,\\\
S_{2}&=a_{2}v-b_{2}v^{2}-c_{2}uv.\end{split}$

The next step is to validate the transversality conditions using the
Sotomayor’s theorem [9]. Now,

$H^{T}R_{f^{*}}(E_{3}^{*},f)=\left(1,0\right)\left(0,0\right)^{T}=0.$

Also,

$\displaystyle H^{T}\left[DS_{f}\left(E_{3},f^{*}\right)G\right]$
$\displaystyle=\left(\begin{array}[]{cc}1&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}-\dfrac{a_{1}v^{*}}{\left(1+fv^{*}\right)^{2}}&0\\\
0&0\\\ \end{array}\right)\left(\begin{array}[]{ccc}w_{1}\\\ w_{2}\\\
\end{array}\right)$
$\displaystyle=\dfrac{a_{1}b_{2}v^{*}}{c_{2}\left(1+fv^{*}\right)^{2}}\neq 0.$

and

$\begin{split}H^{T}\left[D^{2}S\left(E_{3},f^{*}\right)(G,G)\right]&=\left(\begin{array}[]{cc}1&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}\dfrac{2b_{2}}{c_{2}}\left(\dfrac{a_{1}f}{\left(1+fv^{*}\right)^{2}}+c_{1}-\dfrac{b_{1}b_{2}}{c_{2}}\right)\\\
0\\\ \end{array}\right)\\\
&=\dfrac{2b_{2}}{c_{2}}\left(\dfrac{a_{1}f}{\left(1+fv^{*}\right)^{2}}+c_{1}-\dfrac{b_{1}b_{2}}{c_{2}}\right)\neq
0.\end{split}$

Therefore by the Sotomayor’s theorem system (10) experiences a transcritical
bifurcation at some
$f=f^{*}=\dfrac{b_{2}\left(a_{1}b_{2}-a_{2}c_{1}\right)}{a_{2}^{2}c_{1}}$
around $E_{3}^{*}$. ∎

######  Remark 4.

Consider the case of one interior equilibrium, such as in Fig. 6 \- that is,
we are in the weak competition case, when there is no fear or $f=0$. Now
increasing the fear coefficient $f$, results in a transcritical bifurcation,
where the interior equilibrium (the node $E_{4}$) and the boundary saddle
equilibrium $E_{3}=(0,v^{*})$ collide, exchange stability, after which $E_{4}$
now moves to the 2nd quadrant, while $E_{3}$ becomes globally asymptotically
stable. Thus in this setting, a certain critical level of “fear” can move the
system from a weak competition setting of coexistence to a competitive
exclusion type scenario. This transition occurs via a transcritical
bifurcation. This is rigorously proven in Theorem 2.19.

(a)

(b)

Figure 8. Bifurcation diagrams showing the impacts of the intrinsic growth
rate $a_{1}$ and the fear effect parameter $f$. In $(A)$ we observe the
occurrence of a saddle-node bifurcation at $a_{1}=a_{1}^{*}=0.785676$. The
parameters used are $a_{2}=2,b_{1}=2,b_{2}=0.3,c_{1}=0.3,c_{2}=0.05,f=0,k=20$.
Initial condition was chosen as $(u_{0},v_{0})=(0.1,2.5)$. A transcritical
bifurcation is also observed in $(B)$ at $f=f^{*}=0.3333$. The parameters are
chosen as $a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=1$ and $c_{1}=0.3,c_{2}=1.8,k=0$.
Initial condition was chosen as $(u_{0},v_{0})=(1,1)$. (Note: TC=Transcritical
point, SN=Saddle-Node point, BP=Branch Point.)

(a)

(b)

Figure 9. Bifurcation diagrams showing the impact of the fear parameter $f$
with different parameter sets. The parameters are chosen as
$a_{2}=3,b_{2}=1,c_{1}=0.5,c_{2}=1.8,k=0$. In $(A)$, $a_{1}=1.8,b_{1}=1$ and a
transcritical bifurcation is observed at $f=f^{*}=0.0666$. In $(B)$,
$a_{1}=3.6,b_{1}=1.8$ and we observe the occurrence of a saddle-node at
$f=f^{*}=0.3966$ and a transcritical bifurcation at $f=f^{*}=0.4666$. The
initial condition was chosen as $(u_{0},v_{0})=(0.5,3)$ for both $(A)$ and
$(B)$. (Note: TC=Transcritical point, SN=Saddle-Node point.)  Table 1. Effect
of fear on classical competition ODE dynamics for the case of $v$ fearing $u$

| Classical case | $k>0,f=0$
---|---|---
(i) | CE $(u^{*},0)$ | Species $v$ is competitively excluded for both small and large $k$.
(ii) | CE $(0,v^{*})$ | (1) If $\frac{a_{2}b_{1}^{2}\mathbf{k}}{(b_{1}+ka_{1})^{2}}<\frac{b_{1}b_{2}-c_{1}c_{2}}{c_{1}}$, then species $u$ is competitively excluded.
|  | (2) Possibility of one interior saddle equilibrium, with a large level of $k$.
|  | (3) Possibility of two positive interior equilibria, one sink and one saddle, with an intermediate level of $k$.
(iii) | weak competition | (1) If $k<k_{c}=\frac{b_{2}b_{1}-c_{2}c_{1}}{a_{2}c_{1}}$ via Thm 2.1 yields co-existence.
|  | (2) If $k>k_{c},$ then species $v$ is competitively excluded.
(iv) | strong competition | Interior equilibrium always exists and is a saddle.

Table 2. Effect of fear on classical competition ODE dynamics for the case of
$u$ fearing $v$

| Classical case | $k=0,f>0$
---|---|---
(i) | CE $(u^{*},0)$ | (1) Species $v$ is competitively excluded, with a small level of $f$.
|  | (2) Possibility of one interior saddle equilibrium, with a large level of $f$.
|  | (3) Possibility of two positive interior equilibria, one sink and one saddle, with an intermediate level of $f$.
(ii) | CE $(0,v^{*})$ | Species $u$ is competitively excluded for both small and large $f$.
(ii) | weak competition | (1) For small $f$ we have co-existence.
|  | (2) For large $f$, $u$ is competitively excluded.
iv) | strong competition |
|  | Interior equilibrium always exists and is a saddle.

##  3\. The PDE Case

Species diffusion is ubiquitous in spatial ecology [25]. Species disperse to
find mates, food and shelter [79]. Such movement is modeled often via reaction
diffusion systems [25]. The spatially explicit Lokta-Volterra model,
particularly in the case of heterogeneity in spatial resources has been
intensely investigated, [29, 33, 30, 34, 36, 37, 38, 39, 40, 41, 49]. Herein,
we consider the spatially explicit version of (2), resulting in the following
reaction diffusion system,

(15) $\displaystyle\begin{split}u_{t}&=d_{1}\Delta
u+a_{1}u-b_{1}u^{2}-c_{1}uv,\quad x\in\Omega,\\\ v_{t}&=d_{2}\Delta
v+\dfrac{a_{2}u}{1+kv}-b_{2}v^{2}-c_{2}uv,\quad x\in\Omega,\\\ \dfrac{\partial
u}{\partial\nu}&=\dfrac{\partial
v}{\partial\nu}=0,\quad\text{on}\quad\partial\Omega,\\\ u(x,0)&=u_{0}(x),\quad
v(x,0)=v_{0}(x).\end{split}$

Here $u(x,t),v(x,t)$ are the densities of two competing species, where $v$ is
also fearful of $u$. The species diffuse in a bounded domain
$\Omega\subset\mathbb{R}^{n}$, with dispersal speeds $d_{1}$ and $d_{2}$
respectively. We impose no flux Neumann boundary conditions, modeling the
effect that the species do not immigrate or emigrate from the domain $\Omega$.
We now proceed to study the dynamics of the above model, when various forms of
fear are considered.

