The Beddington–DeAngelis Competitive Response: Intra-Species Interference Enhances Coexistence in Species Competition

Abstract

Species coexistence is a major issue in ecology. We disentangled the role of individual interference when competing in the classical interference competition model. For the first time, we considered simultaneously intra- and inter-species interference by introducing the Beddington–DeAngelis competitive response into the classical competition model. We found a trade-off between intra- and inter-species interference that refines in a sense the well-known balance of intra- and inter-species competition coefficients. As a result, we found that (i) global coexistence is possible for a larger range of values of the inter-/intra-species competition coefficients and contributes to explaining the high prevalence of species coexistence in nature. This feature is exclusively due to intra-species interference. (ii) We found multi-stability scenarios previously described in the literature that can be reinterpreted in terms of individuals interference.

1. Introduction

It is a fact that species coexistence in nature is closer to being the rule rather than the exception. This empirical observation is at odds with the classical ecological theory established in the Competitive Exclusion Principle: two species competing for the same limited resource cannot coexist at constant population values. However, ref. [1] posed the so-called Paradox of the Plankton questioning how so many species of marine plankton coexist in a rather homogeneous medium with limited food, seemingly contradicting Gause’s rule of competitive exclusion. This is just one of many examples illustrating how puzzling coexistence is [2]. Since then, much effort has been invested in conciliating observational and theoretical knowledge, describing different frameworks and mechanisms that explain such a contradiction; see [3] or [4] for recent reviews.

Competition can be modeled in different ways [5]. In exploitative (scramble) competition, individuals can take a portion of the resource, which appears explicitly in the model. In contrast, in interference (contest) competition, one, and only one, of the contenders will keep the whole resource, which is not explicit in the equations. The prototype models of interference competition are [6] (the competition counterpart of the Lotka–Volterra predator–prey model) and [7], the first formulated with ordinary differential equations and the latter with difference equations. Both of them were based on laboratory experiments (two species of fungi, Saccharomyces cerevisiae and Schizosaccharomyces kefir, and two species of flour beetle, Tribolium castaneum and Triblium confusum, respectively) and are considered to support the Competitive Exclusion Principle.

The above-mentioned models are dynamic equivalents and exhibit four competition outcomes (apart from the extinction of both species): either species 1 or species 2 wins regardless of the initial number of individuals; species coexistence occurs no matter the initial number of individuals; or the so-called priority effects occur (one species will go extinct, depending on the initial number of individuals). The models share a key feature: their nullclines (the zero-growth curves) are straight lines, which implies the existence of at most one coexistence equilibrium point in the interior of the positive cone. Observational experiments [8,9] show that nullclines can be different from straight lines. In [10], the authors underline that Lotka–Volterra models exhibit per-capita growth rates that are linear functions of density and may bias their predictions (see references therein). Despite this empirical evidence and these theoretical reasons, interference competition models have remained the same for nearly 80 years, and effort has been put into exploitative competition [4]. Recently, several authors have resumed research on interference competition by incorporating into the classical Gause model different features, such as herd behavior [11,12,13], inter-species interference [3], and group defense [14] behavior.

Interestingly, none of the above works (with the exception of [3]) accounted for the way interference takes place, despite being expansions of the classical interference competition model. In [2], the authors discussed the matter of (co)existence by stating the following:

“To fully appreciate how such large numbers of species can coexist, we need to have a thorough understanding of the dynamics of their populations and, in particular, of how individuals within a species’ population interact with each other, and with populations of other species. Indeed, the coexistence of individuals within a population is intriguing, given that interactions within species can in fact be stronger than interactions among the populations of different species.”

Also, Ref. [2] pointed out that clustering is a coexistence mechanism: the strongest competitor faces mainly conspecifics, and the same happens to the weaker competitor.

The above ideas strongly suggest that it is important to account for the balance between intra-species interference and inter-species interference in order to better understand species coexistence. We follow the works [15,16] on predator–prey models to fully incorporate the effect of individual interference on species competition into the classical Gause competition model. When individuals of different species want the same resource, it may take some time to “decide” which of them will obtain the resource, as explained in [3]. Also, when a few individuals of both species want the same resource, individuals interfere with not only heterospecifics but also conspecifics. We incorporate the latter feature into the model analyzed in [3].

The new model that we present here displays a different feature from those observed in previous models: the global species coexistence region is larger than in all the preceding models, which reveals that interference with homospecifics is key to coexistence. We found the dynamic scenarios described in [3] (those allowed by the classical model and conditional coexistence in favor of one or the other species and conditional coexistence), along with bi-coexistence in [14], although the biological reasons leading to each dynamic scenario are different. We derived a closed formula describing the conditions for the above-mentioned features in particular but meaningful settings.

