# Dynamics of a generalized Beverton-Holt competition model subject to Allee
effects

Yun Kang111Sciences and Mathematics Faculty, Arizona State University, Mesa,
AZ 85212, USA. E-mail<EMAIL_ADDRESS>
###### Abstract

In this article, we propose and study a generalized Beverton-Holt competition
model subject to Allee effects to obtain insights on how the interplay of
Allee effects and contest competition affects the persistence and the
extinction of two competing species. By using the theory of monotone dynamics
and the properties of critical curves for non-invertible maps, our analysis
shows that our model has relatively simple dynamics, i.e., almost every
trajectory converges to a locally asymptotically stable equilibrium if the
intensity of intra-specific competition intensity exceeds that of inter-
specific competition. This equilibrium dynamics is also possible when the
intensity of intra-specific competition intensity is less than that of inter-
specific competition but under conditions that the maximum intrinsic growth
rate of one species is not too large. The coexistence of two competing species
occurs only if the system has four interior equilibria. We provide an
approximation to the basins of the boundary attractors (i.e., the extinction
of one or both species) where our results suggests that contest species are
more prone to extinction than scramble ones are at low densities. In addition,
in comparison to the dynamics of two species scramble competition models
subject to Allee effects, our study suggests that (i) Both contest and
scramble competition models can have only three boundary attractors without
the coexistence equilibria, or four attractors among which only one is the
persistent attractor, whereas scramble competition models may have the
extinction of both species as its only attractor under certain conditions,
i.e., _the essential extinction_ of two species due to strong Allee effects;
(ii) Scramble competition models like Ricker type models can have much more
complicated dynamical structure of interior attractors than contest ones like
Beverton-Holt type models have; and (iii) Scramble competition models like
Ricker type competition models may be more likely to promote the coexistence
of two species at low and high densities under certain conditions: At low
densities, weak Allee effects decrease the fitness of resident species so that
the other species is able to invade at its low densities; While at high
densities, scramble competition can bring the current high population density
to a lower population density but is above the Allee threshold in the next
season, which may rescue a species that has essential extinction caused by
strong Allee effects. Our results may have potential to be useful for
conservation biology: For example, if one endangered species is facing
essential extinction due to strong Allee effects, then we may rescue this
species by bringing another competing species subject to scramble competition
and Allee effects under certain conditions.

###### keywords:

Allee effects, Scramble competition models, Contest competition models, Basins
of attractions, Extinction, Coexistence

##  1 Introduction

An Allee effect is a biological phenomenon characterized by a positive
correlation between a population density and its per capita growth rate at
small population densities (Allee _et al._ 1949). A distinction is made
between a _strong Allee effect_ and a _weak Allee effect_ : a _strong Allee
effect_ refers as to a population that exhibits a “critical size or density”,
below which population declines to extinction, and above which it may
increase; while a _weak Allee effect_ refers as to a population that lacks a
“critical density”, but where, at lower densities, the population growth rate
rises with increasing densities (Stephens _et al._ 1999; Lidicker 2010). Allee
effects have been detected in natural populations for a wide array of taxa and
are believed to be strong regulators of extinction, influencing colonization
success of invasive species, disease dynamics, and the long-run population
demise in conservation. Mathematical models can help us understand these
combined effects on persistence, expansion or extinction of biological
species. There is a considerable amount of populations dynamics in the
presence of Allee effects (e.g., Dennis 1989 & 2002; Selgrade and Namkoong
1992; McCarthy 1997; Shigesada and Kawasaki 1997; Greene and Stamps 2001;
Keitt _et al_ 2001; Fagan _et al_ 2002; Wang _et al_ 2002; Liebhold and
Bascompte 2003; Schreiber 2003; Drake 2004; Zhou _et al_ 2004; Petrovskii _et
al_ 2005; Taylor and Hastings 2005; Jang 2006; Aguirre _et al_ 2009; Egami
2009&2010; Thieme _et al._ 2009; Elaydi and Sacker 2010; Wang _et al_ 2010;
Liu _et al_ 2011; Kang and Castillo-Chavez 2012; Kang and Castillo-Chavez
2014a&b; Cai _et al_ 2014; Peng and Kang 2015; Livadiotis _et al_ 2015) as
well as various models in patchy environments (e.g., Amarasekare 1998a &1998b;
Gyllenberg _et al_ 1999; Ackleh _et al_ 2007; Kang and Lanchier 2011; Kang and
Armbruster 2011; Kang and Castillo-Chavez 2014c). In another direction,
evolutionary models with Allee effects have gained some attentions (Cushing
and Hudson 2012; Kang and Udiani 2014; Cushing 2015). For example, Cushing and
Hudson (2012) investigated the global dynamics of an evolutionary model for a
population subject to strong Allee effects. One of their results suggests that
evolution is beneficial in the sense that reduces the possibility of
extinction due to an Allee effect.

Resource competition can be defined in the contest-scramble spectrum
competition according to resource partitioning among competitors (Nicholson
1954). Contest and scramble-type competition are two extreme forms of
competition, which are characterized by resource monopolization and resource
sharing, respectively. Contest competition results in a constant number of
survivors with enough resource gain against an initial density of competitors,
while scramble competition (also called over-compensation) results in an
increased number of survivors with decreased resource intake and an increased
number of competitors (Calow _et al._ 1998). A well-known example for contest
competition is the Beverton-Holt model while the well-known example for
scramble competition is the Ricker model. Even though Allee effects are
expected commonly in nature, they are often ignored in studies of species
coexistence mechanisms. Many species can experience different forms of
competition at high densities and Allee effects at low densities with the
consequence that their populations do not grow optimally at low densities and
individuals compete with one another at high densities (Begon _et al._ 1996;
Etiemme _et al._ 2002; Kang 2013). The potential effects of combining positive
density dependences from Allee effects with negative density dependences
coming from different types of competition are expected to generate rich
dynamics due to the fact that outcomes of varied patterns of competition have
different consequences for population dynamics and the evolution of individual
traits (Godfray 1987; Lomnicki 1988; Ives 1989). For instance, with a species
at a low density, in the invader state, it faces both intra-specific and
inter-specific competition, but obtains a lesser effect of reduced intra-
specific competition due to Allee effects compared with the standard
investigations of models of competing species. As comparisons between intra-
specific and inter-specific density dependences are critical to competitive
coexistence, ignoring the potential that Allee effects greatly modify intra-
specific competition is a critical oversight. For instance, Chesson and Ellner
(1989) has pointed out the potential for major effects such as stochastic
models may have interior invariant regions if low-density growth rates are
depressed by Allee effects. Hopf _et al_ (1993) also point out potential major
implications for community structure of Allee effects uncovered by introducing
them into a competition model.

Recently, Kang and Yakubu (2011) and Kang (2013) have explored the population
dynamics of two competing species when both species experience _scramble
competition_ (i.e., Ricker-type models) and _Allee effects_ induced by
predation saturations. The positive density dependence from weak Allee effects
can decrease the fitness of resident species so that the other species is able
to invade at its low densities. Thus _weak Allee effects_ can promote the
permanence of two competing species at their low densities (Kang and Yakubu
2011). In the presence of strong Allee effects, scramble inter-specific
competition can bring the current hight population density to a lower
population density but being above the Allee threshold. Thus _strong Allee
effects_ can save species from potential _essential extinction_ (Schreiber
2003) by bringing in scramble inter-specific competition (Kang 2013),
therefore promote the existence of two competing species at their high
densities under certain conditions. Understanding how different forms of
competition in the presence of Allee effects affect population dynamics when
species suffer from Allee effects, can advance our understanding of the
extinction and establishment of species in ecological communities, with
implications for conservation programs (Zhou _et al_ 2004; Courchamp _et al._
2009; Kang and Yakubu 2011; Kang 2013; Kang _et al_ 2014 a&b).

A recent work by Livadiotis and Elaydi (2012) considered a general framework
of population models subject to strong Allee effects. We adopt and modify
their definitions on mathematical formulation of strong Allee effects in
single species models. Our main purpose is to investigate the dynamical
outcomes in two species contest competition models when both species have
_Allee effects_. The contest competition models in this article refer as to a
generalized Beverton-Holt competition model while scramble competition models
refer as to species that have one-hump growth functions (e.g., Ricker’s map)
in its single state. More precisely, we would like to explore possible answers
to the following ecological questions:

  1. 1.

What are the dynamical outcomes of a two species competition model when both
species experience both strong Allee effects and contest competition?

  2. 2.

How do the dynamics of models with a strong Allee effect and contest
competition compare to those of models with a strong Allen effect and scramble
competition?

  3. 3.

What are the generic dynamical features of population models subject to strong
Allee effects and different forms of competition at low and high densities?

The paper is organized as follows: In Section 2, we define how we will
mathematically formulate Allee effects, contest and scramble competition for
single species population models and derive a generalized Beverton-Holt
competition model that each species is subject to strong Allee effects. In
addition, we provide the important results of single species models with
strong Allee effects and contest competition that can be applied to two
species competition models. In Section 3, we apply the theory of monotone
dynamics to obtain sufficient conditions when our two species competition
model has equilibrium dynamics. This result can establish some generic
dynamical features of contest competition models subject to Allee effects. We
also give approximated basins of attractions of boundary attractors (i.e.,
initial conditions that lead to the extinction of one or both species).
Moreover, we explore conditions that lead to the coexistence of two competing
species. These results can help us understand the coexistence and the
extinction conditions for two competing species that are subject to contest
competition and Allee effects. In Section 4, we apply the theoretical results
derived in Section 3 to a specific two competing species model with strong
Allee effects and contest competition. Numerical simulations are performed to
obtain more insights into the generic features of two species models with
strong Allee effects and contest competition. The results obtained from
Section 3 &4 can partially answer three questions proposed in the
introduction. In Section 5, we discuss the dynamic outcomes of two competing
species population models with different types of competition (i.e., contest
and scramble competition) where each species has strong Allee effects. In
addition, we summarize our results and mention some potential future studies.
In Appendix, we provide detailed proofs of our theoretical results.

##  2 Model Derivations

In this article, we are interested in the population dynamics of a generalized
Beverton-Holt two species competition model subject to Allee effects. More
specifically, we would like to explore how the interactions of contest
competition and Allee effects affect the establishment and extinction of
species. By understanding population dynamics of such models, we are able to
obtain insights on the impact of different forms of competition combined with
Allee effects on the persistence and extinction of species by comparing the
dynamics to the case when two competing species both suffer from scramble
competition and Allee effects.

###  2.1 Single species population models

First, we would like to introduce the terms of contest competition, scramble
competition and Allee effects for one species population models rigorously.
Let $u_{t}$ be a species population density in season $t$, then its population
density in season $t+1$ can be represented as the following equation:

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
H(u_{t})=u_{t}h(u_{t})$ (1)

where $H(u)\geq 0,u\in\mathbb{R}_{+}$ is the growth function of species $u$
and $h(u)$ is the per capital growth rate that is subject to the condition
$\lim_{u\rightarrow}h(u)=0$ which guarantees the boundedness of population
(Kang and Chesson 2010; Kang 2013). We say that species $u$ has _contest
intra-specific competition_ if $\lim_{u\rightarrow\infty}H(u)=c>0$ which
implies that a constant number of survivors with enough resource gain against
an initial density of competitors. We say that species $u$ has _scramble
intra-specific competition_ if $\lim_{u\rightarrow\infty}H(u)=0$ which implies
that an increased number of survivors with decreased resource intake and an
increased number of competitors. _Contest_ and _scramble_ intra-specific
competition are two extreme forms of competition that can result in very
different consequences for population dynamics. For example, an important
ecological example of contest intra-specific competition is the Beverton-Holt
model (see Figure 1) whose $H$ takes the form of $H(u)=\frac{ru}{1+au}$ while
a well-known ecological example of scramble intra-specific competition is the
Ricker’s model (see Figure 1) whose $H$ takes the form of $H(u)=ue^{r-au}$.
The Beverton-Holt model has relatively simple population dynamics where
population goes to 0 if $r<1$ and population goes to $r-1$ if $r>1$. While the
Ricker’s model has very complicate population dynamics (e.g., periodic orbits,
chaos) depending on the values of $r$. See Figure 1 for examples of single
species models with contest intra-specific competition versus scramble intra-
specific competition.

Figure 1: Examples of single species models with contest competition (a)
versus scramble competition (b).

For convenience, we define the following four conditions with respect to Model
(1):

  * A1: There exists some $\epsilon>0$ such that $\frac{\partial h}{\partial u}>0$ for $u\in(0,\epsilon)$.

  * A2: $h(u)\geq 0$ and $\lim_{u\rightarrow\infty}h(u)=a<1$.

  * A3: There exists unique two numbers $0<A<K$, such that $h(A)=h(K)=1$ and $h^{\prime}(A)>1$. In addition, $h(0)<1$.

  * A4: There exists a unique fixed point $K$, i.e., $h(K)=1$. In addition, $h(0)>1$.

Condition A1 indicates that the per capita growth rate has a positive
correlation with population density for small populations. Condition A2
indicates that the population is positive invariant (i.e., all future
population is nonnegative for any nonegative initial population) and bounded.
Condition A3 indicates that the population has two steady states $A$ and $K$
where $A$, called _Allee threshold_ , is always unstable and $K$, called
_carry capacity_ , can be locally stable under certain conditions. Condition
A4 indicates that species $u$ is persistent and may be locally stable at its
carry capacity $K$. Now we can define the following terms:

  * 1.