####  3.0.1. Notations and preliminary observations

To prove global existence of solutions to (15), it suffices to derive uniform
estimate on the $\mathbb{L}^{p}$ norms of the R.H.S. of (15), for some
$p>\frac{n}{2}$. Classical theory will then yield global existence, [80]. The
usual norms in spaces $\mathbb{L}^{p}(\Omega)$, $\mathbb{L}^{\infty}(\Omega)$
and $\mathbb{C}\left(\overline{\Omega}\right)$ are respectively denoted by

(16) $\left\|u\right\|_{p}^{p}=\int_{\Omega}\left|u(x)\right|^{p}dx,\
\left\|u\right\|_{\infty}\text{=}\underset{x\in\Omega}{max}\left|u(x)\right|.$

To this end, we use standard techniques [57]. We first recall classical
results guaranteeing non-negativity of solutions, local and global existence
[5, 57]:

######  Lemma 3.1.

Let us consider the following $m\times m$ \- reaction diffusion system: for
all $i=1,...,m,$

(17) $\partial_{t}u_{i}-d_{i}\Delta
u_{i}=f_{i}(u_{1},...,u_{m})~{}in~{}\mathbb{R}_{+}\times\Omega,~{}\partial_{\nu}u_{i}=0~{}\text{on}~{}\partial\Omega,u_{i}(0)=u_{i0},$

where $d_{i}\in(0,+\infty)$,
$f=(f_{1},...,f_{m}):\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ is
$C^{1}(\Omega)$ and $u_{i0}\in L^{\infty}(\Omega)$. Then there exists a $T>0$
and a unique classical solution of 17 on $[0,T).$ If $T^{*}$ denotes the
greatest of these $T^{\prime}s$, then

$\Bigg{[}\sup_{t\in[0,T^{*}),1\leq i\leq
m}||u_{i}(t)||_{L^{\infty}(\Omega)}<+\infty\Bigg{]}\implies[T^{*}=+\infty].$

If the nonlinearity $(f_{i})_{1\leq i\leq m}$ is moreover quasi-positive,
which means

$\forall i=1,...,m,~{}~{}\forall u_{1},...,u_{m}\geq
0,~{}~{}f_{i}(u_{1},...,u_{i-1},0,u_{i+1},...,u_{m})\geq 0,$

then

$[\forall i=1,...,m,u_{i0}\geq 0]\implies[\forall i=1,...,m,~{}\forall
t\in[0,T^{*}),u_{i}(t)\geq 0].$

######  Lemma 3.2.

Using the same notations and hypotheses as in Lemma 3.1, suppose moreover that
$f$ has at most polynomial growth and that there exists
$\mathbf{b}\in\mathbb{R}^{m}$ and a lower triangular invertible matrix $P$
with nonnegative entries such that

$\forall
r\in[0,+\infty)^{m},~{}~{}~{}Pf(r)\leq\Bigg{[}1+\sum_{i=1}^{m}r_{i}\Bigg{]}\mathbf{b}.$

Then, for $u_{0}\in L^{\infty}(\Omega,\mathbb{R}_{+}^{m}),$ the system (17)
has a strong global solution.

Under these assumptions, the following local existence result is well known,
see D. Henry [80],

######  Theorem 3.3.

The system (19) admits a unique, classical solution $(u,v)$ on
$[0,T_{\max}]\times\Omega$. If $T_{\max}<\infty$ then

(18) $\underset{t\nearrow
T_{\max}}{\lim}\Big{\\{}\left\|u(t,.)\right\|_{\infty}+\left\|v(t,.)\right\|_{\infty}\Big{\\}}=\infty,$

where $T_{\max}$ denotes the eventual blow-up time in
$\mathbb{L}^{\infty}(\Omega).$

The next result follows from the application of standard theory [78],

######  Theorem 3.4.

Consider the reaction diffusion system (15). For spatially homogenous initial
data $u_{0}\equiv c,v_{0}\equiv d$, with $c,d>0$, then the dynamics of (19)
and its resulting kinetic (ODE) system, when $d_{1}=d_{2}=0$ in (15), are
equivalent.

####  3.0.2. Spatially Heterogeneous Fear

Our objective now is to consider the case of a fear function that may be
heterogeneous in space. A motivation for this comes from several ecological
and sociological settings. For example it is very common for prey to be highly
fearful closer to a predators lair, but less fearful in a region of refuge
[51], or in regions of high density due to group defense [61]. Furthermore, a
conceivably weaker drug cartel, could have certain localized strongholds,
within which they would be more feared by stronger groups. To these ends, it
is conceivable that the fear coefficient $k$ is not a constant, but actually
varies in the spatial domain $\Omega$, so $k=k(x)$, which could take different
forms depending on the application at hand. This is also in line with the LOF
concept [28]. Thus we consider the following spatially explicit version of
(2), with heterogeneous fear function $k(x)$, resulting in the following
reaction diffusion system,

(19)
$\displaystyle\begin{split}u_{t}&=d_{1}(u)_{xx}+a_{1}u-b_{1}u^{2}-c_{1}uv,\quad
x\in\Omega,\\\
v_{t}&=d_{2}(v)_{xx}+\dfrac{a_{2}u}{1+k(x)v}-b_{2}v^{2}-c_{2}uv,\quad
x\in\Omega,\\\ \dfrac{\partial u}{\partial\nu}&=\dfrac{\partial
v}{\partial\nu}=0,\quad\text{on}\quad\partial\Omega.\\\ u(x,0)&=u_{0}(x)\equiv
c>0,\quad v(x,0)=v_{0}(x)\equiv d>0,\end{split}$

where $\Omega\subset\mathbb{R}^{n}$. We assume no flux Neumann boundary
conditions. Also we prescribe spatially homogeneous (flat) initial conditions
$u(x,0)=u_{0}(x)\equiv c>0,\quad v(x,0)=v_{0}(x)\equiv d>0.$ Furthermore, we
impose the following restrictions on the fear function $k(x)$,

(20) $\displaystyle\begin{split}&(i)\quad k(x)\in C^{1}(\Omega),\\\ &(ii)\quad
k(x)\geq 0,\\\ &(iii)\quad\mbox{If}\ k(x)\equiv 0\ \mbox{on}\
\Omega_{1}\subset\Omega,\ \mbox{then}\ |\Omega_{1}|=0.\\\ &(iv)\quad\mbox{If}\
k(x)\equiv 0\ \mbox{on}\ \cup^{n}_{i=1}\Omega_{i}\subset\Omega,\ \mbox{then}\
\Sigma^{n}_{i=1}|\Omega_{i}|=0.\end{split}$

######  Remark 5.

If $k(x)\equiv 0$ on $\Omega_{1}\subset\Omega$, with $|\Omega_{1}|>\delta>0$,
or $k(x)\equiv 0$ on $\cup^{n}_{i=1}\Omega_{i}\subset\Omega$, with
$\Sigma^{n}_{i=1}|\Omega_{i}|>\delta>0$, that is, on non-trivial parts of the
domain, the analysis is notoriously difficult, as one now is dealing with a
_degenerate_ problem. See [16, 17] for results on this problem. This case is
not in the scope of the current manuscript.

Since the nonlinear right hand side of (19) is continuously differentiable on
$\mathbb{R}^{+}\times$ $\mathbb{R}^{+}$, then for any initial data in
$\mathbb{C}\left(\overline{\Omega}\right)$ or
$\mathbb{L}^{p}(\Omega),\;p\in\left(1,+\infty\right)$, it is standard to
estimate the $\mathbb{L}^{p}-$norms of the solutions and thus deduce global
existence. Standard theory will apply even in the case of a bonafide fear
function $k(x)$, because due to our assumptions on the form of $k$, standard
comparison arguments will apply [11]. Thus applying the classical methods
above, via Theorem 3.3, and Lemmas 3.1-3.2, we can state the following lemmas,

######  Lemma 3.5.

Consider the reaction diffusion system (19), for $k(x)$ s.t the assumtions via
(20) hold. Then solutions to (19) are non-negative, as long as they initiate
from positive initial conditions.

######  Lemma 3.6.

Consider the reaction diffusion system (19). For $k(x)$ s.t the assumtions via
(20) hold. The solutions to (19) are classical. That is for
$(u_{0},v_{0})\in\mathbb{L}^{\infty}(\Omega)$, $(u,v)\in
C^{1}(0,T;C^{2}(\Omega))$, $\forall T$.