The manuscript is organized as follows. In Section 2, we derive in detail the above-mentioned model. In Section 3, we assume that intra- and inter-species interference only affects one of the competing species. In Section 4.1, we consider just intra-species interference in both species. In Section 4.2, the full model is analyzed under symmetric competition assumptions. Note that the analytical description of the full model without further restrictions is algebraically feasible but biologically uninterpretable. Finally, we discuss the mathematical results from an ecological viewpoint in Section 5. The mathematical technicalities are gathered in the Appendix A.

2. The Beddington–DeAngelis Competition Model and Preliminary Results

The base model is nothing but the classical Gause competition model [6]:

$$\left\{\begin{array}{cc}\hfill x_1^{\prime }=& r_1{x}_1-a_{11}{x}_1^2-a_{12}{x}_1{x}_2,\hfill \\ \hfill x_2^{\prime }=& r_2{x}_2-a_{22}{x}_2^2-a_{21}{x}_1{x}_2,\hfill \end{array}\right.$$

(1)

where $x_i$ stands for the number of individuals of species $i=1,2$, $r_i>0$ is the intrinsic growth rate of species $i=1,2$, and $a_{ij}>0$ the coefficient accounting for intra-species ($i=j$) and inter-species ($i\ne j$) competition for $i,j=1,2$. We considered the emergent carrying capacities formulation as in [17,18] rather than the usual carrying capacities [19].

The main feature of the classical model (1) is that the per capita growth rate decreases linearly with $x_i$ and $x_j$ ($i\ne j$), i.e.,

$$\frac{x_i^{\prime }}{x_i}=r_i-a_{ii}{x}_i-a_{ij}{x}_j,i\ne j,i,j=1,2.$$

(2)

As we pointed out in [3,14], the hypothesis that the competitive pressure exerted by a fixed number of individuals of species j on species i is the same regardless of the number of individuals of species i does not make sense in reality. For instance, 100 individuals of species j would not affect in the same way, say, 20 or 2000 individuals of species i. At the same time, in Section 1, we justified the interest in disentangling intra-species from inter-species interference when competing.

This task was addressed by Beddington [15] and DeAngelis [16] for predator–prey models. We followed their works to build a competition model that separates the contribution of inter-species interference as well as mutual interference when competing.

Let us consider species i. In the classical model, species interactions do not take time, so that individuals are active for the whole time interval T considered, i.e., $T=T_{act,i}$. In [3], we assumed that inter-species competition takes time, so that $T>T_{act,i}$ plus the time spent facing species j. We further assume the intra-species interference of individuals of species i when competing with species j. That is to say, a few individuals of each species seek the same resource, and only one individual can obtain it. Then, there is also intra-species interference when disputing a given resource with heterospecifics. Specifically, we partition T as follows:

$$T=T_{act,i}+T_{inte{r}_{ji}}{N}_j+T_{intr{a}_i}\widetilde{N_i},$$

(3)

where $T_{act,i}$ is the time for which each individual of species $i\ne j$ is active; $T_{inte{r}_{ji}}$ and $T_{intr{a}_i}$ stand for the average time taken by an interaction between one individual of each or the same competing species, respectively; and $N_j$ and ${\tilde{N}}_i$ are the numbers of competitors from species $j\ne i$ and i, respectively, that become extinct due to the interference of a single individual of species $i\ne j$. Indeed, $N_j$ can be expressed as follows:

$$N_j=k_j{T}_{act,i}{x}_j$$

(4)

where $x_j$ is the total number of individuals of species j, and $k_j$ is the product of the resource finding rate of species j times the probability of meeting a competitor of species $i\ne j$. Also, the number of encounters between individuals of species $i\ne j$ is given by

$${\tilde{N}}_i=b_i{T}_{act,i}(x_i-1)$$

(5)

where $b_i$ is the rate of encounters between equals within species $i\ne j$ and is related to both their speed of movement and the range at which they sense each other. Note that $x_i-1$ is used instead of $x_i$ since each individual cannot interfere with itself.