Species $u$ has _Allee effects_ if its population model (1) satisfies
Condition A1, A2.

  * 2.

Species $u$ has _strong Allee effects_ if its population model (1) satisfies
Condition A1, A2, A3 (see blue curves in Figure 2-3).

  * 3.

Species $u$ has _weak Allee effects_ if its population model (1) satisfies
Condition A1, A2, A4 (see black curves in Figure 2-3).

Our definition of _Allee effects_ is more general than Livadiotis and Elaydi
(2012) since they only define _strong Allee effects_ as the following three
conditions: i) $h^{\prime}(u)>0$ for $u\in(0,\epsilon)$ for some $\epsilon>0$;
ii) $h(0)<1$ and iii) There exists a unique $K>0$, such that
$h(K)=1,h^{\prime}(K)<0$. These two definitions can be considered the same.
When species $u$ is subject to _strong Allee effects_ , i.e., its population
model (1) satisfies Condition A1, A2, A3, its population dynamics can be
summarized as follows: When initial population density is below $A$, the
population goes extinct; When initial population density is above $A$, the
population may be able to sustain; when $h(0)$ is large enough, then (1) may
have _the essential extinction_ (i.e., for any positive initial condition, the
population converges to 0 with probability 1) due to strong Allee effects
(Schreiber 2003). See Figure 2-3 for an illustration of strong Allee effects
(blue curves in both figures) versus weak Allee effects (black curves in both
figures).

Figure 2: An illustration of strong Allee effects versus weak Allee effects of
the growth function $H(u)=uh(u)$: The red curve is $H=u$ or $h=1$; the green
curve is the case when (1) has no Allee effects; the blue curve is the case
when (1) has strong Allee effects and the black curve is the case when (1) has
weak Allee effects. Figure 3: An illustration of strong Allee effects v.s.
weak Allee effects of the per capita growth rate
$h(u)=\frac{H(u)}{u}$:corresponding to its growth function $H(u)$ in Figure 2:
The red curve is $H=u$ or $h=1$; the green curve is the case when (1) has no
Allee effects; the blue curve is the case when (1) has strong Allee effects
and the black curve is the case when (1) has weak Allee effects.

Outcomes of varied patterns of intra-specific competition (e.g., _contest_
competitions or _scramble_ competitions) have different consequences for
population dynamics (1) when species experiences _Allee effects_. A well-known
contest competition model introduced by Thomson (1993) for fisheries is called
the Sigmoid Beverton-Holt model that can be described as

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
H(u_{t})=u_{t}h(u_{t})=u_{t}\frac{ru^{\delta-1}_{t}}{1+u_{t}^{\delta}}$ (2)

where $r,\delta>0$. Model (2) is a depensatory generalization of the Beverton-
Holt stock-recruitment relationship used to develop a set of constraints
designed to safeguard against overfishing (Thomson 1993; Myers _et al_ 1995;
Myers and Mertz 1998; Stoner and Ray-Culp 2000; Myers 2001; Gascoigne and
Lipcius 2004; Harry _et al_ 2012). One important feature of (2) is that it
exhibits the _Allee effects_ if $\delta>1$. The dynamical properties of (2)
have been proposed by Harry _et al_ (2012). Here we summarize the important
dynamics of (2) that will be used later for our two-species model as the
following proposition:

######  Proposition 2.1 (Dynamical of (2)).

Let $r,\delta>0$ and (2) has an initial condition $x_{0}>0$. If $\delta>1$, we
define $r_{crit}=\delta(\delta-1)^{\frac{1}{\delta}-1}$. Then the following
statements are true:

  1. 1.

If $\delta\in(0,1)$, then Model (2) has two non-negative equilibria: an
unstable equilibrium 0 and a stable positive equilibrium $K$.

  2. 2.

If $\delta>1$ and $r<r_{crit}$, then the only equilibrium of Model (2) is 0
and it is locally stable.

  3. 3.

If $\delta>1$ and $r>r_{crit}$, then Model (2) has three non-negative
equilibria: $0,\,A$ and $K$ such that $0<A<\frac{r(\delta-1)}{\delta}<K$. The
0 equilibrium is stable with the basin of attraction $[0,A)$; $A$ is unstable
(repellor), while $K$ is stable with the basin of attraction $(A,\infty)$.

Notes: Harry _et al_ (2012) gave similar results of Proposition 2.1 as their
Proposition 1but without rigorous proof. Here, we provide a detailed proof in
the Appendix. Proposition 2.1 implies that Model (2) has _Allee effects_ for
any $r,\delta>0$. Precisely, Model (2) has _weak Allee effects_ if
$\delta\in(0,1)$ while Model (2) has _strong Allee effects_ if $\delta>1$ and
$r>r_{crit}$. In addition, we would like to point out that when $\delta=2$,
Elaydi and Sacker (2009) have extended (2) to the following generalized form:

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
u_{t}\frac{du_{t}+e}{u_{t}^{2}+bu_{t}+c}$ (3)

where $b$ is called the _shock recovery_ parameter and $c$ can be considered
as a carrying capacity. The dynamics of (3) is similar to (2) when $\delta=2$.
Define a single species population model as follows

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
H(u_{t})=u_{t}h(u_{t})=u_{t}\frac{ru^{\delta-1}_{t}}{a+u_{t}^{\delta}}$ (4)

where $r$ is the maximum intrinsic growth rate and $a$ can be considered as a
carrying capacity as the parameter $c$ in Model (3). Then the dynamics of
Model (4) can be obtained as the following corollary by directly applying the
results from Proposition 2.1 if we define
$r_{crit}=a^{\delta}\delta(\delta-1)^{\frac{1}{\delta}-1}$:

######  Corollary 2.1 (Dynamical of (4)).

Let $r,a,\delta>0$ and (4) has an initial condition $x_{0}>0$. If $\delta>1$,
we define $r_{crit}^{a}=a^{1/\delta}\delta(\delta-1)^{\frac{1}{\delta}-1}.$
Then the results of Proposition 2.1 can directly apply to Model (4). In
particular, if $a,\delta>1$ and $r>r_{crit}^{a}$, then Model (4) has two
interior equilibria $A^{a}$ and $K^{a}$ such that $0<A<A^{a}<K^{a}<K$ where
$A,K$ are defined in Proposition 2.1.

The scramble intra-specific competition models subject to _Allee effects_ can
take a form of the Ricker’s model, for example,

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
u_{t}h(u_{t})=u_{t}e^{r(1-u_{t})-\frac{m}{1+bu_{t}}}$ (5)

where $r$ represent the intrinsic growth rate (also the scramble intra-
specific competition coefficient); $m$ represents predation intensities and
$b$ represents the product of the proportional to the handling time and the
carrying capacity. For Model (5), we have

$h(u)=e^{r(1-u)-\frac{m}{1+bu}}\mbox{ and
}h^{\prime}(u)=\frac{[bm-r(1+bu)^{2}]e^{r(1-u)-\frac{m}{1+bu}}}{(1+bu)^{2}}.$

Thus, according to the definitions above, Model (5) has _Allee effects_ if
$mb>0$; Model (5) has _weak Allee effects_ if $bm>r>m$; and Model (5) has
_strong Allee effects_ if $r<m<\frac{r(1+b)^{2}}{4b},b>1$. In addition, Model
(5) has more complicate dynamics (e.g., chaos, essential extinctions) than the
contest competition model subject to Allee effects (2). The detailed dynamics
of (5) has been studied by Schreiber (2003).

Another example of scramble competition with Allee effects is a generalized
Maynard Smith-Slatkin model that can be described as follows:

$\displaystyle u_{t+1}$ $\displaystyle=$ $\displaystyle
H(u_{t})=u_{t}h(u_{t})=u_{t}\frac{ru_{t}^{\delta-1}}{1+u_{t}^{\delta}+bu_{t}^{d}}$
(6)

where all parameters are positive; $r,\delta$ have the same biological
meanings as in Model (2) and $b,d$ have the same biological meaning as in
Model (3). Model (6) can exhibit similar complicated dynamics as Model (5) due
to the generic feature of _scramble competition_. Here we summarize the basic
dynamics of (6) that will be used in the later sections as the following
theorem:

######  Theorem 2.1 (Dynamics of Model (6)).

If $\delta>1$, we define $r_{crit}$ as follows

$r_{crit}=\left(\frac{\delta}{b(d-\delta)}\right)^{(1-\delta)/d}\left[\frac{d}{d-\delta}+\left(\frac{\delta}{b(d-\delta)}\right)^{\delta/d}\right].$

Depending on the values of $r,b,\delta,d$, the basic dynamics of (6) can be
classified as the following cases:

  1. 1.

If $\delta>d$, then the trajectory of Model (6) with initial condition
$x_{0}>0$ converges to one of its equilibria. More specifically, we have the
following two cases:

    1. (a)

If $\delta<1$, then Model (6) has two non-negative equilibria 0 and $K$ where
0 is unstable and $K$ is globally stable.

    2. (b)

If $\delta>1$, then $u=0$ is unstable and Model (6) can have 0 or two or even
four positive equilibria. Moreover, Model (6) is globally stable at 0 when it
has no positive interior equilibrium.

  2. 2.

If $\delta=d$, then Model (6) is reduced to Model (2) where its dynamics has
been stated in Proposition 2.1.

  3. 3.

If $\delta<d$, then $u=0$ is always locally asymptotically stable. In
addition,

    1. (a)

If $\delta<1$, then Model (6) has two non-negative equilibria 0 and $K$ where
0 is unstable.

    2. (b)

If $\delta>1$ and $r>r_{crit}$, then Model (6) has at least two distinct
positive roots $A,K$ such that $0<A<u_{c}<K$.

  4. 4.

If both $\delta$ and $d$ are positive integers, then Model (6) has at most two
distinct positive interior equilibria.

Notes: Theorem 2.1 implies that Model (6) can exhibit _strong Allee effects_
if $\delta>\max\\{1,d\\}$ or $1<\delta<d,r>r_{crit}.$ Model (2) can exhibit
_strong Allee effects_ only if $\delta>1$ and $r>r_{crit}.$ Even though Part 1
(b) and Part 3 (b) of Theorem 2.1 do not state that Model (6) has exactly two
interior equilibria, numerical simulations suggest that Model (6) has at most
two interior equilibria. In addition, if $1<\delta<d$ and $r>r_{crit}$, then
Model (6) is a unimodal map that can have very complicated dynamics as the
Ricker’s map including _the essential extinction_ due to _strong Allee
effects_. Numerical simulations suggest that increasing values of $r,b,\delta$
can destabilize the system while increasing value of $d$ can stabilize the
system (see Figure 5-4 as examples).

Figure 4: Periodic time series of Model (6) when $\delta=2;d=5;b=0.1$ and
$r=7.5$. Figure 5: Chaotic time series of Model (6) when $\delta=2;d=5;b=0.1$
and $r=25$.

###  2.2 A generalized Beverton-Holt competition model with Allee effects

A generalized Beverton-Holt two species competition model with Allee effects
from single species models (2) and (6) can be represented by the following two
equations:

$\displaystyle x_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{1}x_{t}^{\delta_{1}}}{1+x_{t}^{\delta_{1}}+b_{1}y_{t}^{\delta_{3}}}$
(7) $\displaystyle y_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{2}y_{t}^{\delta_{2}}}{1+y_{t}^{\delta_{2}}+b_{2}x_{t}^{\delta_{4}}}$
(8)

where $r_{i},i=1,2$ describes the maximum intrinsic growth rate of each
species; $b_{i},i=1,2$ describes the relative inter-specific competition
coefficient to its intra-specific competition; $\delta_{i},i=1,2$ describes
the nonlinearity of the intra-specific competition, while $\delta_{i},i=3,4$
describes the nonlinearity of the inter-specific competition. Define Condition
H1:

  * H1: $\delta_{i}>1,\,r_{i}>r^{i}_{crit},\,i=1,2$ where $r^{i}_{crit}=\delta_{i}(\delta_{i}-1)^{\frac{1}{\delta_{i}}-1},i=1,2$.

According to Proposition 2.1, each species of Model (7)-(8) suffers both
contest intra-specific competition and strong Allee effects if Condition H1 is
satisfied. Due to our interests, we study the population dynamics of (7)-(8)
when Condition H1 is satisfied for most cases.

##  3 Dynamical properties of a generalized Beverton-Holt competition model
with Allee effects

In this section, we explore the basic dynamical property of Model (7)-(8) in
terms of its monotonicity, the extinction and coexistence of two competing
species. If Condition H1 is satisfied, Model (7)-(8) always has the following
five boundary equilibria:

$E_{0}=(0,0),\,E_{x_{a}0}=(A_{1},0),\,E_{x_{k}0}=(K_{1},0),\,E_{0y_{a}}=(0,A_{2})\mbox{
and }E_{0y_{k}}=(0,K_{2})$

where $A_{i},i=1,2$ is the _Allee threshold_ of species $i$ and $K_{i},i=1,2$
is the carrying capacity of species $i$, thus, $0<A_{i}<K_{i},i=1,2$. Simple
calculations indicate that $E_{0},E_{x_{k}0}$ and $E_{0y_{k}}$ are locally
asymptotically stable, thus these three boundary equilibria are also
attractors for Model (7)-(8). This gives the following lemma:

######  Lemma 3.1 (Basic properties).

With respect the map defined by Model (7)-(8) the positive cone
$\mathbb{R}^{2}_{+}$ is forward invariant and the rectangular
$[0,r_{1}]\times[0,r_{2}]$ is absorbing. If Condition H1 is satisfied, then it
has five boundary equilibria $E_{0},E_{x_{a}0},E_{x_{k}0},E_{0y_{a}}$ and
$E_{0y_{k}}$ where $E_{0},E_{x_{k}0}$ and $E_{0y_{k}}$ are locally
asymptotically stable and $E_{x_{a}0},E_{y_{a}0}$ are unstable. In addition,
let $X_{1}=\\{(x,0)\in\mathbb{R}^{2}_{+}\\}$ and
$X_{2}=\\{(0,y)\in\mathbb{R}^{2}_{+}\\}$, then both $X_{i},i=1,2$ are positive
invariant sets for Model (7)-(8).