Our goal in this section is to investigate the dynamics of (19). Herein we
will us the comparison technique, and compare to the ODE cases of classical
competition, or the constant fear function case, where the dynamics are well
known.

######  Remark 6.

The analysis in this section are primarily focused on the choice of spatially
homogenous (flat) initial data.

We begin by defining the following systems of PDEs,

(21)
$\displaystyle\begin{split}\overline{u}_{t}&=d_{1}(\overline{u})_{xx}+a_{1}\overline{u}-b_{1}\overline{u}^{2}-c_{1}\overline{u}\overline{v},\\\
\overline{v}_{t}&=d_{2}(\overline{v})_{xx}+a_{2}\overline{v}-b_{2}\overline{v}^{2}-c_{2}\overline{u}\overline{v},\end{split}$

(22)
$\displaystyle\begin{split}\widehat{u_{t}}&=d_{1}(\widehat{u_{t}})_{xx}+a_{1}\widehat{u}-b_{1}\widehat{u}^{2}-c_{1}\widehat{u}\widehat{v},\\\
\widehat{v}_{t}&=d_{2}(\widehat{v})_{xx}+\dfrac{a_{2}\widehat{v}}{1+\mathbf{\widehat{k}}\widehat{u}}-b_{2}\widehat{v}^{2}-c_{2}\widehat{u}\widehat{v},\end{split}$

(23)
$\displaystyle\begin{split}\widetilde{u}_{t}&=d_{1}(\widetilde{u})_{xx}+a_{1}\widetilde{u}-b_{1}\widetilde{u}^{2}-c_{1}\widetilde{u}\widetilde{v},\\\
\widetilde{v}_{t}&=d_{2}(\widetilde{v})_{xx}+\dfrac{a_{2}\widetilde{v}}{1+\mathbf{\widetilde{k}}\widetilde{u}}-b_{2}\widetilde{v}^{2}-c_{2}\widetilde{u}\widetilde{v},\end{split}$

(24)
$\displaystyle\begin{split}\tilde{u}_{t}&=d_{1}(\tilde{u})_{xx}+a_{1}\tilde{u}-b_{1}\tilde{u}^{2}-c_{1}\tilde{u}\widetilde{v},\\\
\tilde{v}_{t}&=d_{2}(\tilde{v})_{xx}+\dfrac{a_{2}\tilde{v}}{1+\mathbf{\tilde{k}}\frac{a_{1}}{b_{1}}}-b_{2}\tilde{v}^{2}-c_{2}\tilde{u}\tilde{v},\end{split}$

where

(25)
$\displaystyle\mathbf{\widehat{k}}=\min_{x\in\Omega}k(x),\quad\,\quad\mathbf{\widetilde{k}}=\max_{x\in\Omega}k(x).$

We assume no flux Neumann boundary conditions for all of the reaction
diffusion systems (21) - (24). Also in each of the systems we prescribe
spatially homogenous (flat) initial conditions $u(x,0)=u_{0}(x)\equiv
c>0,\quad v(x,0)=v_{0}(x)\equiv d>0.$

We now state the following lemma,

######  Lemma 3.7.

Consider the reaction diffusion system (19), as well as the reaction diffusion
systems (21) - (24). Then the following point wise comparison holds,

$\displaystyle\tilde{v}\leq\widetilde{v}\leq
v\leq\widehat{v}\leq\overline{v}.$

###### Proof.

Note via positivity of solutions to (19), (21) - (24), the definitions via
(25), and the upper bound on $u$, which is a solution to (19) of
$\frac{a_{1}}{b_{1}}$, (derived via comparison to the logistic equation), we
have

(26)
$\displaystyle\dfrac{a_{2}}{1+\mathbf{\widetilde{k}}\frac{a_{1}}{b_{1}}}\leq\dfrac{a_{2}}{1+\mathbf{\widetilde{k}}u(x)}\leq\dfrac{a_{2}}{1+k(x)u(x)}\leq\dfrac{a_{2}}{1+\mathbf{\widehat{k}}u(x)}\leq
a_{2},\quad\forall x\in\Omega.$

Thus the result follows via standard comparison theory [11]. ∎

###  3.1. The Competitive Exclusion Case

######  Theorem 3.8.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, and

$\mathbf{\widetilde{k}}>\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}\quad\text{and}\quad\dfrac{a_{1}}{a_{2}}>\max\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}}.$

Then the solution $(u,v)$ to (19) converges uniformly to the spatially
homogenous state $(\frac{a_{1}}{b_{1}},0)$ as $t\to\infty$.

###### Proof.

From the classical theory of competition [20], we know the dynamics for (21),
that is in the competitive exclusion case, when

$\dfrac{a_{1}}{a_{2}}>\max\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}},$

we have

$(\overline{u},\overline{v})\to\Big{(}\dfrac{a_{1}}{b_{1}},0\Big{)}.$

Moreover, under the assumption

$\mathbf{\widetilde{k}}>\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}},$

and making use of Lemma 2.5, along with the use of Theorem 3.4, we have
$(\widetilde{u},\widetilde{v})\to\Big{(}\dfrac{a_{1}}{b_{1}},0\Big{)}$. Now
using Lemma 3.7 we have,

$\displaystyle\widetilde{v}\leq v\leq\overline{v},$

which entails,

$\displaystyle\lim_{t\rightarrow\infty}(\widetilde{u},\widetilde{v})\leq\lim_{t\rightarrow\infty}(u,v)\leq\lim_{t\rightarrow\infty}(\overline{u},\overline{v}),$

subsequently,

$\displaystyle\left(\frac{a_{1}}{b_{1}},0\right)\leq\lim_{t\rightarrow\infty}(u,v)\leq\left(\frac{a_{1}}{b_{1}},0\right).$

Now using a squeezing argument, in the limit that $t\rightarrow\infty$, we
have uniform convergence of solutions of (19), i.e.,

$(u,v)\to\Big{(}\dfrac{a_{1}}{b_{1}},0\Big{)}$

as $t\rightarrow\infty$. ∎

Using the positivity of solutions the requirement on $k(x)$ and so in turn on
$\mathbf{\widetilde{k}}$, can be weakened to derive a stronger result,

######  Theorem 3.9.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, and
$\dfrac{a_{1}}{a_{2}}>\max\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}}$.
Then the solution $(u,v)$ to (19) converges uniformly to the spatially
homogeneous state $(\frac{a_{1}}{b_{1}},0)$ as $t\to\infty$.

###### Proof.

From the classical theory of competition [20], we know the dynamics for (21),
that is in the competitive exclusion case, when

$\dfrac{a_{1}}{a_{2}}>\max\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}},$

we have

$(\overline{u},\overline{v})\to\Big{(}\dfrac{a_{1}}{b_{1}},0\Big{)}.$

Now using Lemma 3.7, we have

$\displaystyle v\leq\overline{v}.$

Using the non negativity of solutions to (19) via Lemma 3.5, entails,

$\displaystyle
0\leq\lim_{t\rightarrow\infty}v\leq\lim_{t\rightarrow\infty}(\overline{v})=0,$

subsequently,

$\displaystyle\lim_{t\rightarrow\infty}(u,v)\rightarrow\left(\frac{a_{1}}{b_{1}},0\right).$

∎

######  Theorem 3.10.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, and

$\dfrac{a_{2}b_{1}^{2}\mathbf{\widetilde{k}}}{(b_{1}+\mathbf{\widetilde{k}}a_{1})^{2}}<\dfrac{b_{1}b_{2}-c_{1}c_{2}}{c_{1}}\quad\text{and}\quad\dfrac{a_{1}}{a_{2}}<\min\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}}.$

Then the solution $(u,v)$ to (19) converges uniformly to the spatially
homogeneous state $(0,\frac{a_{2}}{b_{2}})$ as $t\to\infty$.