Notes: It is easy to check that the set $[0,r_{1}]\times[0,r_{2}]$ is forward
invariant and all orbits in the positive cone $\mathbb{R}^{2}_{+}$ lie in this
set after one time step. The stability of the boundary can be obtained by
straight forward calculations, i.e., calculating the eigenvalues of the
Jacobian matrix evaluated at the boundary equilibria. Thus, we omit the
details here. Also, we would like to point out that $E_{x_{a}0}$ and
$E_{0y_{a}}$ are saddle nodes: $E_{x_{a}0}$ is a saddle node with the unstable
manifold lying on the x-axis and the stable manifold tangent to the vertical
eigenvector emanating from $E_{x_{a}0}$; similarly, $E_{0y_{a}}$ is a saddle
node with the unstable manifold lying on the y-axis and the stable manifold
tangent to the horizontal eigenvector emanating from $E_{0y_{a}}$.

###  3.1 Competitive systems

Monotonicity properties have been used successfully to study the dynamical
behavior of solutions to differential equations (e.g., see [Smith 1995] for an
overview) and difference equations (e.g., Selgrade and Ziehe 1987; Dancer and
Hess 1991; Hess and Lazer 1991; Smith 1998; Wang and Jiang 2001; Clark _et al_
2003). A competitive map is one type of monotone systems.

To continue our study, we first define a partial order on $\mathbb{R}^{2}$ so
that the positive cone in this new partial order is the fourth quadrant.
Specifically, for $u=(u_{1},u_{2}),v=(v_{1},v_{2})\in\mathbb{R}^{2}_{+}$, we
say that $u\leq_{K}v$ if $u_{1}\leq v_{1}$ and $v_{2}\leq u_{2}$. Two points
$u,v\in\mathbb{R}^{2}_{+}$ are said to be _related_ (or $K$-ordered) if
$u\leq_{K}v$ or $v\leq_{K}u$. A strict inequality between points can be
defined as $u<_{K}v$ if $u<_{K}v$ and $u\neq v$. A stronger inequality can be
defined as $u<<_{K}v$ if $u_{1}<v_{1}$ and $v_{2}<u_{2}$. A map
$T:Int\,\mathbb{R}^{2}_{+}\rightarrow Int\,\mathbb{R}^{2}_{+}$ is
_competitive_ (or $K$-order preserving) if $u\leq_{K}v$ implies that
$T(u)\leq_{K}T(v)$ for all $u,v\in Int\,\mathbb{R}^{2}_{+}$. A map
$T:Int\,\mathbb{R}^{2}_{+}\rightarrow Int\,\mathbb{R}^{2}_{+}$ is _strictly
competitive_ (_strongly competitive_) if $u<_{K}v$ implies that
$T(u)<_{K}T(v)$ ($T(u)<<_{K}T(v)$) for all $u,v\in Int\,\mathbb{R}^{2}_{+}$.
Clearly, being related is an invariant under iteration of a strongly
competitive map. Let $J$ be the Jacobian matrix of a map $T$, then $J$ is
$K$-positive ($K$-strongly positive) if its diagonal entries are nonnegative
(positive) and its off-diagonal entries are nonpositive (negative).

######  Theorem 3.1 (Competitive systems case 1).

If $\delta_{1}\delta_{2}\geq\delta_{3}\delta_{4}$, then Model (7)-(8) is
strongly competitive and all orbits in the positive cone $\mathbb{R}^{2}_{+}$
converge to an equilibrium.

Notes: The proof of Theorem 3.1 use the monotone properties of Model (7)-(8),
which is provided in the Appendix. This theorem indicates that Model (7)-(8)
has equilibrium dynamics (e.g., no nontrivial periodic orbits) whenever
$\delta_{1}\delta_{2}\geq\delta_{3}\delta_{4}$. Recall that $\delta_{i},i=1,2$
measures the intensity of the intra-specific competition of species, and
$\delta_{i},i=3,4$ represents the inter-specific competition. Then we can
define $\delta_{1}\delta_{2},\delta_{3}\delta_{4}$ as intensities of the
inter- and intra-specific competition, respectively. Then the result is that
the dynamics equilibration when the intensity of intra-specific competition
intensity exceeds that of inter-specific competition.

From the proof of Theorem 3.1, we can see that Model (7)-(8) is _strictly
competitive_ in $\mathbb{R}^{2}_{+}$ and is _strongly competitive_ in
$Int\,\mathbb{R}^{2}_{+}$. For the case that
$\delta_{1}\delta_{2}<\delta_{3}\delta_{4}$, we have the following theorem:

######  Theorem 3.2 (Competitive systems case 2).

Assume that $\delta_{1}\delta_{2}<\delta_{3}\delta_{4}$. If

$r_{1}<\left(\frac{\delta_{1}\delta_{2}}{b_{2}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{4}}\mbox{
or
}r_{2}<\left(\frac{\delta_{1}\delta_{2}}{b_{1}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{3}},$

then every orbit of Model (7)-(8) converges to an equilibrium.

Notes: The proof of Theorem 3.2 use the monotone properties of Model (7)-(8)
as well as the properties of non-invertible maps. The detailed of proof has
been given in the Appendix.In the case that
$\delta_{1}\delta_{2}<\delta_{3}\delta_{4}$, Theorem 3.2 indicates that the
dynamics of (7)-(8) can still have equilibrium dynamics as the case when
$\delta_{1}\delta_{2}\geq\delta_{3}\delta_{4}$ (Theorem 3.1) if the maximum
intrinsic growth rate of one species is not too large. However, when this
additional condition is not satisfied, Model (7)-(8) can have non-equilibrium
attractors as suggested by results of Terescák (1996) and Smith (1998). For
example, if we let
$\delta_{1}=\delta_{2}=\delta,\delta_{3}=\delta_{4}=d,r_{1}=r_{2},b_{1}=b_{2}$,
then the dynamics of (7)-(8) on the invariant manifold
$\Omega_{x=y}=\\{(x,y)\in\mathbb{R}^{2}_{+}:x=y\\}$ can be reduced to Model
(6) which can have complicated dynamics. See Figure 4-5 as examples for
different dynamics on $\Omega_{x=y}$ when $\delta=2;d=5;b=0.1$ and $r=7.5$
(Figure 4) and $r=25$ (Figure 5).

###  3.2 Basins of attractions of extinction equilibria

Assume that Condition H1 is satisfied for Model (7)-(8). We can define the
following sets:

$O_{0}=[0,A_{1}]\times[0,A_{2}],\,\,O_{ex}=\\{(x,y)\in\mathbb{R}^{2}_{+}:x<A_{1}\\}\mbox{
and }O_{ey}=\\{(x,y)\in\mathbb{R}^{2}_{+}:y<A_{2}\\}$

and Condition H2-H3 as follows:

  * H2: $\delta_{1}>1$ and $r_{1}>r^{a_{1}}_{crit}$ where $r^{a_{1}}_{crit}=(a_{1})^{1/\delta_{1}}\delta_{1}(\delta_{1}-1)^{\frac{1}{\delta_{1}}-1}$ and $a_{1}=1+b_{1}(A_{2})^{\delta_{3}}$.

  * H3: $\delta_{2}>1$ and $r_{2}>r^{a_{2}}_{crit}$ where $r^{a_{2}}_{crit}=(a_{2})^{1/\delta_{2}}\delta_{2}(\delta_{2}-1)^{\frac{1}{\delta_{2}}-1}$ and $a_{2}=1+b_{2}(A_{1})^{\delta_{4}}$.

Condition H2 or H3 indicates that species suffers from strong Allee effects.
Assume that Model (4) satisfies Condition H2, then according to Corollary 2.1,
it has two interior equilibria $A_{1}^{a_{1}}$ and $K_{1}^{a_{1}}$ where
$A_{1}^{a_{1}}$ is the Allee threshold and $K_{1}^{a_{1}}$ is the carrying
capacity. Similarly, if Model (4) satisfies Condition H3, then it has two
interior equilibria $A_{1}^{a_{2}}$ and $K_{1}^{a_{2}}$ where $A_{2}^{a_{2}}$
is the Allee threshold and $K_{2}^{a_{2}}$ is the carrying capacity. Let

$O_{x}=\\{(x,y)\in\mathbb{R}^{2}_{+}:x>A_{1}^{a_{1}},\,y<A_{2}\\}\mbox{ and
}O_{y}=\\{(x,y)\in\mathbb{R}^{2}_{+}:x<A_{2},\,y>A_{2}^{a_{2}}\\},$

then we can show the following theorem:

######  Theorem 3.3 (Basins of attractions of boundary attractors I).

Assume that Model (7)-(8) satisfies Condition H1, then

  * 1.

For any initial value in $O_{ex}$, we have $\lim_{t\rightarrow\infty}x_{t}=0.$

  * 2.

For any initial value in $O_{ey}$, we have $\lim_{t\rightarrow\infty}y_{t}=0.$

  * 3.

For any initial value in $O_{0}$, we have
$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0}.$

If Model (4) satisfies Condition H2, then for any initial value in $O_{x}$, we
have

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{x_{k_{1}}0}.$

If Model (4) satisfies Condition H3, then for any initial value in $O_{y}$, we
have

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0y_{k_{2}}}.$

Notes: Theorem 3.3 (see the proof in the Appendix) provides an approximation
of the boundary attractor’s basins of attractions for Model (7)-(8) when each
species suffers from contest competition and strong Allee effects. In the
absence of other species, each species’ extinction region is $[0,A_{i}]$ while
in the presence of other species, the extinction region becomes larger due to
the inter-specific competition, thus, competition cannot save species from
extinction at their low densities. In order to maintain species x’s
population, its initial condition should be large enough and the other species
population should be small enough, e.g., the initial condition should be
larger than $A_{1}^{a_{1}}>A_{1}$ while species y’s initial condition should
be less than $A_{2}$. In addition, Theorem 3.3 implies that the basins of
attractions of the boundary attractors $E_{x_{k_{1}}0}$ and $E_{0y_{k_{2}}}$
are unbounded (see Figure 12, 14 and 16).

Define

$\begin{array}[]{lcl}O_{ex}^{l}&=&\bigcup_{k>0}\\{(x,kx)\in
Int\,\mathbb{R}^{2}_{+}:x>\left(\frac{r_{1}-A_{1}}{b_{1}k^{\delta_{3}}}\right)^{\frac{1}{\delta_{3}-\delta_{1}}}\\}\\\
O_{ey}^{l}&=&\bigcup_{k>0}\\{(ky,y)\in
Int\,\mathbb{R}^{2}_{+}:y>\left(\frac{r_{2}-A_{2}}{b_{2}k^{\delta_{4}}}\right)^{\frac{1}{\delta_{4}-\delta_{2}}}\\}\end{array}$

and let $O_{0}^{l}=O_{ex}^{l}\cap O_{ey}^{l}.$ Then we have the following
theorem:

######  Theorem 3.4 (Basins of attractions of boundary attractors II).

Assume that Model (7)-(8) satisfies Condition H1.

  * 1.

If $\delta_{3}>\delta_{1}$, then for any initial value in $O_{ex}\cup
O_{ex}^{l}$, we have $\lim_{t\rightarrow\infty}x_{t}=0.$

  * 2.

If $\delta_{4}>\delta_{2}$, then for any initial value in $O_{ey}\cup
O_{ey}^{l}$, we have $\lim_{t\rightarrow\infty}y_{t}=0.$

  * 3.

If $\delta_{3}>\delta_{1}$ and $\delta_{4}>\delta_{2}$, then for any initial
value in $O_{0}\cup O_{0}^{l}$, we have
$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0}.$

Notes: The detailed proof of Theorem 3.4 is provided in the Appendix. This
theorem indicates that if a species’ relative competition degree is less than
1 (i.e., $\delta_{3}>\delta_{1}$ or $\delta_{4}>\delta_{2}$), then its
extinction region (i.e., the initial conditions that lead to the extinction of
two species) consists of two distinct components, of which one is stated in
Theorem 3.3 and the other one is the area that when its competitor’s
population is large enough. This phenomenon is is a typical property of non-
invertible maps which is caused by the large degree of the inter-specific
competition, i.e., $\delta_{3}>\delta_{1}$ or $\delta_{4}>\delta_{2}$. Theorem
3.4 also indicates that the basins of attractions of $E_{0}$ may be unbounded.
In fact, the basins of attractions of $E_{0}$ is a connected and unbounded
region when there is no interior equilibrium (see Figure 11 and 12). While the
basins of attractions of $E_{0}$ consists of two parts when there is an
interior equilibrium: one part is a bounded region including the neighborhood
of $E_{0}$ while the other part is a connected unbounded region including very
large initial values of both species (see Figure 14 and 16).