###### Proof.

From the classical theory of competition [20], we know the dynamics for (21),
that is in the competitive exclusion case, when

$\dfrac{a_{1}}{a_{2}}<\min\Big{\\{}\dfrac{b_{1}}{c_{2}},\dfrac{c_{1}}{b_{2}}\Big{\\}},$

we have

$(\overline{u},\overline{v})\to\Big{(}0,\dfrac{a_{2}}{b_{2}}\Big{)}.$

Consider the nullclines of (23). If

$\dfrac{d}{du}\Big{[}\dfrac{a_{1}-b_{1}u}{c_{1}}\Big{]}<\dfrac{d}{du}\Big{[}\dfrac{1}{b_{2}}\Big{(}\dfrac{a_{2}}{1+\mathbf{\widetilde{k}}u}-c_{2}u\Big{)}\Big{]},$

it follows via the geometry of the nullclines, that the $v$-nullcline remains
above the $u$-nullcline, and thus
$(\widetilde{u},\widetilde{v})\to\Big{(}0,\dfrac{a_{2}}{b_{2}}\Big{)}$. On
simplification, and using the upper bound estimate for $u$, we have

$\dfrac{a_{2}b_{1}^{2}\mathbf{\widetilde{k}}}{(b_{1}+ka_{1})^{2}}<\dfrac{a_{2}\mathbf{\widetilde{k}}}{(1+\mathbf{\widetilde{k}}u)^{2}}<\Big{(}\dfrac{b_{1}b_{2}-c_{1}c_{2}}{c_{1}}\Big{)}.$

Now using Lemma 3.7 we have,

$\displaystyle\widetilde{v}\leq v\leq\overline{v},$

which entails,

$\displaystyle\lim_{t\rightarrow\infty}(\widetilde{u},\widetilde{v})\leq\lim_{t\rightarrow\infty}(u,v)\leq\lim_{t\rightarrow\infty}(\overline{u},\overline{v}),$

subsequently,

$\displaystyle\left(0,\frac{a_{2}}{b_{2}}\right)\leq\lim_{t\rightarrow\infty}(u,v)\leq\left(0,\frac{a_{2}}{b_{2}}\right).$

Now using a squeezing argument, in the limit that $t\rightarrow\infty$, we
have uniform convergence of solutions of (19), i.e.,

$(u,v)\to\Big{(}0,\dfrac{a_{2}}{b_{2}}\Big{)}$

as $t\rightarrow\infty$. ∎

(a) $f=0$ and $k=0.5+\sin^{2}(x)$

(b) $f=0$ and $k=10+\sin^{2}(x)$

(c) $f=0$ and $k=10x^{5}$

(d) $f=0$ and $k=x^{5}+5x-\sin(x)$

Figure 10. Numerical simulation of $(\ref{eq:PDE})$ for the case of
competition exclusion in $\Omega=[0,1]$. The parameters are chosen as
$[u_{0},v_{0}]=[0.1,0.2],d_{1}=1,d_{2}=1,a_{1}=3,a_{2}=1,b_{1}=b_{2}=1$ and
$c_{1}=c_{2}=0.5$.

(a) $f=0$ and $k=4\sin^{2}(x)$

(b) $f=0$ and $k=4+x^{2}$

Figure 11. Numerical simulation of $(\ref{eq:PDE})$ for the case of
competition exclusion in $\Omega=[0,1]$. The parameters are chosen as
$[u_{0},v_{0}]=[0.1,0.2],d_{1}=1,d_{2}=1,a_{1}=1,a_{2}=2,b_{1}=2,b_{2}=1,c_{1}=1$
and $c_{2}=1$.

###  3.2. The Weak Competition Case

We state the following result,

######  Lemma 3.11.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, with

$\mathbf{\widetilde{k}}>\dfrac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}\quad\text{and}\quad
b_{1}b_{2}>2c_{1}c_{2}.$

Then the solution $(u,v)$ to (19) converges uniformly to the spatially
homogeneous state $(u^{*},0)$ as $t\to\infty$.

###### Proof.

Via Lemma 2.2, and the parametric restrictions assumed we have,

$\displaystyle\widetilde{v}\leq v\leq\widehat{v},$

and the result follows via similar analysis as in Theorem 3.8. ∎

######  Lemma 3.12.

Consider the reaction diffusion system (19), s.t. we are in the weak
competition case when $k(x)\equiv 0$. Then given $0<\epsilon<<1$, there exists
a fear function $k_{\epsilon}(x)$, for which the assumptions via (20) are met,
s.t. the solution $(u,v)$ to (19) with the fear function $k_{\epsilon}(x)$,
converges uniformly to a spatially homogeneous state $(u^{*},v^{*})$ as
$t\to\infty$.

###### Proof.

Given $0<\epsilon<<1$, we can always construct a $k_{\epsilon}$, s.t
$k_{c}-\epsilon\leq\widehat{k}_{\epsilon}$, whereas
$\widetilde{k}_{\epsilon}\leq k_{c}+\epsilon$. Thus via Lemma 3.7 we have,

$\displaystyle\widetilde{v}\leq v_{\epsilon}\leq\widehat{v}.$

Lemma 2.7 ensures we have $(\widetilde{u},\widetilde{v})\to(u^{*},v^{*})$ and
$(\widehat{u},\widehat{v})\to(u^{**},v^{**})$, where the spatially homogeneous
solutions may be different. Hence, via squeezing argument, we can take
$\epsilon\rightarrow 0$, to yield the uniform convergence of solutions, i.e.,

$\lim_{\epsilon\rightarrow
0}\lim_{t\rightarrow\infty}(u_{\epsilon},v_{\epsilon})\to(u^{*},v^{*}).$

This proves the lemma. ∎

(a) $[u_{0},v_{0}]=[1,2]$

(b) $[u_{0},v_{0}]=[0.1,0.2]$

(c) $[u_{0},v_{0}]=[0.1,0.2]$

(d) $[u_{0},v_{0}]=[0.1,0.2]$

Figure 12. Numerical simulation of $(\ref{eq:PDE})$ for the case of weak-
competition in $\Omega=[0,1]$. The parameters are chosen as
$d_{1}=1,d_{2}=1,a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=2,c_{1}=0.3$ and $c_{2}=1.8$.

The numerical simulations above motivate the following conjecture,

######  Conjecture 1.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, with

$\mathbf{\widetilde{k}}<\frac{1}{a_{2}}\left(\frac{b_{1}b_{2}}{c_{1}}-c_{2}\right),$

and the parameters follow $(\ref{one_post_weak})$ and Theorem 2.1, then the
solution $(u,v)$ to (19) converges uniformly to the spatially homogenous state
$(u^{*},v^{*})$ as $t\to\infty$.

######  Remark 7.

We see via Lemma 2.5, and Lemma 2.7, that in the ODE case, if we are in the
weak competition setting without fear, then a critical amount of fear $k_{c}$
is _both_ sufficient and necessarily required to change the system’s dynamics
to a competitive exclusion type scenario. In the PDE case, where the fear
function $k(x)$ can be spatially heterogeneous, this requirement is certainly
sufficient, as seen via Theorem 3.13, but _not_ necessary, in a point wise
sense.

This result is stated next,

######  Theorem 3.13.

Consider the reaction diffusion system (19), with initial data $u_{0}(x)\geq
1$, s.t $u\nearrow u^{*}$, s.t. we are in the weak competition setting when
$k(x)\equiv 0$. For a fear function $k(x)$, s.t the assumptions via (20) are
met, the solution $(u,v)$ converges uniformly to the spatially homogeneous
state $(\frac{a_{1}}{b_{1}},0)$ as $t\to\infty$, if the following condition
holds,

(27)
$\displaystyle\frac{C_{1}}{|\Omega|}\int_{\Omega}\frac{1}{1+k(x)}dx<\left(\frac{1}{1+k_{c}\left(\frac{a_{1}}{b_{1}}\right)}\right),$

where $k_{c}$ is as defined in Lemma 2.5.