###  3.3 Persistent equilibria

Assume that Model (7)-(8) satisfies Condition H1, then according to Theorem
3.3, it always has three boundary attractors. This suggests that the
coexistence of two species is possible only if there exists an interior
equilibrium. We are going to prove the following theorem regarding the number
of interior equilibria.

######  Theorem 3.5 (Interior equilibria).

Assume that Model (7)-(8) satisfies Condition H1. Let

$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}\mbox{
and
}F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}},$

then we have the following statements:

  * 1.

If $F_{1}(x_{c})<A_{2}$ or $F_{2}(y_{c})<A_{1}$, then Model (7)-(8) has no
interior equilibrium.

  * 2.

If $A_{2}<F_{1}(x_{c})<K_{2},\,K_{1}<F_{2}(y_{c})$ or
$A_{1}<F_{2}(y_{c})<K_{1},\,K_{2}<F_{1}(x_{c})$, then Model (7)-(8) has two
interior equilibria.

  * 3.

If $F_{1}(x_{c})>K_{2}$ and $F_{2}(y_{c})>K_{1}$, then Model (7)-(8) has four
interior equilibria.

where $x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$ and
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.

Notes: See Figure 6-10 regarding the generic nullclines of Model (7)-(8).
Theorem 3.5 (its proof stated in the Appendix) gives us two scenarios when
Model (7)-(8) has interior equilibria where coexistence occurs only if Model
(7)-(8) has four interior equilibria (see Figure 15 and Figure 16). Sufficient
conditions guaranteeing four interior equilibria is that $F_{1}(x_{c})>K_{2}$
and $F_{2}(y_{c})>K_{1}$, i.e.

$x_{c}<\frac{r_{1}x_{c}^{\delta_{1}}}{1+x_{c}^{\delta_{1}}+b_{1}K_{2}^{\delta_{3}}}\mbox{
and
}y_{c}<\frac{r_{2}y_{c}^{\delta_{2}}}{1+y_{c}^{\delta_{2}}+b_{2}K_{1}^{\delta_{4}}}$

which indicates that any initial condition in
$[x_{c},K_{1}]\times[y_{c},K_{2}]$ lead to a locally asymptotically stable
interior equilibrium according to the monotonicity of the system.
Biologically, this implies that both species are able to persist under proper
initial conditions if $F_{1}(x_{c})>K_{2}$ and $F_{2}(y_{c})>K_{1}$ holds. In
the next subsection, we explore the local stability of a symmetric system.

Figure 6: Schematic nullclines when Model (7)-(8) has no interior equilibrium
of Case 1, i.e., $F_{2}(y_{c})<A_{1}$: The red curve is
$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}$
where its maximum $F_{1}(x_{c})$ occurs at
$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$; While the blue curve is
$F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}$
where its maximum $F_{2}(y_{c})$ occurs at
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.  Figure 7: Schematic
nullclines when Model (7)-(8) has no interior equilibrium of Case 2, i.e.,
$F_{1}(x_{c})<A_{2}$: The red curve is
$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}$
where its maximum $F_{1}(x_{c})$ occurs at
$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$; While the blue curve is
$F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}$
where its maximum $F_{2}(y_{c})$ occurs at
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.  Figure 8: Schematic
nullclines when Model (7)-(8) has two interior equilibria of Case 1, i.e.,
$K_{2}<F_{1}(x_{c}),\,A_{1}<F_{2}(y_{c})<K_{1}$: The red curve is
$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}$
where its maximum $F_{1}(x_{c})$ occurs at
$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$; While the blue curve is
$F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}$
where its maximum $F_{2}(y_{c})$ occurs at
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.  Figure 9: Schematic
nullclines when Model (7)-(8) has two interior equilibria of Case 2, i.e.,
$A_{2}<F_{1}(x_{c})<K_{2},\,K_{1}<F_{2}(y_{c})$: The red curve is
$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}$
where its maximum $F_{1}(x_{c})$ occurs at
$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$; While the blue curve is
$F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}$
where its maximum $F_{2}(y_{c})$ occurs at
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.  Figure 10: Schematic
nullclines when Model (7)-(8) has four interior equilibria, i.e.,
$F_{1}(x_{c})>K_{2}$ and $F_{2}(y_{c})>K_{1}$: The red curve is
$F_{1}(x)=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}$
where its maximum $F_{1}(x_{c})$ occurs at
$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}$; While the blue curve is
$F_{2}(y)=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}$
where its maximum $F_{2}(y_{c})$ occurs at
$y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}$.

###  3.4 A symmetric competition model with Allee effects

In order to obtain more insights on the coexistence of two competing species,
we focus on a symmetric case of (7)-(8) where
$r_{1}=r_{2}=r,b_{1}=b_{2}=b,\delta_{1}=\delta_{2}=\delta,\delta_{3}=\delta_{4}=d$
which gives the following model

$\displaystyle x_{t+1}$ $\displaystyle=$
$\displaystyle\frac{rx_{t}^{\delta}}{1+x_{t}^{\delta}+by_{t}^{d}}$ (9)
$\displaystyle y_{t+1}$ $\displaystyle=$
$\displaystyle\frac{ry_{t}^{\delta}}{1+y_{t}^{\delta}+bx_{t}^{d}}.$ (10)

Understanding the dynamics of (9)-(10) also can help us obtain a better
understanding of the dynamics of asymmetric cases due to the robust property
of our system (7)-(8).

It is easy to verify that (9)-(10) is reduced to the 1-D model described by
(6) if $x_{0}=y_{0}$, i.e., the manifold defined by
$\Omega_{y=x}=\\{(x,y)\in\mathbb{R}^{2}_{+}:x=y\\}$ is invariant. According to
Theorem 2.1, Model (9)-(10) has at least two symmetric equilibria $(A,A)$ and
$(K,K)$ such that $0<A<\left(\frac{r(d-\delta)}{\delta}\right)^{1/d}<K$ if

$\displaystyle d>\delta>1\mbox{ and
}r>r_{crit}=\left(\frac{\delta}{b(d-\delta)}\right)^{(1-\delta)/d}\left[\frac{d}{d-\delta}+\left(\frac{\delta}{b(d-\delta)}\right)^{\delta/d}\right].$
(11)

Now we have the following theorem regarding the stability of $(K,K)$.

######  Theorem 3.6 (Stability of the symmetric equilibrium).

Assume that Model (9)-(10) satisfies the inequalities (11). Then (9)-(10) has
at least two symmetric equilibria $(A,A)$ and $(K,K)$ such that
$0<A<\left(\frac{r(d-\delta)}{\delta}\right)^{1/d}<K$. Moreover, the symmetric
interior equilibrium $(x^{*},x^{*})$ is locally asymptotically stable if

$\displaystyle\Big{|}\frac{\delta+b(\delta-d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}\Big{|}<1\mbox{
and }\frac{\delta+b(\delta+d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}<1.$
(12)

While the symmetric interior equilibrium $(x,x)$ is unstable if

$\displaystyle\Big{|}\frac{\delta+b(\delta-d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}\Big{|}>1\mbox{
or }\frac{\delta+b(\delta+d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}>1.$
(13)

Notes: Theorem 3.6 indicates that it is impossible for the interior
equilibrium of the symmetric model (9)-(10) to have Neimak-Sacker bifurcation
from the invariant $\Omega_{x=y}$ since the eigenvalue that determines the
points moving towards or away from $\Omega_{x=y}$ is always positive (see the
detailed proof in the Appendix). This suggests that (9)-(10) may have
relatively simple dynamics.

##  4 Application to an example

In this section, we let $\delta_{1}=\delta_{2}=2$, then Model (7)-(8) can be
rewritten as follows:

$\displaystyle x_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{1}x_{t}^{2}}{1+x_{t}^{2}+b_{1}y_{t}^{\delta_{3}}}$ (14)
$\displaystyle y_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{2}y_{t}^{2}}{1+y_{t}^{2}+b_{2}x_{t}^{\delta_{4}}}.$
(15)

In the absence of species y, i.e., $y_{0}=0$, we have (14)-(15) is reduced to
a single species model (2) when $r=r_{1}$ and $\delta=2$, i.e.,

$\displaystyle x_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{1}x_{t}^{2}}{1+x_{t}^{2}}.$ (16)

Model (16) is a well-known fisheries model when fish is subject to contest
intra-specific competition and strong Allee effects. The dynamics of (16) is
very simple: 1. If $r_{1}<2$, then (16) converges to 0 for any $x_{0}>0$. 2.
If $r_{1}>2$, then (16) has two interior equilibria $A_{1}$ and $K_{1}$ where

$A_{1}=\frac{r_{1}-\sqrt{r_{1}^{2}-4}}{2}\mbox{ and
}K_{1}=\frac{r_{1}+\sqrt{r_{1}^{2}-4}}{2}$

such that (16) converges to 0 if $x_{0}<A_{1}$ while (16) converges to $K_{1}$
if $x_{0}>A_{1}$.

Similarly, in the absence of species x, i.e., $x_{0}=0$, (14)-(15) is reduced
to a single species model (2) for the case that $r=r_{1}$ and $\delta=2$,
i.e.,

$\displaystyle y_{t+1}$ $\displaystyle=$
$\displaystyle\frac{r_{2}y_{t}^{2}}{1+y_{t}^{2}}$ (17)

where it also has two interior equilibria $A_{2}$ and $K_{2}$ where

$A_{2}=\frac{r_{2}-\sqrt{r_{2}^{2}-4}}{2}\mbox{ and
}K_{2}=\frac{r_{2}+\sqrt{r_{2}^{2}-4}}{2}.$

Due to our interests, we focus on the case that $r_{i}>2,i=1,2$ for Model
(14)-(15). Then apply Theorem 3.1, Theorem 3.3, Theorem 3.2, Theorem 3.4 and
Theorem 3.5, we obtain the following corollary regarding the dynamics of
(14)-(15):

######  Corollary 4.1 (Dynamics of (14)-(15)).

Assume that $r_{i}>2$. Define

$r^{a_{1}}_{crit}=\sqrt{1+b_{1}\left(\frac{r_{2}-\sqrt{r_{2}^{2}-4}}{2}\right)^{1/\delta_{3}}}\mbox{
and
}r^{a_{2}}_{crit}=\sqrt{1+b_{2}\left(\frac{r_{1}-\sqrt{r_{1}^{2}-4}}{2}\right)^{1/\delta_{4}}}.$

Then the following statements are true:

  1. 1.

If $\delta_{3}\delta_{4}\leq 4$, then every orbit of Model (14)-(15) with any
initial condition in $\mathbb{R}^{2}_{+}$ converges to one of its equilibria.

  2. 2.

If $\delta_{3}\delta_{4}>4$ and

$r_{1}<\left(\frac{4}{b_{2}[\delta_{3}\delta_{4}-4]}\right)^{1/\delta_{4}}\mbox{
or
}r_{2}<\left(\frac{4}{b_{1}[\delta_{3}\delta_{4}-4]}\right)^{1/\delta_{3}},$

then every orbit of Model (14)-(15) with any initial condition in
$\mathbb{R}^{2}_{+}$ converges to one of its equilibria.

  3. 3.

Model (14)-(15) always has three boundary attractors
$E_{0},E_{x_{k_{1}}0},E_{0y_{k_{2}}}$ where

$E_{0}=(0,0),\,E_{x_{k_{1}}0}=(\frac{r_{1}+\sqrt{r_{1}^{2}-4}}{2},0)\mbox{ and
}E_{0y_{k_{2}}}=(0,\frac{r_{2}+\sqrt{r_{2}^{2}-4}}{2}).$

Moreover,

     * (a)

For any initial value in $O_{ex}$, we have $\lim_{t\rightarrow\infty}x_{t}=0.$
While if $\delta_{3}>2$, then for any initial value in $O_{ex}\cup
O_{ex}^{l}$, we have $\lim_{t\rightarrow\infty}x_{t}=0.$

     * (b)

For any initial value taken in $O_{ey}$, we have
$\lim_{t\rightarrow\infty}y_{t}=0.$ While if f $\delta_{4}>2$, then for any
initial value in $O_{ey}\cup O_{ey}^{l}$, we have
$\lim_{t\rightarrow\infty}y_{t}=0.$

     * (c)

For any initial value taken in $O_{0}$, we have
$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0}.$ While if $\delta_{3}>2$ and
$\delta_{4}>2$, then for any initial value in $O_{0}\cup O_{0}^{l}$, we have
$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0}.$

     * (d)

If $r_{1}>r^{a_{1}}_{crit}$, then for any initial value in $O_{x}$, we have

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{x_{k_{1}}0}.$

While if $r_{2}>r^{a_{2}}_{crit}$, then for any initial value in $O_{y}$, we
have

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0y_{k_{2}}}.$

  4. 4.