###### Proof.

Consider (19), integrating the equation over $\Omega$ yields,

$\displaystyle\frac{d}{dt}\int_{\Omega}vdx$
$\displaystyle=\int_{\Omega}\left(\dfrac{a_{2}v}{1+k(x)u}-b_{2}v^{2}-c_{2}uv\right)dx$
$\displaystyle\leq\int_{\Omega}\left(\dfrac{a_{2}v}{1+k(x)u_{0}}-b_{2}v^{2}-c_{2}uv\right)dx$
$\displaystyle=\frac{1}{1+k(x^{*})u_{0}}\int_{\Omega}a_{2}vdx-\int_{\Omega}\left(b_{2}v^{2}+c_{2}uv\right)dx$
$\displaystyle\leq\frac{1}{1+k(x^{*})}\int_{\Omega}a_{2}vdx-\int_{\Omega}\left(b_{2}v^{2}+c_{2}uv\right)dx$
$\displaystyle\leq\left(C_{1}\int_{\Omega}\frac{1}{1+k(x)}dx\right)\int_{\Omega}a_{2}vdx-\int_{\Omega}\left(b_{2}v^{2}+c_{2}uv\right)dx$
$\displaystyle\leq\left(\frac{1}{1+k_{c}\frac{a_{1}}{b_{1}}}\right)\int_{\Omega}a_{2}vdx-\int_{\Omega}\left(b_{2}v^{2}+c_{2}uv\right)dx$
$\displaystyle\leq\int_{\Omega}\left(\frac{a_{2}}{1+k_{c}\frac{a_{1}}{b_{1}}}v-b_{2}v^{2}-c_{2}uv\right)dx.$

This follows via the mean value theorem for integrals. We can now compare,

(29) $\displaystyle\int_{\Omega}vdx<\int_{\Omega}\tilde{v}dx,$

where $\tilde{v}$ solves
$\frac{d\tilde{v}}{dt}=\left(\frac{a_{2}}{1+k_{c}\frac{a_{1}}{b_{1}}}\tilde{v}-b_{2}\tilde{v}^{2}-c_{2}u\tilde{v}\right).$
Thus using Theorem 3.4, Lemma 2.2 and the positivity of solutions, this
entails

(30) $\displaystyle
0\leq\lim_{t\rightarrow\infty}\int_{\Omega}vdx\leq\lim_{t\rightarrow\infty}\int_{\Omega}\tilde{v}dx=0.$

The bounds on $v$, standard Lebesgue convergence theorems, and squeezing
argument entail,

(31) $\displaystyle\int_{\Omega}(\lim_{t\rightarrow\infty}v)dx\rightarrow 0,$

which implies the uniform convergence,

$\displaystyle\lim_{t\rightarrow\infty}(u,v)\rightarrow\left(\frac{a_{1}}{b_{1}},0\right).$

∎

######  Remark 8.

The $C_{1}$ in Theorem 3.13 is a pure constant, that will depend on the size
of the domain $\Omega$, and the other problem parameters, but not on the
initial data or the spatial variable $x$.

######  Remark 9.

Clearly $k(x)$ could be chosen s.t. it lies below $k_{c}$ for a portion of the
domain, and above $k_{c}$ on some portion of the domain - thus $k_{min}$ does
not lie uniformly above $k_{c}$. Yet via Theorem 3.13, we see that, one can
change the system’s dynamics and bring it to a competitive exclusion type
scenario, from a coexistence situation.

The next result gives a lower estimate for the fear function $k$,

######  Lemma 3.14.

Consider the fear function $k(x)$ in Theorem 3.13, then we have,

$\displaystyle\int_{\Omega}k(x)dx\geq|\Omega|\left(1-\frac{1}{C_{1}}\left(\frac{1}{1+k_{c}\left(\frac{a_{1}}{b_{1}}\right)}\right)\right).$

###### Proof.

We have that for any fear function $k(x)$ satisfying (20),

(32)
$\displaystyle\frac{1}{1+k(x)}>1-k(x)\implies\int_{\Omega}\frac{1}{1+k(x)}dx>|\Omega|-\int_{\Omega}k(x)dx.$

Now using Theorem 3.13 the result follows. ∎

###  3.3. The Strong Competition Case

######  Theorem 3.15.

Consider the reaction diffusion system (19), for a fear function $k(x)$, s.t
the assumptions via (20) are met, with $b_{1}b_{2}<c_{1}c_{2}$. Then there
exists sufficiently large positive initial data $[u_{0},v_{0}]$ for which the
solution $(u,v)$ to (19) converges uniformly to the spatially homogenous state
$(0,\frac{a_{2}}{b_{2}})$ as $t\to\infty$, while there also exists
sufficiently small positive initial data $[u_{1},v_{1}]$ for which solution
$(u,v)$ to (19) converges uniformly to the spatially homogenous state
$(\frac{a_{1}}{b_{1}},0)$ as $t\to\infty$.

###### Proof.

Consider the system (21). From the classical strong competition parametric
restrictions, $b_{1}b_{2}<c_{1}c_{2}$ that are assumed, as well as Lemma 3.7,
we can make use of standard competition theory and use the stable manifold
theorem, i.e., $\exists W_{s}(E_{4})\in C^{1}$ separatrix, such that for
initial data $(\overline{u}_{0},\overline{v}_{0})$ chosen above $W_{s}(E_{4})$
the solution $(\overline{u},\overline{v})\to(0,v^{*})$ and for initial data
chosen below $W_{s}(E_{4})$, $(\overline{u},\overline{v})\to(u^{*},0)$. Here
$E_{4}$ is the interior saddle equilibrium to the kinetic (ODE) system for
(21). Moreover, since
$\dfrac{a_{2}}{1+\mathbf{\widetilde{k}}\frac{a_{1}}{b_{1}}}\leq a_{2}$, and
$b_{1}b_{2}<c_{1}c_{2}$, we have that for (25) we still remain in the strong
competition case, and via standard theory again, $W^{1}_{s}(E^{*}_{4})\in
C^{1}$ separatrix, such that for initial data
$(\widehat{u}_{0},\widehat{v}_{0})$ chosen above $W^{1}_{s}(E^{*}_{4})$ the
solution $(\widehat{u},\widehat{v})\to(0,v^{*})$ and for initial data chosen
below $W^{1}_{s}(E_{4})$, $(\widehat{u},\widehat{v})\to(u^{*},0)$. Here
$E^{*}_{4}$ is the interior saddle equilibrium to the kinetic (ODE) system for
(25). Now since
$\dfrac{a_{2}}{1+\mathbf{\widetilde{k}}\frac{a_{1}}{b_{1}}}\leq a_{2}$, the
$v$ component of $E^{*}_{4}$ is higher than the $v$ component of $E_{4}$. Now
using the $C^{1}$ property of the separatricies
$W^{1}_{s}(E^{*}_{4}),W_{s}(E_{4})$, we have the existence of a wedge $\nu$
emanating from $E_{4}$, s.t within $\nu$ we have $W^{1}_{s}(E^{*}_{4})\geq
W_{s}(E_{4})$. Note via Lemma 3.7 we have $\tilde{v}\leq v\leq\widehat{v}$.
Let us consider positive initial data $(u_{0},v_{0})$ chosen large enough,
within $\nu$ s.t. $(u_{0},v_{0})>>W^{1}_{s}(E^{*}_{4})>W_{s}(E_{4})$, we will
have

$\displaystyle\Big{\\{}(0,v^{*})\Big{\\}}\leq\Big{\\{}(u,v)\Big{\\}}\leq\Big{\\{}(0,v^{*})\Big{\\}}.$

On the other hand, for positive initial data $(u_{1},v_{1})$ chosen small
enough s.t. $(u_{1},v_{1})<<W^{1}_{s}(E_{4})<W_{s}(E_{4})$, we will have

$\displaystyle\Big{\\{}(u^{*},0)\Big{\\}}\leq\Big{\\{}(u,v)\Big{\\}}\leq\Big{\\{}(u^{*},0)\Big{\\}}.$

This proves the theorem.

∎

(a) $[u_{0},v_{0}]=[0.2,0.2].$

(b) $[u_{0},v_{0}]=[2,0.5].$

(c) $[u_{0},v_{0}]=[0.2,0.2].$

(d) $[u_{0},v_{0}]=[2,0.5].$

Figure 13. Numerical simulation of $(\ref{eq:PDE})$ for the case of strong-
competition in $\Omega=[0,1]$. The parameters are chosen as
$d_{1}=1,d_{2}=1,a_{1}=0.5,a_{2}=2,b_{1}=0.5,b_{2}=0.8$ and $c_{1}=c_{2}=4$.