Model (14)-(15) has no interior equilibrium if

$\left(\frac{r_{1}^{2}/4-1}{b_{1}}\right)^{1/\delta_{3}}<\frac{r_{2}-\sqrt{r_{2}^{2}-4}}{2}\mbox{
or
}\left(\frac{r_{2}^{2}/4-1}{b_{2}}\right)^{1/\delta_{4}}<\frac{r_{1}-\sqrt{r_{1}^{2}-4}}{2}.$

Model (14)-(15) has two interior equilibria if

$\frac{r_{2}-\sqrt{r_{2}^{2}-4}}{2}<\left(\frac{r_{1}^{2}/4-1}{b_{1}}\right)^{1/\delta_{3}}<\frac{r_{2}+\sqrt{r_{2}^{2}-4}}{2}\mbox{
and
}\left(\frac{r_{2}^{2}/4-1}{b_{2}}\right)^{1/\delta_{4}}>\frac{r_{1}+\sqrt{r_{1}^{2}-4}}{2}$

or

$\frac{r_{1}-\sqrt{r_{1}^{2}-4}}{2}<\left(\frac{r_{2}^{2}/4-1}{b_{2}}\right)^{1/\delta_{4}}<\frac{r_{1}+\sqrt{r_{1}^{2}-4}}{2}\mbox{
and
}\left(\frac{r_{1}^{2}/4-1}{b_{1}}\right)^{1/\delta_{3}}>\frac{r_{2}+\sqrt{r_{2}^{2}-4}}{2}.$

Model (7)-(8) has four interior equilibria if

$\left(\frac{r_{1}^{2}/4-1}{b_{1}}\right)^{1/\delta_{3}}>\frac{r_{2}+\sqrt{r_{2}^{2}-4}}{2}\mbox{
and
}\left(\frac{r_{2}^{2}/4-1}{b_{2}}\right)^{1/\delta_{4}}>\frac{r_{1}+\sqrt{r_{1}^{2}-4}}{2}.$

The corollary above does not give us information on the dynamic patterns of
the system. Due to the complexity of the model, we use numerical simulations
to investigate their dynamical patterns. According to simulations, we
summarize its dynamical patterns as follows:

  * 1.

No interior equilibrium: When Model (14)-(15) has no interior equilibrium (see
Figure 11), it has exactly three boundary attractors, i.e., (0,0), $(x_{2},0)$
and $(0,y_{2})$, and no interior attractor. The basins of attractions of these
three attractors are unbounded (shown in Figure 12) where the extinction area
(i.e., basins of attractions of $E_{0}=(0,0)$) is an open and unbounded region
in red. This implies that the inter-specific competition make both two
competing species prone to extinction.

Figure 11: Nucllines of (14)-(15) when $r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=15$ and
$\delta_{3}=\delta_{4}=5$. Figure 12: Basins of attractions of (14)-(15) when
$r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=15$ and $\delta_{3}=\delta_{4}=5$: The black
region is the basins of attractions of $(0,y_{2})$; The cyan region is the
basins of attractions of $(x_{2},0)$; The blue region is the basins of
attractions of the interior attractor which is an interior equilibrium and the
red region is the basins of attractions of $(0,0)$.

  * 2.

Two interior equilibrium: When Model (14)-(15) has two interior equilibrium
(see Figure 13), it has exactly three boundary attractors, i.e., (0,0),
$(x_{2},0)$ and $(0,y_{2})$, and no interior attractor. The basins of
attractions of these three boundary attractors are unbounded (shown in Figure
14): The basins of attractions of $(x_{2},0)$ and $(0,y_{2})$ are connected
and unbounded while the basins of attractions of $E_{0}$ consists of a
bounded-connected and an unbounded-connected region.

Figure 13: Nucllines of (14)-(15) when $r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=.05$
and $\delta_{3}=\delta_{4}=5$. Figure 14: Basins of attractions of (14)-(15)
when $r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=.05$ and $\delta_{3}=\delta_{4}=5$: The
black region is the basins of attractions of $(0,y_{2})$; The cyan region is
the basins of attractions of $(x_{2},0)$; The blue region is the basins of
attractions of the interior attractor which is an interior equilibrium and the
red region is the basins of attractions of $(0,0)$.

  * 3.

Four interior equilibrium: This is the only case when Model (14)-(15) has an
interior attractor (i.e., the coexistence of two species). When Model
(14)-(15) has four interior equilibrium (see Figure 15), it has four
attractors of which three are boundary attractors: (0,0), $(x_{2},0)$ and
$(0,y_{2})$, and one is an interior attractor $(x^{*},y^{*})$ whose basins of
attractions is a bounded and connected region. The basins of attractions of
these four attractors are shown in Figure 16 where the features of basins of
attractions of boundary attractors are the same as the two interior equilibria
case.

Figure 15: Nucllines of (14)-(15) when $r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=.01$
and $\delta_{3}=\delta_{4}=5$. Figure 16: Basins of attractions of (14)-(15)
when $r_{1}=r_{2}=2.5$, $b_{1}=b_{2}=.01$ and $\delta_{3}=\delta_{4}=5$: The
black region is the basins of attractions of $(0,y_{2})$; The cyan region is
the basins of attractions of $(x_{2},0)$; The blue region is the basins of
attractions of the interior attractor which is an interior equilibrium and the
red region is the basins of attractions of $(0,0)$.

The observations above for Model (14)-(15) should be able to apply to a
general model (7)-(8) due to the robustness of the model. In addition, we
would like to point out that Model (7)-(8) can have coexistence of two species
only if it has four interior equilibria.

##  5 Discussion

Competition is an important ecological process which, in the short run, can
cause a reduction in the number of species living within an area, by
preventing very similar species from co-occurring; while in the long term, is
likely to increase species diversity, by acting as a force for specialization
and divergence. Competition can vary between two extreme forms: contest versus
scramble. Allee effect is an another important ecological process that leads
to a positive correlation of the per capita growth rate of a species and its
low population density. The interplay of Allee effects and intra-specific
competition can result in a critical threshold, called Allee threshold, below
which species goes to extinction; while above which species may persist. This
is referred to strong Allee effects. When there is no such threshold, it is
referred as to weak Allee effects. Many species in nature are subject to both
Allee effects and competition. The combinations of Allee effects and different
type of competition can generate varied dynamical outcomes, especially for the
coexistence of species.

Including Allee effects in competition models can have huge potential impacts
on diversity and community structure (Chesson and Ellner 1989; Hopf _et al_
1993). Allee effects impose a cost of rarity (Mechanism R). When communities
of organisms are subject to a cost of rarity imposed by Allee effects, they
should be comprised of distinct species with differences in resource use that
are greater than expected by chance (Hopf _et al_ 1993). Kang and Yakubu
(2011) and Kang (2013) studied the population dynamics of a two competing
species model where each species suffers from Allee effects and scramble
competition: Kang and Yakubu (2011) showed that weak Allee effects may promote
the permanence of two competing species at their low densities. The biological
explanation for this is that weak Allee effects decrease the fitness of
resident species so that the other species is able to invade at its low
densities. Kang (2013) showed that strong Allee effects may save two competing
species from essential extinctions at their high densities. The biological
explanation for this is that scramble competition can bring the current high
population density to a lower population density but is above the Allee
threshold in the next season with the consequence that both competing species
are able to persist. To extend the study by Kang and Yakubu (2011) and Kang
(2013), this article studies the population dynamics of a generalized
Beverton-Holt two species competition model where each species has strong
Allee effects to explore the establishment and the extinction of two competing
species subject to contest competition and Allee effects.

###  5.1 Two dimensional strong Allee effects of symmetric competition models

Adopting the definition of Allee effects for single species, we are able to
define Allee effects of a symmetric two species competition model that can be
represented as

$\displaystyle x_{t+1}$ $\displaystyle=$ $\displaystyle
F(x_{t},y_{t})=x_{t}f(x_{t},y_{t})$ (18) $\displaystyle y_{t+1}$
$\displaystyle=$ $\displaystyle F(y_{t},x_{t})=y_{t}f(y_{t},x_{t}).$ (19)

where $\frac{\partial f(x,y)}{\partial y}<0$. Then we say two dimensional
system (18)-(19) has _two dimensional strong Allee effects_ if both $f(x,0)$
and $f(x,x)$ satisfy Condition A1, A2, A3. Model (9)-(9) subject to
inequalities (11) provides an example of two competing species suffering from
_contest competition_ and _two dimensional strong Allee effects_. The
ecological examples when two competing species suffering from _scramble
competition_ and _two dimensional strong Allee effects_ are the following two
examples:

Model I: Scramble competition with strong Allee effects due to mating
limitations

$\displaystyle x_{t+1}$ $\displaystyle=$ $\displaystyle
x_{t}e^{r(1-x_{t})-ay_{t}}\frac{bx_{t}}{1+bx_{t}}$ (20) $\displaystyle
y_{t+1}$ $\displaystyle=$ $\displaystyle
y_{t}e^{r(1-y_{t})-ax_{t}}\frac{by_{t}}{1+by_{t}}$ (21)

where $r$ represent the intrinsic growth rates as well as intra-specific
competition coefficient; $b$ represent the product of an individual’s
searching efficiency and the carrying capacities and $a$ represents the inter-
specific competition coefficient. If

$\frac{e^{r-1}}{r+a}>\frac{1}{r+a}+\frac{1}{b},$

then Model I has _two dimensional strong Allee effects_.

Model II: Scramble competition with strong Allee effects due to predation
saturations

$\displaystyle x_{t+1}$ $\displaystyle=$ $\displaystyle
x_{t}e^{r(1-x_{t})-ay_{t}-\frac{m}{1+bx_{t}}}$ (22) $\displaystyle y_{t+1}$
$\displaystyle=$ $\displaystyle y_{t}e^{r(1-y_{t})-ax_{t}-\frac{m}{1+by_{t}}}$
(23)

where $r$ represents the intrinsic growth rates as well as intra-specific
competition coefficient; $m$ represents predation intensities; $b$ represent
the product of the proportional to the handling time and the carrying
capacities and $a$ represents the inter-specific competition coefficient. If

$r<m<\frac{(br+r+a)^{2}}{4b(a+r)}\mbox{ and }b>1+a/r,$

then Model I has _two dimensional strong Allee effects_.

Kang and Yakubu (2011) and Kang (2013) have studied the detailed dynamics of
Model I & II introduced above. By comparing our results to their study on a
two competing species model where each species is subject to scramble
competition and strong Allee effects (i.e., Model I & II), we are able to
obtain some generic dynamical features of a two-species competition model
subject to strong Allee effects and the different dynamical outcomes due to
different types of competition introduced in the models as follows:

Generic dynamic features of scramble v.s. contest competition models with
strong Allee effects:

  1. 1.

Both contest and scramble competition models with strong Allee effects can
have either tri-stability (i.e., three boundary attractors
$E_{0},E_{x_{k_{1}}0},E_{0y_{k_{1}}}$) or four attractors (i.e., three
boundary attractors plus an interior attractor) depending on certain
conditions.

  2. 2.

The coexistence of two competing species is possible only if the system has at
least four interior equilibria.

  3. 3.

Both models can exhibit _two dimensional strong Allee effects_ in the
symmetric case.

  4. 4.

The basins of attractions of $E_{0}$ are unbounded, which suggests that two
competing species are not able to coexist if their population densities are
too low or too high.

Different dynamical outcomes of scramble v.s. contest competition models with
strong Allee effects:

  1. 1.

Our generalized Beverton-Holt two species (representing contest competition)
model always has three boundary attractors
$E_{0},E_{x_{k_{1}}0},E_{0y_{k_{1}}}$. While two-species Ricker models
(representing scramble competition) subject to strong Allee effects can have
either one (i.e., extinction of two species) or three boundary attractors
where the extinction is caused by strong Allee effects (i.e., each species has
_essential extinction_ at its single state).

  2. 2.

Scramble competition models with strong Allee effects can promote the
coexistence of two competing species at their high densities under certain
condition. While this is not the case for contest competition models with
strong Allee effects.

  3. 3.

Scramble competition models can have complicated structures of attractors
(e.g., chaos, _essential extinctions_) while the contest competition model may
have relative simple dynamics (e.g., from performed simulations, we only
observed locally asymptotically stable equilibriua as attractors).

  4. 4.

The basins of attractions of attractors that single species persists are
bounded and may consist of different distinct parts for scramble competition
models (see black and cyan areas in Figure 3&4 in Kang 2013) while the basins
of attractions of attractors that single species persists are unbounded and
connected for contest competition models (see black and cyan areas in Figure
12, 14 and 16).

The discussion above regarding contest v.s. scramble competition models with
strong Allee effects are based on our study on the generalized Beverton-Holt
competition model (7)-(8) and Ricker-type competition models with Allee
effects studied in Kang and Yakubu (2011) and Kang (2013). The comparison may
provide us useful insights of general two species competition models where
each species has _strong Allee effects_ in its single state. However, in order
to obtain more general results, we need to classify the different types of
intra- and inter-specific competition rigorously and study how their
combinations with Allee effects lead to different dynamical outcomes. For
example, the following example can be considered as a combination of contest
inter-specific competition and scramble intra-specific competition subject to
Allee effects:

$\begin{array}[]{lcl}x_{t+1}&=&\frac{x_{t}e^{r_{1}(1-x_{t})}I^{x}(x_{t})}{1+c_{1}y_{t}}\\\
y_{t+1}&=&\frac{y_{t}e^{r_{2}(1-y_{t})}I^{y}(y_{t})}{1+c_{2}x_{t}}\end{array}$

where $I^{i},i=x,y$ represents positive density dependence that can lead to
Allee effects and $c_{i},i=1,2$ are relatively inter-specific competition
coefficients. This could be our future research project.

###  5.2 Conclusion

In this article, we propose a generalized Beverton-Holt competition model to
investigate how the interactions of Allee effects and contest competition
affect the species’s establishment and extinction. More precisely, we would
like to answer the following questions addressed in the introduction: 1. What
are the population dynamical outcomes of a two competing species when each of
them suffers from both strong Allee effects and contest competition? 2. How
does contest competition combined with strong Allee effects affect species’
persistence and extinction by comparing to models subject to strong Allee
effects and scramble competition? 3. What are the generic dynamical features
of population models subject to strong Allee effects and competition? Here, we
summarize our main results from the study as follows:

  1. 1.

Theorem 3.1 and Theorem 3.2 reveal that a two species contest competition
model with Allee effects has equilibrium dynamics, under the conditions that
the intensity of intra-specific competition intensity either exceeds that of
inter-specific competition or is less than that of inter-specific competition
but the maximum intrinsic growth rate of one species is not too large. This
partially answers the first question.

  2. 2.

Theorem 3.3 and Theorem 3.4 provide us an approximations of basins of
attractions of boundary attractors (i.e., initial conditions that lead to the
extinction of one or both species). This indicates that the inter-specific
competition can make species prone to extinction if it suffers from both
strong Allee effects and contest competition, and the situation is worsen if
species’ relative competition degree is less than 1. In addition, Theorem 3.5,
3.6 combined with numerical simulations suggest that decreasing the values of
the inter-specific competition coefficient and degree may promote the
coexistence of two species. These results may answer the second question.

  3. 3.

The third question can be partially answered by comparing with dynamics of two
species scramble competition models subject to strong Allee effects in Kang
(2013) to our Theorem 3.5 combined with numerical simulations. The comparison
suggests that (i) Both contest and scramble competition models can have either
three boundary attractors or four attractor with only one interior attractor,
however, scramble competition model may have only one attractor of the
extinction of both species due to strong Allee effects; (ii) Scramble
competition models represented by Ricker-type models may have more complicated
structure of interior attractor than contest competition ones; and (iii)
Scramble competition may be more likely to promote the coexistence of two
species at low or high densities under certain conditions.

Our study combined with the results from Kang and Yakubo (2011) as well as
Kang (2013) provide us useful insights on how Allee effects interacting with
different forms of competition have an impact on the persistence and
extinction of species. One direct application of these results is that we may
be able to save rare species from _essential extinction_ due to strong Allee
effects by introducing another similar competing species under proper
conditions. Thus, our study may help us understand the diversity of nature and
make better policy for conservation program, especially for rare species that
suffers from Allee effects. In the future, it will be interesting to
investigate the following questions: 1. What are population dynamics of two
competing species when each of them or both have different form of competition
in their different life stages as well as Allee effects? 2. What are necessary
conditions for a general two species population model to generate two
dimensional strong Allee effects? 3. How can we predict more precise initial
conditions that lead to the extinction and the coexistence of two species?

## Acknowledgement

The author would like to thank the useful discussions with Dr. Peter Chesson,
Dr. Elaydi Saber, Dr. Lidicker William, Dr. Livadiotis George and Dr. Kwessi
Eddy for the early draft of the manuscript. Y.K.’s research is partially
supported by NSF-DMS (1313312), Simons Collaboration Grants for Mathematicians
(208902) and the research scholarship from College of Letters and Sciences
(CLS) at Arizona State University.

## Appendix

### Proof of Proposition 2.1

###### Proof.

First, it is easy to see that Model (2) is positively invariant and bounded in
$\mathbb{R}_{+}$. Notice that

$H(0)=0,\,\frac{dH(u)}{du}=\frac{r\delta
u^{\delta-1}}{(1+u^{\delta})^{2}}>0\mbox{ and
}\lim_{u\rightarrow\infty}H(u)=r>0,$

we can apply Proposition 2.1 of Kang and Dieter (2011) to obtain the following
conclusion: Assume that $u=H(u)$ has $n+1$ consecutive, distinct and non-
degenerate solutions ${\bar{u}^{i}},i=0,1,...,n$ with the following property

$0=\bar{u}^{0}<\bar{u}^{1}<\cdot\cdot\cdot<\bar{u}^{n}.$

then the even ${\bar{u}^{i}},i\geq 2$ is locally asymptotical stable with
$(\bar{u}^{i-1},\bar{u}^{i+1})$ as its basin of attraction when
${\bar{u}^{0}=0}$ is locally asymptotical stable
(i.e.,$\frac{dH(u)}{du}\big{|}_{u=0}<1$) and the odd ${\bar{u}^{i}},i\geq 1$
is locally asymptotical stable with $(\bar{u}^{i-1},\bar{u}^{i+1})$ as its
basin of attraction when ${\bar{u}^{0}=0}$ is unstable
(i.e.,$\frac{dH(u)}{du}\big{|}_{u=0}>1$).

According $\frac{dH(u)}{du}=\frac{r\delta u^{\delta-1}}{(1+u^{\delta})^{2}}$,
we can conclude that ${\bar{u}^{0}=0}$ is unstable if $\delta<1$ and
${\bar{u}^{0}=0}$ is locally asymptotical stable if $\delta>1$. Therefore, we
have the following statement: i) If $\delta\in(0,1)$ and Model (2) has two
non-negative equilibria $0,K$, then 0 is an unstable and and $K$ is a stable
positive equilibrium that attracts all points in $(0,\infty)$. ii) If
$\delta>1$ and the only equilibrium of Model (2) is 0, then $0$ is globally
stable. iii) If $\delta>1$ and Model (2) has three non-negative distinct
equilibria: $0,\,A$ and $K$ such that $0<A<K$, then 0 equilibrium is locally
asymptotical stable with the basin of attraction $[0,A)$; $A$ is unstable
(repellor), while $K$ is locally asymptotical stable with the basin of
attraction $(A,\infty)$.

Now we only need to derive the conditions when (2) has only one, or two or
three equilibria. As we know, 0 is always a equilibrium of (2). If $u^{*}$ is
a nontrivial positive equilibrium of (2), then we have $h(u^{*})=1$, i.e.,
$u^{*}$ is a positive root of $F(u)=u^{\delta}-ru^{\delta-1}+1$. If
$\delta<1$, then we have

$\lim_{u\rightarrow
0^{+}}F(u)=-\infty,\,\lim_{u\rightarrow\infty}F(u)=\infty\mbox{ and
}\frac{dF}{du}=\delta u^{\delta-1}-r(\delta-1)u^{\delta-2}>0.$

This implies that $F(u)=u$ has a unique positive solution $u^{*}$ and

$\frac{dH}{du}\big{|}_{u=u^{*}}=\frac{r\delta(u^{*})^{\delta-1}}{(1+(u^{*})^{\delta})^{2}}=\frac{\delta}{1+(u^{*})^{\delta}}<\delta<1.$

Therefore, if $\delta\in(0,1)$, Model (2) has only two non-negative equilibria
$0,K$ where 0 is an unstable and and $K$ is a stable positive equilibrium that
attracts all points in $(0,\infty)$.

If $\delta>1$, then $\frac{dF}{du}=\delta
u^{\delta-1}-r(\delta-1)u^{\delta-2}=u^{\delta-2}[\delta u-r(\delta-1)]=0$
when $u=u_{c}=\frac{r(\delta-1)}{\delta}$. Notice that

$F(0)=1>0,\,\frac{dF}{du}\big{|}_{u<u_{c}}<0\mbox{ and
}\frac{dF}{du}\big{|}_{u>u_{c}}>0,$

thus $F(u)=0$ (i.e., $h(u)=1$) have no positive root if

$F(u_{c})>0\Rightarrow
h(u_{c})<1\Rightarrow\frac{ru_{c}^{\delta-1}}{1+u_{c}^{\delta}}=\frac{r\left(\frac{r(\delta-1)}{\delta}\right)^{\delta-1}}{1+\left(\frac{r(\delta-1)}{\delta}\right)^{\delta}}<1\Rightarrow
r<r_{crit}=\delta(\delta-1)^{\frac{1}{\delta}-1}$

and $F(u)=0$ have two non-negative distinct equilibria $0<A<u_{c}<K$ if

$F(u_{c})<0\Rightarrow
h(u_{c})>1\Rightarrow\frac{ru_{c}^{\delta-1}}{1+u_{c}^{\delta}}=\frac{r\left(\frac{r(\delta-1)}{\delta}\right)^{\delta-1}}{1+\left(\frac{r(\delta-1)}{\delta}\right)^{\delta}}>1\Rightarrow
r>r_{crit}=\delta(\delta-1)^{\frac{1}{\delta}-1}.$

Therefore the statements of Proposition 2.1 hold. ∎

### Proof of Theorem 2.1

###### Proof.

First, we can easily verify that Model (6) is positively invariant and bounded
in $\mathbb{R}_{+}$. If $u^{*}$ is a positive equilibrium of (6), then it is a
positive root of $F(u)=bu^{d}+u^{\delta}-ru^{\delta-1}+1$. Since

$F^{\prime}(u)=bdu^{d-1}+\delta
u^{\delta-1}-r(\delta-1)u^{\delta-2}=u^{\delta-2}[bdu^{d-\delta+1}+\delta
u-r(\delta-1)]=u^{\delta-2}G(u)$

where $G(u)=bdu^{d-\delta+1}+\delta u-r(\delta-1).$ If $\delta<1$, then we
have

$\lim_{u\rightarrow 0^{+}}F(u)=-\infty,\,F^{\prime}(u)>0,\,\lim_{u\rightarrow
0^{+}}F^{\prime}(u)=\infty\mbox{ and }\lim_{u\rightarrow\infty}F(u)=\infty,$

thus, $F(u)$ has a unique positive root $K>0$.

Moreover, we have $H(0)=0$ and
$H^{\prime}(u)=\frac{ru^{\delta-1}[\delta+bu^{d}(\delta-d)]}{(1+u^{\delta}+bu^{d})^{2}}$.
This indicates the following two cases:

  1. 1.

If $\delta\geq d$, then $H^{\prime}>0$, thus, Model (6) is a monotone system
where its dynamics can be classified into the following two cases:

    1. (a)

If $\delta<1$, then Model (6) has two non-negative equilibria 0 and $K$ where
0 is unstable and $K$ is globally stable.

    2. (b)

If $\delta>1+d$, then we have $u=0$ is locally stable and there exists some
unique positive $u^{c}$ such that $G^{\prime}(u)|_{u<u^{c}}<0\mbox{ and
}G^{\prime}(u)|_{u>u^{c}}>0.$ Since $\lim_{u\rightarrow 0^{+}}G(u)=\infty$ and
$\lim_{u\rightarrow\infty}G(u)=\infty$, therefore, $G(u)$ has exactly two
positive roots if $G(u_{c})<0$. Also, we can conclude that $G(u)>0$ if
$G(u_{c})>0$. This implies that if $G(u_{c})<0$, then $F(u)$ has exactly two
positive distinct critical points $0<u_{c}^{1}<u^{c}<u_{c}^{2}$. Therefore, we
can conclude that 1) $F(u)$ has exact four distinct positive roots if
$F(u_{c}^{1})<0$ and $F(u_{c}^{2})<0$; 2) $F(u)$ has exact two distinct
positive roots if $F(u_{c}^{1})F(u_{c}^{2})<0$; and 3) $F(u)$ has no positive
root if $F(u_{c}^{1})>0$ and $F(u_{c}^{2})>0$. In the case that $F(u)$ has no
positive root, we can conclude that $u=0$ is globally stable. While if
$G(u_{c})>0$, then $F(u)<u$ for all $u>0$, therefore, we have $u=0$ is
globally stable.

    3. (c)

If $1+d>\delta>1$, then we have $u=0$ is locally stable and

$G(0)=-r(\delta-1),\,G^{\prime}(u)>0\mbox{ and
}\lim_{u\rightarrow\infty}G(u)=\infty,$

thus, $F^{\prime}(u)$ has a unique positive point $u_{c}$ such that
$G(u_{c})=0\Rightarrow F^{\prime}(u_{c})=0$ and

$F^{\prime}(u)|_{u<u_{c}}>0\mbox{ and }F^{\prime}(u)|_{u>u_{c}}<0.$

Therefore, if $h(u_{c})>1$, then Model (6) has as least two consecutive,
distinct and non-degenerate equilibria $A,K$ such that $0<A<u_{c}<K$.

  2. 2.

If $\delta<d$, then there exists a unique critical point
$u_{c}=(\frac{r(d-\delta)}{\delta})^{1/d}$ such that

$H^{\prime}(u)|_{u<u_{c}}>0,\,\,H^{\prime}(u_{c})=0\mbox{ and
}H^{\prime}(u)|_{u>u_{c}}<0.$

We consider the following two cases:

    1. (a)

If $\delta<1$, then Model (6) is a unique hump model that is similar to the
Ricker’s map, thus, it has two non-negative equilibria 0 and $K$ where 0 is
unstable.

    2. (b)

If $\delta>1$, then $H^{\prime}(0)=0$, thus $u=0$ is locally asymptotically
stable. If $r>r_{crit}$, then we have $H(u_{c})>u_{c}$. This implies that
Model (6) has at least two distinct positive roots $A,K$ such that
$0<A<u_{c}<K$.

If both $\delta$ and $d$ are positive integers, then the number of interior
equilibrium of (6) is the same as the number of positive roots of the
following polynomial:

$F(x)=bx^{d}+x^{\delta}-rx^{\delta-1}+1.$

Descartes’ rule of signs (Smith and Latham 1954; Meserve 1982) states that if
the terms of a single-variable polynomial with real coefficients are ordered
by descending variable exponent, then the number of positive roots of the
polynomial is either equal to the number of sign differences between
consecutive nonzero coefficients, or is less than it by a multiple of 2.
Therefore, the statement of Part 4 holds.