(a) $[u_{0},v_{0}]=[0.2,0.2].$

(b) $[u_{0},v_{0}]=[0.002,0.2].$

Figure 14. Numerical simulation of $(\ref{eq:PDE})$ for the case of two-
positive interior equilibria in $\Omega=[0,1]$. The parameters are chosen as
$d_{1}=1,d_{2}=1,a_{1}=1,a_{2}=2,b_{1}=2,b_{2}=0.3$ and
$c_{1}=0.3,c_{2}=0.05$.

Figure 15. Numerical simulation for the case $(v^{*}<u^{*})$. The parameters
are chosen as
$[u_{0},v_{0}]=[0.1,0.2],d_{1}=1,d_{2}=1,a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=2,c_{1}=0.3$
and $c_{2}=1.8$. The interior equilibrium $[u^{*},v^{*}]=[0.958904,0.136986]$
and the fear threshold
$k_{c}=\frac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}=0.11$ and
$C_{1}=1$.

Figure 16. Numerical simulation for the case $(u^{*}<v^{*})$. The parameters
are chosen as
$[u_{0},v_{0}]=[0.1,0.2],d_{1}=1,d_{2}=1,a_{1}=1.5,a_{2}=1.8,b_{1}=1,b_{2}=1,c_{1}=0.8$
and $c_{2}=0.5$. The interior equilibrium $[u^{*},v^{*}]=[0.1,1.75]$ and the
fear threshold
$k_{c}=\frac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}=0.933$ and
$C_{1}=0.8$.

Figure 17. The parameters are chosen as
$[u_{0},v_{0}]=[0.1,0.2],d_{1}=1,d_{2}=1,a_{1}=1,a_{2}=2,b_{1}=1,b_{2}=2,c_{1}=0.3$
and $c_{2}=1.8$. The interior equilibrium $[u^{*},v^{*}]=[0.958904,0.136986]$
and the fear threshold
$k_{c}=\frac{a_{2}b_{1}^{2}-c_{2}a_{1}b_{1}}{a_{1}^{2}c_{2}}=0.11$ and
$C_{1}=1$.

From the observations in the above numerical simulations, we can state a
conjecture concerning positive interior equilibrium,

######  Conjecture 2.

Consider the system $\eqref{eq:PDE}$. For any non-negative fear function
$k(x)$ such that

(33)
$\displaystyle\Big{(}\dfrac{b_{2}b_{1}}{c_{1}}-c_{2}\Big{)}<\frac{ka_{2}b_{1}^{2}}{(b_{1}+ka_{1})^{2}}$

and parametric restrictions given by (20) and Theorem 2.1 hold true, for
$\mathbf{k}=\mathbf{\widetilde{k}},\mathbf{\hat{k}}$. There exists some data
$[u_{0},v_{0}]$, for which the solution $(u,v)\to(u^{*},v^{*})$, and for some
choice of data $[u_{1},v_{1}]$, the solution converges to the boundary
equilibrium $(0,v^{*})$.

##  4\. Discussion

In this section we discuss several aspects of the lemmas and theorems
described in the current manuscript.

(a) $f=0.5$

(b) Various heterogeneous fear functions

Figure 18. (A) Phase plots showing dynamics under strong competition
parametric restriction with $k=0$. Parameters used are
$a_{1}=1,a_{2}=1,b_{1}=1,b_{2}=1,c_{1}=2,c_{2}=2$ and $f=0.5$ (B) Illustration
of various heterogeneous fear functions.
$k_{1}(x)=0.8e^{-x},k_{2}(x)=0.01+\sin^{2}(10x),k_{3}(x)=x+0.2x^{5}$ and
$k_{4}(x)=x$

We see that in the competitive exclusion case, where $(u^{*},0)$ is globally
stable, a small amount of fear in $u$, can bring about a strong competition
type situation - via a transcritical bifurcation. Where after bifurcation
there appears an interior equilibrium, which is a saddle, and one has initial
condition dependent attraction to $(u^{*},0)$ or $(0,v^{*})$. See Figs.
[4,5,9]. However, a slightly larger level of fear, can bring down the
separatrix drastically see Fig. 18(a). Furthermore, here (depending on
parametric restrictions) only a very small quantity of fear can create two
interior equilibrium, see Figs. [5,9], where there are both weak and strong
competition dynamics at play, where most initial conditions are attracted to
an interior equilibrium, but certain initial conditions $(u_{0},v_{0})$ (where
$u_{0}<<1,v_{0}>>1$), are attracted to $(0,v^{*})$. It is interesting to think
about this in an invasion setting. If $u$ were an invasive species, and $v$ a
resident species, then in the absence of fear in $u$, it can invade and thus
exclude $v$. However, if the resident $v$ can instill just a small amount of
fear in the invader $u$, coexistence for most initial conditions is possible.
If one plays devil’s advocate, and switches the role of $u$ and $v$, ($u$
resident, $v$ invader), then we see in the absence of fear in the resident, it
will exclude the invader, but if it can instill some fear, then coexistence is
possible, as earlier alluded to. However, having too much fear in either case,
yields only one interior equilibrium, which is a saddle and we are in a strong
competition type setting, with initial condition dependent attraction to
either $(u^{*},0)$ or $(0,v^{*})$. Thus if coexistence (for most initial
conditions) is sought after, it is advantageous to induce a little fear - but
not to much fear. Dynamically, this is seen because in the former case we have
a saddle-node bifurcation occurring first, followed by a transcritical
bifurcation, see Fig. 9(b), as opposed to the later case, where only a
transcritical bifurcation occurs, see Fig. 9(a).

This phenomenon becomes even more interesting in sociological or political
settings, particularly when we are in the spatially explicit scenario.
Consider the case of law enforcement trying to control crime in high crime
areas, where traditionally control efforts have failed, indicative of the
competitive exclusion scenario, or a coexistence scenario, where the criminal
groups have not been managed so coexist at high levels, and have not been
eradicated. The question becomes, what is the optimal design of the fear
function $k(x)$, that could now yield a co-existence scenario - or one that
can yield a competitive exclusion scenario. We see via Theorem 3.13, that this
function could be very small in some areas of the spatial domain, and large
enough in others, so that it would change dynamics - this is also seen
numerically, where “hotspot” type fear functions have been utilised to
generate a competitive exclusion type scenario, see Figs. [15,16,17]. Similar
ideas for control and policing activities have been explored in [31, 32].
Essentially, the fear function can be modeled through various functional
types, that also preserve spatial features of the underlying domain. For
instance, consider competition between two warring drug cartels, where the
weaker cartel has certain territorial strongholds. This situation can be
modeled through oscillatory functions like $\sin^{2}(nx)$, where
$n\in\mathbb{N}$. Moreover, for complex ecological models, the result of these
theorems also holds, where we need to model the fear responses by a sequence
of functions or in the form of piece-wise functions. We display some of these
functions in Fig. 18(b). Many of the relevant theorem have been tested by
choosing such functions (See Figs. 10,12,13,14). Note, all of our PDE
simulations for the case of a spatially heterogeneous fear function were
performed in `MATLAB` R2021b, using the `MATLAB` inbuilt function pdepe, which
is used in solving $1-$D parabolic and elliptic PDEs. The simulations were run
on an $8-$core CPU, Apple $M1$ pro-chip-based workstation. In this
configuration, a typical simulation takes between $5-7$ seconds when the
spatial domain is taken to be $[0,1]$, and is partitioned into $1000$ sub-
intervals.