∎

### Proof of Theorem 3.1

###### Proof.

Denote by $T:\mathbb{R}^{2}_{+}\rightarrow\mathbb{R}^{2}_{+}$ the map defined
by the right side of Model (7)-(8), then we have

$\displaystyle T$ $\displaystyle=$
$\displaystyle(T_{1}(x,y),T_{2}(x,y))=\left(\frac{r_{1}x^{\delta_{1}}}{1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}},\frac{r_{2}y^{\delta_{2}}}{1+y^{\delta_{2}}+b_{2}x^{\delta_{4}}}\right).$
(24)

Thus, $T$ is $C^{1}$ and $T_{i}<r_{i},i=1,2$. The Jacobian matrix of $T$ can
be represented as follows:

$\displaystyle J$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{lcl}\frac{r_{1}\delta_{1}x^{\delta_{1}-1}(1+b_{1}y^{\delta_{3}})}{\left(1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}\right)^{2}}&-\frac{a_{1}b_{1}\delta_{3}x^{\delta_{1}}y^{\delta_{3}-1}}{\left(1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}\right)^{2}}\\\
-\frac{a_{2}b_{2}\delta_{4}y^{delta_{2}}x^{\delta_{4}-1}}{\left(1+y^{\delta_{2}}+b_{2}x^{\delta_{4}}\right)^{2}}&\frac{a_{2}\delta_{2}y^{\delta_{2}-1}(1+b_{2}x^{\delta_{4}})}{\left(1+y^{\delta_{2}}+b_{2}x^{\delta_{4}}\right)^{2}}\end{array}\right]$
(27)

which gives

$det(J)=\frac{r_{1}r_{2}[\delta_{1}\delta_{2}(x^{\delta_{1}-1}y^{\delta_{2}-1}+b_{2}x^{\delta_{1}+\delta_{4}-1}y^{\delta_{2}-1}+b_{1}x^{\delta_{1}-1}y^{\delta_{2}+\delta_{3}-1})+b_{1}b_{2}x^{\delta_{1}+\delta_{4}-1}y^{\delta_{2}+\delta_{3}-1}(\delta_{1}\delta_{2}-\delta_{3}\delta_{4})]}{\left(1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}\right)^{2}\left(1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}\right)^{2}}$
(28)

According to Lemma 3.1, we know that the map $T$ is positively invariant in
$Int\,\mathbb{R}^{2}_{+}$, i.e., $T:Int\,\mathbb{R}^{2}_{+}\rightarrow
Int\,\mathbb{R}^{2}_{+}$. Thus, for $(x,y)\in Int\,\mathbb{R}^{2}_{+}$, we
have $det(J)\geq 0$ (where equality occurs when
$\delta_{1}\delta_{2}=\delta_{3}\delta_{4}$) and $J$ has a constant sign
configuration as $J=\left[\begin{array}[]{lcl}+&-\\\ -&+\end{array}\right].$
This indicates that (27) is $K$-positive in $Int\,\mathbb{R}^{2}_{+}$ if and
it is $K$-strongly positive if $\delta_{1}\delta_{2}>\delta_{3}\delta_{4}$.

Take $u,v\in Int\,\mathbb{R}^{2}_{+}$ such that $u<_{K}v$, then by the mean
value theorem and (27), we have

$\begin{array}[]{lcl}T_{1}(v)-T_{1}(u)&=&\frac{\partial T_{1}}{\partial
x}(v_{1}-u_{1})+\frac{\partial T_{1}}{\partial y}(v_{2}-u_{2})>0\\\
T_{2}(v)-T_{2}(u)&=&\frac{\partial T_{2}}{\partial
x}(v_{1}-u_{1})+\frac{\partial T_{2}}{\partial y}(v_{2}-u_{2})<0\end{array}$

where the partial derivatives are evaluated at a point on the line segment
connecting $u$ and $v$. This implies that $T(u)<<_{K}T(v)$, i.e., the map $T$
is strongly competitive.

Choose any $(x_{0},y_{0})\in T(Int\,\mathbb{R}^{2}_{+})$, we have
$x_{0}<r_{1}$ and $y_{0}<r_{2}$. Let $(x,y)$ be a preimage of $(x_{0},y_{0})$,
then it satisfies the following equation:

$\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\frac{r_{1}x^{\delta_{1}}}{1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}}$
(29) $\displaystyle y_{0}$ $\displaystyle=$
$\displaystyle\frac{r_{2}y^{\delta_{2}}}{1+y^{\delta_{2}}+b_{2}x^{\delta_{4}}}$
(30)

This indicates that

$\displaystyle x_{0}$ $\displaystyle=$
$\displaystyle\frac{r_{1}x^{\delta_{1}}}{1+x^{\delta_{1}}+b_{1}(\frac{y_{0}(1+b_{2}x^{\delta_{4}})}{r_{2}-y_{0}})^{\delta_{3}/\delta_{2}}}=f(x)$
(31) $\displaystyle y_{0}$ $\displaystyle=$
$\displaystyle\frac{r_{2}y^{\delta_{2}}}{1+y^{\delta_{2}}+b_{2}(\frac{x_{0}(1+b_{1}y^{\delta_{3}})}{r_{1}-x_{0}})^{\delta_{4}/\delta_{1}}}=g(y).$
(32)

Since $\delta_{1}\delta_{2}\geq\delta_{3}\delta_{4}$, we have

$\displaystyle\begin{array}[]{lcl}f^{\prime}(x)&=&\frac{r_{1}(\delta_{1}\delta_{2}[x^{\delta_{1}}+b_{2}x^{\delta_{1}+\delta_{4}}+b_{1}x^{\delta_{1}}(\frac{y_{0}(1+b_{2}x^{\delta_{4}})}{r_{2}-y_{0}})^{\delta_{3}/\delta_{2}}]+b_{1}b_{2}x^{\delta_{1}+\delta_{4}}(\frac{y_{0}(1+b_{2}x^{\delta_{4}})}{r_{2}-y_{0}})^{\delta_{3}/\delta_{2}}[\delta_{1}\delta_{2}-\delta_{3}\delta_{4}])}{\delta_{2}x(1+b_{2}x^{\delta_{4}})(1+x^{\delta_{1}}+b_{1}(\frac{y_{0}(1+b_{2}x^{\delta_{4}})}{r_{2}-y_{0}})^{\delta_{3}/\delta_{2}})^{2}}>0\\\
g^{\prime}(y)&=&\frac{r_{2}(\delta_{1}\delta_{2}[y^{\delta_{2}}+b_{1}y^{\delta_{2}+\delta_{3}}+b_{2}y^{\delta_{2}}(\frac{x_{0}(1+b_{1}y^{\delta_{3}})}{r_{1}-x_{0}})^{\delta_{4}/\delta_{1}}]+b_{1}b_{2}y^{\delta_{2}+\delta_{3}}(\frac{x_{0}(1+b_{1}y^{\delta_{3}})}{r_{1}-x_{0}})^{\delta_{4}/\delta_{1}}[\delta_{1}\delta_{2}-\delta_{3}\delta_{4}])}{\delta_{1}y(1+b_{1}y^{\delta_{3}})(1+y^{\delta_{2}}+b_{2}(\frac{x_{0}(1+b_{1}y^{\delta_{3}})}{r_{1}-x_{0}})^{\delta_{4}/\delta_{1}})^{2}}>0.\end{array}$
(35)

This implies that both $f^{-1}(x_{0})$ and $g^{-1}(y_{0})$ have only one
point, i.e., $(x_{0},y_{0})\in T(Int\,\mathbb{R}^{2}_{+})$ has only one
preimage. Thus, the map $T$ is injective in $Int\,\mathbb{R}^{2}_{+}$.

It is easy to check that $Int\,\mathbb{R}^{2}_{+}$ contains order intervals
and is $\leq_{K}$-convex (i.e., it contains the line segment jointing any two
of its points that are $K$-ordering). Therefore, we can apply Corollary 4.4 of
Smith (1998) to obtain that $\\{T^{n}(x_{0},y_{0})=(x_{n},y_{n})\\}$ is
eventually componentwise monotone for every $(x_{0},y_{0})\in
Int\,\mathbb{R}^{2}_{+}$, i.e., there exists a positive integer $N$ such that
either $x_{n}\leq x_{n+1}$ for all $n>N$ or $x_{n+1}\leq x_{n}$ for all $n>N$
and similarly for $y_{n}$. Notice that

$T(Int\,\mathbb{R}^{2}_{+})\subset
T(\mathbb{R}^{2}_{+})\subset[0,r_{1}]\times[0,r_{2}],$

thus, $\\{T^{n}(x_{0},y_{0})=(x_{n},y_{n})\\}$ converges to an equilibrium of
Model (7)-(8) in $[0,r_{1}]\times[0,r_{2}]$. ∎

### Proof of Theorem 3.2

###### Proof.

If $\delta_{1}\delta_{2}<\delta_{3}\delta_{4}$, then (28) vanishes at the
following curves :

$\begin{array}[]{lcl}LC_{-1}^{1}:&&xy=0\\\
LC_{-1}^{2}:&&\delta_{1}\delta_{2}[1+b_{2}x^{\delta_{4}}+b_{1}y^{\delta_{3}}]+b_{1}b_{2}y^{\delta_{3}}x^{\delta_{4}}[\delta_{2}\delta_{1}-\delta_{4}\delta_{3}]=0.\end{array}$

Thus, we have

$LC_{-1}^{2}:\,y=\gamma(x)=\left(\frac{\delta_{1}\delta_{2}[1+b_{2}x^{\delta_{4}}]}{b_{1}[b_{2}x^{\delta_{4}}(\delta_{4}\delta_{3}-\delta_{2}\delta_{1})-\delta_{1}\delta_{2}]}\right)^{1/\delta_{3}},\mbox{
provided that
}x>\left(\frac{\delta_{1}\delta_{2}}{b_{2}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{4}}$

or

$LC_{-1}^{2}:\,x=\gamma^{-1}(y)=\left(\frac{\delta_{1}\delta_{2}[1+b_{1}y^{\delta_{3}}]}{b_{2}[b_{1}y^{\delta_{3}}(\delta_{4}\delta_{3}-\delta_{2}\delta_{1})-\delta_{1}\delta_{2}]}\right)^{1/\delta_{4}},\mbox{
provided that
}y>\left(\frac{\delta_{1}\delta_{2}}{b_{1}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{3}}$

which gives that

$\gamma^{\prime}(x)=-\frac{b_{2}\delta_{1}\delta_{2}\delta_{4}^{2}x^{\delta_{4}-1}}{b_{1}[b_{2}x^{\delta_{4}}(\delta_{4}\delta_{3}-\delta_{2}\delta_{1})-\delta_{1}\delta_{2}]^{2}}\left(\frac{\delta_{1}\delta_{2}[1+b_{2}x^{\delta_{4}}]}{b_{1}[b_{2}x^{\delta_{4}}(\delta_{4}\delta_{3}-\delta_{2}\delta_{1})-\delta_{1}\delta_{2}]}\right)^{1/\delta_{3}-1}<0$

provided that
$x>\left(\frac{\delta_{1}\delta_{2}}{b_{2}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{4}}.$
In addition, by straight forward calculations, we have

$\gamma^{\prime\prime}(x)>0\mbox{ if }\delta_{3}\geq\delta_{4}\geq 1.$

The curve $\gamma(x)$ separates $Int\,\mathbb{R}^{2}_{+}$ into two connected
unbounded components. The one unbounded component containing
$(0,\epsilon)\times(0,\epsilon)$ for very small $\epsilon$ is denoted by
$\Omega_{+}$ and it has $det(J)>0$. While the other unbounded component is
denoted by $\Omega_{-}$ and it has $det(J)<0$.

Let $T$ be the map describing Model (7)-(8). The image of $\gamma(x)$ under
the map $T$, i.e., $T(\gamma)$, called the critical curves in (Mira _et al_
1996; Smith 1998; Kang 2011; Kang and Smith 2011), together with the portions
of the coordinate axes which it cuts off, form the boundary of
$T(Int\,\mathbb{R}^{2}_{+})$. Because the curve $\gamma$ is linearly order by
$<<_{K}$ (i.e., $\gamma^{\prime}(x)<0$ indicates that any two points in
$\gamma$ are related in the sense of strictly $K$-ordering) and $T$ preserves
the strong ordering, $T(\gamma)$ is linearly ordered by $<<_{K}$ and connects
$(0,r_{2})$ to $(r_{1},0)$. Let $B$ denote the bounded component of
$Int\,\mathbb{R}^{2}_{+}\setminus T(\gamma)$. Then
$T(\Omega_{+})=T(\Omega_{-})=B$, i.e., the map $T$ folds
$Int\,\mathbb{R}^{2}_{+}$ over at $\gamma$ and maps both unbounded components
onto $B$.