In the weak competition case without fear, once there is a critical level of
fear, in either $u$ or $v$, a competitive exclusion type phenomenon will
occur, see Lemma 2.5. This occurs again via a transcritical bifurcation, see
Theorem 2.19 and Fig. [5,9]. This again is interesting in the spatially
explicit setting, as via Theorem 3.8, where the fear function need not be
above $k_{c}$ so as to induce competitive exclusion. We have explored certain
types of fear functions herein, see Fig.[10,15,16,17], but the effect of
functions such as in Fig. 18(b), remain to be explored. Here one could look at
the effect of fear in fragmented domains vs domains that are not, via deriving
or enforcing conditions relating the fear function $k(x)$ to the resource
function $m(x)$, see similar ideas explored in [27]. It would be interesting
to explore applications of these sorts to competing political parties, where a
nationally weaker party with smaller voter bank, could have several local
strongholds. What levels of fear need to be instilled in those strongholds so
as to enforce competitive advantage, is an apt question. Herein estimates on
the $\mathbb{L}^{1}(\Omega)$ norm of $k(x)$, would be interesting to derive.
Currently Lemma 3.14, only gives lower estimates - but sharp upper estimates
or even just upper estimates are unknown, and would make for interesting
future work. The effect of fear on the strong competition setting is perhaps
the least interesting dynamically. Herein, having fear in $u$ or $v$, only
shifts the interior saddle equilibrium - but qualitatively the dynamics remain
the same, as in a strong competition type scenario persists, no matter what
level of fear ($f$ or $k$) is chosen. Also it would be of interest to
rigorously prove Conjecture 1 and Conjecture 2. Our current numerical evidence
clearly motivates their validity.

Also, in [50], it is found that various monotonically decreasing fear
functions essentially yield the same dynamics. This has not been tested in the
current work, in the case of competitive systems. Neither have we investigated
rigorosly the case of both competitors being fearful of each other (see 6 for
some preliminaries), with possible different fear functions, indicative of
different LOF for each competitor. Here again, upper estimates on the fear
functions would be useful and we could allude to methods and techniques
explored in [46] and in [81]. Here, one could consider a spatially dependent
growth function as well, such as in [81], and attempt to derive conditions
relating the function describing the resources to the fear function.
Furthermore, it would make for interesting future work if certain choices of
(density) dependence or fear functions which lead to degenerate dynamics, can
cause periodic orbits. In the current scenario Lemma 2.10 and Lemma 2.17, do
not allow limit cycle dynamics. All in all, we hope these questions lead to
future investigations of the fear effect in competitive systems, as a host of
rich applications exist.

##  5\. Acknowledgement

VS and RP acknowledge valuable summer support from the National Science
Foundation via DMS 1715044.

##  6\. Appendix

###  6.1. The case of $u$ fearing $v$

######  Lemma 6.1.

The trivial steady state $E_{1}$ is locally unstable.

###### Proof.

Evaluating $J^{*}$ at $E_{1}$ yields the following characteristic equation:

$\left(a_{1}-\lambda\right)\left(a_{2}-\lambda\right)=0.$

Clearly, $\lambda_{1}=a_{1}>0$ and $\lambda_{2}=a_{2}>0$ and hence $E_{0}$ is
locally unstable. ∎

######  Lemma 6.2.

The boundary equilibrium point $E_{2}$ is locally stable iff
$a_{2}b_{1}<c_{2}a_{1}$.

###### Proof.

On evaluating the $J^{*}$ at $E_{2}$ , we have

$\widehat{J}^{*}(E_{2})=\left(\begin{array}[]{cc}-a_{1}&-\dfrac{a_{1}^{2}f}{b_{1}}-\dfrac{c_{1}a_{1}}{b_{1}}\\\
0&a_{2}-\dfrac{c_{2}a_{1}}{b_{1}}\\\ \end{array}\right).$

Being a triangular matrix, we know that the above matrix has two eigenvalues,
$\lambda_{1}=-a_{1}$ and $\lambda_{2}=a_{2}-\dfrac{c_{2}a_{1}}{b_{1}}$. Under
the assumed parametric restriction

$a_{2}b_{1}<c_{2}a_{1}\iff\lambda_{2}:=a_{2}-\dfrac{c_{2}a_{1}}{b_{1}}<0.$

Hence, the equilibrium point $E_{2}$ is locally stable

∎

######  Lemma 6.3.

The boundary equilibrium point $E_{3}$ is locally stable iff
$f>\dfrac{a_{1}b_{2}^{2}-a_{2}b_{2}c_{1}}{a_{2}^{2}c_{1}}$.

###### Proof.

Let $f>\dfrac{a_{1}b_{2}^{2}-a_{2}b_{2}c_{1}}{a_{2}^{2}c_{1}}$. A similar
evaluation of $J^{*}$ at $E_{3}$ gives

(34)
$J^{*}(E_{3})=\left(\begin{array}[]{cc}\dfrac{a_{1}b_{2}}{a_{2}f+b_{2}}-\dfrac{a_{2}c_{1}}{b_{2}}&0\\\
-\dfrac{a_{2}\left(c_{2}\right)}{b_{2}}&-a_{2}\\\ \end{array}\right)$

and its corresponding characteristic equation is

$\dfrac{\left(a_{2}+\lambda\right)\left(a_{2}b_{2}c_{1}-a_{1}b_{2}^{2}+a_{2}^{2}c_{1}f+\lambda(b_{2}^{2}+a_{2}b_{2}f)\right)}{b_{2}\left(a_{2}f+b_{2}\right)}=0.$

The associated eigenvalues are $\lambda_{1}=-a_{2}<0$ and
$\lambda_{2}=\dfrac{-a_{2}b_{2}c_{1}+a_{1}b_{2}^{2}-a_{2}^{2}c_{1}f}{b_{2}\left(a_{2}f+b_{2}\right)}$.
Since

$f>\dfrac{a_{1}b_{2}^{2}-a_{2}b_{2}c_{1}}{a_{2}^{2}c_{1}}\iff\lambda_{2}<0,$

and hence $E_{3}$ is locally stable. ∎

######  Lemma 6.4.

The interior equilibrium $E_{4}$ exists and is locally stable if

$f<\frac{1}{a_{1}}\Big{(}\frac{b_{1}b_{2}}{c_{2}}-c_{1}\Big{)}.$

###### Proof.

On evaluating $J^{*}$ at $E_{4}$ , we have

$\widehat{J}^{*}(E_{4})=\left(\begin{array}[]{cc}-b_{1}u^{*}&-\dfrac{a_{1}fu^{*}}{(fv^{*}+1)^{2}}-c_{1}u^{*}\\\
-c_{2}v^{*}&-b_{2}v^{*}\\\ \end{array}\right).$

To claim local stability of interior equilibrium $E_{4}$, we need to show
$Trace(J^{*}(E_{4}))<0$ and $Det(J^{*}(E_{4}))>0$. Simple calculation yields

$Trace(J^{*}(E_{4}))=-b_{1}u^{*}-b_{2}v^{*}<0$

and

$\displaystyle\begin{split}Det(J^{*}(E_{4}))&=u^{*}v^{*}\Big{[}b_{1}b_{2}-c_{2}\Big{(}\dfrac{a_{1}f}{(fv^{*}+1)^{2}}+c_{1}\Big{)}\Big{]}.\end{split}$

Note,

$a_{1}f+c_{1}>\dfrac{a_{1}f}{(fv^{*}+1)^{2}}+c_{1}.$

Hence, under the assumption
$f<\frac{1}{a_{1}}\Big{(}\frac{b_{1}b_{2}}{c_{2}}-c_{1}\Big{)}$, we have

$c_{2}\Big{(}\dfrac{a_{1}f}{(fv^{*}+1)^{2}}+c_{1}\Big{)}<c_{2}\Big{(}a_{1}f+c_{1}\Big{)}<b_{1}b_{2}\implies
Det(J^{*}(E_{4}))>0,$

and the result follows. ∎

######  Lemma 6.5.

The interior equilibrium $E_{4}$ exits and is saddle if

$\Big{(}\dfrac{b_{1}b_{2}}{c_{2}}-c_{1}\Big{)}<\dfrac{a_{1}c_{2}^{2}f}{(fa_{2}+c_{2})^{2}}.$

###### Proof.

To claim local stability of interior equilibrium $E_{4}$, we need to show
$Trace(J^{*}(E_{4}))<0$ and $Det(J^{*}(E_{4}))<0$. Consider,

$Trace(J^{*}(E_{4}))=-b_{1}u^{*}-b_{2}v^{*}<0$

whereas

$\displaystyle\begin{split}Det(J^{*}(E_{4}))&=u^{*}v^{*}\Big{[}b_{1}b_{2}-c_{2}\Big{(}\dfrac{a_{1}fu^{*}}{(fv^{*}+1)^{2}}+c_{1}\Big{)}\Big{]}.\end{split}$