If

$r_{1}<\left(\frac{\delta_{1}\delta_{2}}{b_{2}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{4}}\mbox{
or
}r_{2}<\left(\frac{\delta_{1}\delta_{2}}{b_{1}[\delta_{3}\delta_{4}-\delta_{1}\delta_{2}]}\right)^{1/\delta_{3}},$

then $B$ is disjoint from $\gamma$ and is totally contained in $\Omega_{+}$.
This implies that $det(T)_{B}>0$. Note that the bounded component of
$Int\,\mathbb{R}^{2}_{+}\setminus T(\gamma)$ contains the point without pre-
image (e.g., choosing a point with x-coordinate larger than $r_{1}$ and
y-coordinate larger than $r_{2}$), then all the points in
$Int\,\mathbb{R}^{2}_{+}\setminus T(\gamma)$ has zero pre-image. This because
that the cardinality of $T^{-1}(w),w\in Int\,\mathbb{R}^{2}_{+}\setminus
T(\gamma)$ is finite and constant according to Lemma 3.4 of Chow and Hale
(1982). Similarly, we can show that the cardinality of $T^{-1}(w),w\in B$ is
two where one is located in $B$ and the other one is located in
$Int\,\mathbb{R}^{2}_{+}\setminus T(\gamma)$. This implies that

$T:B\rightarrow B\mbox{ is injective }.$

Therefore, we apply Corollary 4.4 of Smith (1998) to obtain that
$\\{T^{n}(x_{0},y_{0})=(x_{n},y_{n})\\}$ is eventually componentwise monotone
for every $(x_{0},y_{0})\in Int\,\mathbb{R}^{2}_{+}$. This indicates that
every orbit of Model (7)-(8) with any initial condition in
$\mathbb{R}^{2}_{+}$ converges to one of its equilibria. ∎

### Proof of Theorem 3.3

###### Proof.

For any initial value $(x_{0},y_{0})$ taken in $O_{ex}$, we have
$x_{0}<A_{1}$, thus we have

$x_{1}=T_{1}(x_{0},y_{0})=\frac{r_{1}x_{0}^{\delta_{1}}}{1+x_{0}^{\delta_{1}}+b_{1}y_{0}^{\delta_{3}}}\leq
T_{1}(x_{0},0).$

Since $\frac{\partial T_{1}}{\partial x}|_{x>0}>0$, thus we have

$x_{t+1}=T_{1}^{t}(x_{0},y_{0})\leq T_{1}^{t}(x_{0},0).$

According to Proposition 2.1, we have
$\lim_{t\rightarrow\infty}T_{1}^{t}(x_{0},0)=0$. Since Model (7)-(8) is
positively invariant in $\mathbb{R}^{2}_{+}$ by Lemma 3.1, thus,
$\lim_{t\rightarrow\infty}x_{t}=0$ holds for Model (7)-(8) if Condition H1 is
satisfied and its initial condition is taken in $O_{ex}$.

Similarly, we can use the same argument to show that
$\lim_{t\rightarrow\infty}y_{t}=0$ holds for Model (7)-(8) if Condition H1 is
satisfied and its initial condition is taken in $O_{ey}$. Therefore,

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=(0,0)=E_{0}\mbox{ provided
}(x_{0},y_{0})\in O_{0}.$

Based on our arguments above, we can conclude that for any initial value
$(x_{0},y_{0})$ taken in $O_{x}$, we have $\lim_{t\rightarrow\infty}y_{t}=0.$
In addition, we have

$x_{1}=T_{1}(x_{0},y_{0})=\frac{r_{1}x_{0}^{\delta_{1}}}{1+x_{0}^{\delta_{1}}+b_{1}y_{0}^{\delta_{3}}}\geq
T_{1}(x_{0},A_{2})=\frac{r_{1}x_{0}^{\delta_{1}}}{1+x_{0}^{\delta_{1}}+b_{1}(A_{2})^{\delta_{3}}}.$

Then by induction, we have $x_{t+1}=T_{1}^{t}(x_{0},y_{0})\geq
T_{1}^{t}(x_{0},A_{2}).$ Since Model (7)-(8) satisfies Condition H2 and
$x_{0}>A_{1}^{a_{1}}$, then according to Corollary 2.1, we have

$\lim_{t\rightarrow\infty}T_{1}^{t}(x_{0},A_{2})=K_{1}^{a_{1}}>A_{1}^{a_{1}}>A_{1}.$

This indicates that for Model (7)-(8), we have

$\liminf_{t\rightarrow\infty}x_{t}\geq K_{1}^{a_{1}}>A_{1}^{a_{1}}>A_{1}.$

Notice that $\lim_{t\rightarrow\infty}y_{t}=0$, thus, for any $\epsilon>0$,
there exists a $N$ large enough, such that for any $t>N$, we have

$x_{t+1}\geq\frac{r_{1}x_{t}^{\delta_{1}}}{1+x_{t}^{\delta_{1}}+\epsilon}\geq
A_{1}^{a_{1}}>A_{1}.$

Let $\epsilon\rightarrow 0$, then the limiting system of Model (7)-(8) becomes
$T_{1}(x,0)$. Since Condition H1 is satisfied and the initial value is larger
than $A_{1}$, thus according to Proposition 2.1, we have
$\lim_{t\rightarrow\infty}x_{t}=K_{1}.$

Similarly, we can apply the same argument above to show the case when an
initial value $(x_{0},y_{0})$ is taken in $O_{x}$. Therefore, the statement of
Theorem 3.3 holds. ∎

### Proof of Theorem 3.4

###### Proof.

Assume that Model (7)-(8) satisfies Condition H1. Let $y=kx$, then from (24)
any point $(x,kx)\in Int\,\mathbb{R}^{2}_{+}$ maps to the following point

$T(x,kx)=\left(T_{1}(x,kx),T_{2}(x,kx)\right)=\left(\frac{r_{1}x^{\delta_{1}}}{1+x^{\delta_{1}}+b_{1}(kx)^{\delta_{3}}},\frac{r_{2}(kx)^{\delta_{2}}}{1+(kx)^{\delta_{2}}+b_{2}x^{\delta_{4}}}\right).$

If $\delta_{3}>\delta_{1}$, then we have

$\frac{\partial T_{1}(x,kx)}{\partial
x}=\frac{r_{1}x^{\delta_{1}-1}[\delta_{1}+b_{1}(kx)^{\delta_{3}}(\delta_{1}-\delta_{3})]}{(1+x^{\delta_{1}}+b_{1}(kx)^{\delta_{3}})^{2}}$

which indicates that

$\frac{\partial T_{1}(x,kx)}{\partial x}|_{x<x_{c}}>0\mbox{ and
}\frac{\partial T_{1}(x,kx)}{\partial
x}|_{x>x_{c}}<0,x_{c}=\left(\frac{\delta_{1}}{b_{1}k^{\delta_{3}}(\delta_{3}-\delta_{1})}\right)^{1/\delta_{3}}.$

Let
$h(x)=\frac{r_{1}x^{\delta_{1}}}{x^{\delta_{1}}+b_{1}(kx)^{\delta_{3}}}=\frac{r_{1}}{1+b_{1}k^{\delta_{3}}x^{\delta_{3}-\delta_{1}}}>T_{1}(x,kx)$.
Since $h(x)>T_{1}(x,kx)$ and $h^{\prime}(x)<0$ provided
$\delta_{3}>\delta_{1}$, thus we have

$T_{1}(x,kx)<h(x)<A_{1}\Rightarrow
x>\left(\frac{r_{1}-A_{1}}{b_{1}k^{\delta_{3}}}\right)^{\frac{1}{\delta_{3}-\delta_{1}}}.$

This gives

$T_{1}(x,kx)<A_{1}\mbox{ whenever
}x>\left(\frac{r_{1}-A_{1}}{b_{1}k^{\delta_{3}}}\right)^{\frac{1}{\delta_{3}-\delta_{1}}}.$

Define

$O_{ex}^{l}=\bigcup_{k>0}\\{(x,kx)\in
Int\,\mathbb{R}^{2}_{+}:x>\left(\frac{r_{1}-A_{1}}{b_{1}k^{\delta_{3}}}\right)^{\frac{1}{\delta_{3}-\delta_{1}}}\\}.$

Then we can conclude that if Model (7)-(8) satisfies Condition H1 and
$\delta_{3}>\delta_{1}$, then for any initial value taken in $O_{ex}^{l}$, we
have

$\lim_{t\rightarrow\infty}x_{t}=0.$

Similarly, we can conclude that if Model (7)-(8) satisfies Condition H1 and
$\delta_{4}>\delta_{2}$, then for any initial value taken in $O_{ey}^{l}$
where

$O_{ey}^{l}=\bigcup_{k>0}\\{(ky,y)\in
Int\,\mathbb{R}^{2}_{+}:y>\left(\frac{r_{2}-A_{2}}{b_{2}k^{\delta_{4}}}\right)^{\frac{1}{\delta_{4}-\delta_{2}}}\\},$

we have

$\lim_{t\rightarrow\infty}y_{t}=0.$

Let $O_{0}^{l}=O_{ex}^{l}\cap O_{ey}^{l}.$Then based on the arguments above,
we can conclude that if Model (7)-(8) satisfies Condition H1 and
$\delta_{3}>\delta_{1},\delta_{4}>\delta_{2}$, then for any initial value
taken in $O_{0}^{l}$, we have

$\lim_{t\rightarrow\infty}(x_{t},y_{t})=E_{0}.$

The argument above combined with the results of Theorem 3.3, we can conclude
that the statement is hold. ∎

### Proof of Theorem 3.5

###### Proof.

Since Model (7)-(8) satisfies Condition H1, thus if $(x,y)$ is an interior
equilibrium of (7)-(8), we have the the following two equations:

$\begin{array}[]{lcl}1&=&\frac{r_{1}x^{\delta_{1}-1}}{1+x^{\delta_{1}}+b_{1}y^{\delta_{3}}}\Rightarrow
y=\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}}=F_{1}(x),\,0<A_{1}<x<K_{1}\\\
1&=&\frac{r_{2}y^{\delta_{2}-1}}{1+y^{\delta_{2}}+b_{2}x^{\delta_{4}}}\Rightarrow
x=\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}}=F_{2}(y),\,0<A_{2}<y<K_{2}.\end{array}$

Since

$\begin{array}[]{lcl}F_{1}^{\prime}(x)&=&\frac{x^{\delta_{1}-2}\left(\frac{r_{1}x^{\delta_{1}-1}-x^{\delta_{1}}-1}{b_{1}}\right)^{1/\delta_{3}-1}}{b_{1}\delta_{3}}\left[r_{1}(\delta_{1}-1)-\delta_{1}x\right]\\\
F_{2}^{\prime}(y)&=&\frac{y^{\delta_{2}-2}\left(\frac{r_{2}y^{\delta_{2}-1}-y^{\delta_{2}}-1}{b_{2}}\right)^{1/\delta_{4}-1}}{b_{2}\delta_{4}}\left[r_{2}(\delta_{2}-1)-\delta_{2}y\right],\end{array}$

thus we have,

$\frac{dF_{1}}{dx}|_{x<x_{c}}>0,\,\frac{dF_{1}}{dx}|_{x=x_{c}}=0\mbox{ and
}\frac{dF_{1}}{dx}|_{x>x_{c}}<0,$

and

$\frac{dF_{2}}{dy}|_{y<y_{c}}>0,\,\frac{dF_{2}}{dy}|_{y=y_{c}}=0\mbox{ and
}\frac{dF_{2}}{dy}|_{y>y_{c}}<0$

where

$x_{c}=\frac{r_{1}(\delta_{1}-1)}{\delta_{1}}\mbox{ and
}y_{c}=\frac{r_{2}(\delta_{2}-1)}{\delta_{2}}.$

Therefore, according to the geometry of $y=F_{1}(x)$ and $x=F_{2}(y)$, we have
follows:

  * 1.

If $F_{1}(x_{c})<A_{2}$ or $F_{2}(y_{c})<A_{1}$, then Model (7)-(8) has no
interior equilibrium.

  * 2.

If $A_{2}<F_{1}(x_{c})<K_{2},\,K_{1}<F_{2}(y_{c})$ or
$K_{2}<F_{1}(x_{c}),\,A_{1}<F_{2}(y_{c})<K_{1}$, then Model (7)-(8) has two
interior equilibria.

  * 3.

If $F_{1}(x_{c})>K_{2}$ and $F_{2}(y_{c})>K_{1}$, then Model (7)-(8) has four
interior equilibria.

∎

### Proof of Theorem 3.6

###### Proof.

Let $(x^{*},x^{*})$ be a symmetric interior equilibrium of Model (9)-(10),
then we have

$r(x^{*})^{\delta-1}=1+(x^{*})^{\delta}+b(x^{*})^{d}.$

This indicates that the Jacobian matrix of Model (9)-(10) evaluated at the
symmetric interior equilibrium $(x^{*},x^{*})$ can be represented as

$\displaystyle J_{(x^{*},x^{*})}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{lcl}\frac{\delta(1+b(x^{*})^{d})}{1+(x^{*})^{\delta}+b(x^{*})^{d}}&-\frac{bd(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}\\\
-\frac{bd(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}&\frac{\delta(1+b(x^{*})^{d})}{1+(x^{*})^{\delta}+b(x^{*})^{d}}\end{array}\right].$
(38)

The eigenvalues of (38) are

$\lambda_{1}(x^{*})=\frac{\delta+b(\delta-d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}\mbox{
and
}\lambda_{2}(x^{*})=\frac{\delta+b(\delta+d)(x^{*})^{d}}{1+(x^{*})^{\delta}+b(x^{*})^{d}}$

where $\lambda_{1}(x^{*})$ denotes the stability of $(x^{*},x^{*})$ on the
invariant manifold $\Omega_{y=x}$ and the sign of $\lambda_{2}(x^{*})-1$
denotes whether the eigenvector associated with $\lambda_{2}$ is pointing
towards or away from $\Omega_{y=x}$.

Therefore, the statement of Theorem 3.6 holds. ∎

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