From the density of reals and nullclines of $v$, we have

$\dfrac{a_{1}c_{2}^{2}f}{(fa_{2}+c_{2})^{2}}<\dfrac{a_{1}f}{\Big{[}f(\frac{a_{2}}{c_{2}}-c_{2}u^{*})+1\Big{]}^{2}}=\dfrac{a_{1}f}{(fv^{*}+1)^{2}}.$

Moreover, under the assumption, we have

$\dfrac{b_{1}b_{2}}{c_{2}}-c_{1}<\dfrac{a_{1}c_{2}^{2}f}{(fa_{2}+c_{2})^{2}}<\dfrac{a_{1}f}{(fv^{*}+1)^{2}}.$

On further rearrangement,

$b_{1}b_{2}-c_{2}\Big{(}\dfrac{a_{1}f}{(fv^{*}+1)^{2}}+c_{1}\Big{)}<0\implies
Det(J^{*}(E_{4}))<0,$

and the result follows. ∎

###  6.2. Case of both species $u$ and $v$ fearing each other

We consider the case of the both the competitor $v$ and $u$ are fearful to
each other. Thus in the classical model (1), we model the fear effect as in
[50], where the growth rate of both the competitor $v$ and $u$, is not
constant but rather density dependent. Essentially, the growth rate is
decreased by a factor $\approx\frac{1}{1+ku}$, and $\approx\frac{1}{1+fv}$,
where $k,f\geq 0$ is a fear coefficient. When $k,f=0$, the assumption is there
is no fear, and one recovers the classical model (1). If fear is present, we
obtain the following ODE model for two competing species $u$ and $v$:

(35) $\begin{split}\dfrac{du}{dt}&=\dfrac{a_{1}u}{1+fv}-b_{1}u^{2}-c_{1}uv,\\\
\dfrac{dv}{dt}&=\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv.\\\ \end{split}$

Equilibria: In the event that there is no fear present, or $f=k=0$, (35)
reduces to the classical competition model (1). The dynamical analysis of (35)
when $f,k>0$, leads to 4th order polynomial analysis $\eqref{eq: vpoly}.$ The
system (35) possesses the following biologically feasible equilibria. These
are

  * •

$\mathbf{E}_{1}=(0,0)$,

  * •

$\mathbf{E}_{2}=\left(\dfrac{a_{1}}{b_{1}},0\right)$,

  * •

$\mathbf{E}_{3}=\left(0,\dfrac{a_{2}}{b_{2}}\right)$,

  * •

$\mathbf{E}_{4}=\left(u^{*},v^{*}\right)$,

where

(36) $\displaystyle
u^{*}=\dfrac{1}{b_{1}}\left(\dfrac{a_{1}}{1+fv^{*}}-c_{1}v^{*}\right)$

and $v^{*}$ is a positive root of the following fourth-order polynomial

(37) $A(v^{*})^{4}+B(v^{*})^{3}+C(v^{*})^{2}+D(v^{*})+E=0$

where

(38) $\begin{split}A&=c_{1}f^{2}k\left(b_{1}b_{2}-c_{1}c_{2}\right),\\\
B&=-f\left(b_{1}b_{2}-c_{1}c_{2}\right)\left(b_{1}f-2c_{1}k\right),\\\
C&=b_{2}b_{1}\left(-a_{1}fk-2b_{1}f+c_{1}k\right)+c_{1}c_{2}\left(2f\left(a_{1}k+b_{1}\right)-c_{1}k\right)+a_{2}b_{1}^{2}f^{2},\\\
D&=c_{2}\left(c_{1}\left(2a_{1}k+b_{1}\right)-a_{1}b_{1}f\right)+2a_{2}b_{1}^{2}f-b_{2}b_{1}\left(a_{1}k+b_{1}\right),\\\
E&=a_{2}b_{1}^{2}-a_{1}c_{2}\left(a_{1}k+b_{1}\right).\end{split}$

The Jacobian matrix of system $(\ref{eq: Fearmodel1})$ is given by

(39)
$\mathbf{J}(u^{*},v^{*})=\left(\begin{array}[]{cc}\dfrac{a_{1}}{fv^{*}+1}-2b_{1}u^{*}-c_{1}v^{*}&-\dfrac{a_{1}fu^{*}}{(fv^{*}+1)^{2}}-c_{1}u^{*}\\\
-\dfrac{a_{2}kv^{*}}{(ku^{*}+1)^{2}}-c_{2}v^{*}&\dfrac{a_{2}}{ku^{*}+1}-2b_{2}v^{*}-c_{2}u^{*}\\\
\end{array}\right).$

###  6.3. Transcritical bifurcation

######  Theorem 6.6.

The model (2) undergoes a transcritical bifurcation around $E_{3}^{*}$ when
$a_{1}=a_{1}^{*}=\dfrac{c_{1}a_{2}}{b_{2}}$ and
$c_{1}\neq\dfrac{b_{1}b_{2}}{ka_{2}+c_{2}}$.

###### Proof.

The Jacobian matrix for system (2) evaluated at $E_{3}$ with
$a_{1}^{*}=\dfrac{c_{1}a_{2}}{b_{2}}$ is given as

(40) $JJ^{*}=\left(\begin{array}[]{cc}0&0\\\
-\dfrac{a_{2}}{b_{2}}\left(a_{2}k+c_{2}\right)&-a_{2}\\\ \end{array}\right).$

The corresponding eigenvalues to the Jacobian of (2) in Eq. (40) are
$\lambda_{1}=0$ and $\lambda_{2}=-a_{2}.$ Clearly, there is a zero eigenvalue
at $a_{1}=a_{1}^{*}=\dfrac{c_{1}a_{2}}{b_{2}}$. Next, we let
$W=(w_{1},w_{2})^{T}$ and $Z=(z_{1},z_{2})^{T}$ represent the eigenvectors
related to the zero eigenvalue of the matrices $JJ^{*}$ and $JJ^{*T}$
respectively.

We obtain $W=\left(-\dfrac{b_{2}}{ka_{2}+c_{2}},1\right)^{T}$ and
$Z=\left(1,0\right)^{T}$. Now, let $R=(R_{1},R_{2})^{T}$ where

$\begin{split}R_{1}&=a_{1}u-b_{1}u^{2}-c_{1}uv,\\\
R_{2}&=\dfrac{a_{2}v}{1+ku}-b_{2}v^{2}-c_{2}uv.\end{split}$

Presently, we validate the transversality conditions using the Sotomayor’s
theorem [9]. Now,

$Z^{T}R_{a_{1}^{*}}(E_{3}^{*},a_{1})=\left(1,0\right)\left(0,0\right)^{T}=0.$

Also,

$\displaystyle Z^{T}\left[DR_{a_{1}}\left(E_{3},a_{1}^{*}\right)W\right]$
$\displaystyle=\left(\begin{array}[]{cc}1&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}1&0\\\ 0&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}w_{1}\\\ w_{2}\\\
\end{array}\right)$ $\displaystyle=-\dfrac{b_{2}}{ka_{2}+c_{2}}\neq 0.$

and

$\begin{split}Z^{T}\left[D^{2}R\left(E_{3},a_{1}^{*}\right)(W,W)\right]&=\left(\begin{array}[]{cc}1&0\\\
\end{array}\right)\left(\begin{array}[]{ccc}\dfrac{2b_{2}}{ka_{2}+c_{2}}\left(c_{1}-\dfrac{b_{1}b_{2}}{ka_{2}+c_{2}}\right)\\\
\dfrac{2a_{2}^{2}k^{2}b_{2}}{\left(ka_{2}+c_{2}\right)^{2}}\\\
\end{array}\right)\\\
&=\dfrac{2b_{2}}{ka_{2}+c_{2}}\left(c_{1}-\dfrac{b_{1}b_{2}}{ka_{2}+c_{2}}\right)\neq
0.\end{split}$

Therefore by the Sotomayor’s theorem system (2) experiences a transcritical
bifurcation at $a_{1}=a_{1}^{*}=\dfrac{c_{1}a_{2}}{b_{2}}$ around $E_{3}^{*}$.
∎

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