Substituting (4) and (5) into (3) and solving the corresponding equation in $T_{act,i}$ yields

$$T_{ac{t}_i}=\frac{T}{1+k_j{T}_{inte{r}_{j}i}{x}_j+b_i{T}_{intr{a}_i}(x_i-1)}j\ne i,j,i=1,2.$$

(6)

Recall that $N_j$ is the number of individuals of j ruled out by a single individual of species i that is given by (4). Then, the total number of individuals of j ruled out by species i is given by

$$N_j{x}_i=\frac{k_{j}T{x}_j{x}_i}{1+k_j{T}_{inte{r}_{j}i}{x}_j+b_i{T}_{intr{a}_i}(x_i-1)}j\ne i,j,i=1,2.$$

(7)

Plugging (7) into System (1) and relabeling the coefficients as $a_{ij}=k_{i}T$, $a_i=k_i{T}_{inte{r}_{i}j}$, and $\widetilde{a_i}=b_i{T}_{intr{a}_i}$ yields

$$\frac{{x_i}^{\prime }}{x_i}=r_i-a_{ii}{x}_i-\frac{a_{ij}{x}_j}{1+a_i{x}_i+\widetilde{a_j}(x_j-1)}i\ne j,i,j=1,2.$$

(8)

Therefore, $a_i$ and ${\tilde{a}}_j$ account for the effect of interference between heterospecifics and homospecifics on species i when both competing species interact.

Remark 1.

Competition is related to but different from interference. In intra-species competition ($a_{ii}$), individuals of the same species interact to obtain a resource in the absence of the other species. In inter-species competition ($a_{ij}$), individuals of different species interact to obtain a resource, and its effect on the other species is density-independent.

Interference models the density-dependent effects of inter-species competition. This takes place, for instance, when few individuals of each species compete for the same resource and face, annoy, disturb, etc., individuals of the other species (${\tilde{a}}_j$) or the same species ($a_i$).

Interference is also different from delayed effects (delay ODEs), as in the latter the effect of competition is felt with a certain delay, but the inter-species competition coefficient can remain density-independent.

We will refer to the term on the right-hand side in Equation (8) as the Beddington–DeAngelis competitive response. Let us rewrite System (8) in a suitable way by setting $u_i=a_{ii}{x}_i/r_i$, $c_{ij}=a_{ij}{r}_j/\left(r_i{a}_{jj}\right)$, and $K_i=r_i/a_{ii}$. This yields

$$\left\{\begin{array}{cc}\hfill u_1^{\prime }=& r_1\left(u_1-u_1^2-{\displaystyle \frac{c_{12}{u}_1{u}_2}{1+a_1{K}_1{u}_1+{\tilde{a}}_2(K_2{u}_2-1)}}\right),\hfill \\ \hfill u_2^{\prime }=& r_2\left(u_2-u_2^2-{\displaystyle \frac{c_{21}{u}_2{u}_1}{1+a_2{K}_2{u}_2+{\tilde{a}}_1(K_1{u}_1-1)}}\right).\hfill \end{array}\right.$$

(9)

Note that $K_i=r_i/a_{ii}>1$, as it is the carrying capacity of species i in the absence of species $j\ne i$. Thus, $u_i$ stands for the ratio $x_i$ to its carrying capacity. Furthermore, $c_{ij}$ is the ratio of the carrying capacity of species j over $a_{ij}/r_i$.

System (1) is a particular case of (9) when $a_i=0={\tilde{a}}_i$. Note that requiring ${\tilde{a}}_i\in [0,1)$ is mandatory to prevent the denominator of Equation (8) from being equal to or less than zero at low population densities. We also assume $a_i\in [0,1]$. Heuristically, on the one hand, the effect of inter-species interference is $a_i{K}_i$, and it should not be larger than the whole carrying capacity. On the other hand, $a_i$ should be comparable to ${\tilde{a}}_j\in [0,1)$. Note that considering the above conditions for the parameter values, the right-hand side of System (9) is smooth, ensuring the existence and uniqueness of solutions for any initial values in the non-negative cone $[0,+\infty )\times [0,+\infty )$.

2.1. First Properties of System (9)

System (9) is well behaved.

Theorem 1.

Consider System (9). Then, the following are true:

1. 

The axes are forward-invariant.

2. 

The solutions are bounded from above.

3. 

The positive cone is forward-invariant.

Proof. 

We just provide an outline of the proof. For statement 1, setting $u_j=0$ reduces System (9) to the logistic equation. Statement 2 follows from the fact that the solutions of System (9) are bounded from above by the solutions of a logistic equation (let $c_{ij}=0$, $i\ne j$). Finally, statement 3 follows from 1 and 2. □

Theorem 2.

Any solution of System (9) eventually converges monotonically to an equilibrium point.

Proof. 

Direct calculations allow us to apply the results in [20]. □

As a consequence, the flow of System (9) strictly decreases outside the rectangle $[0,1]\times [0,1]$ on the positive cone. Thus, there is no equilibrium point for System (9) in $(1,+\infty )\times (1,+\infty )$.

2.2. Remarks on the Classical Competition Model

System (1) is the scaffolding for Systems (8) and (9). Thus, let us present some basic known properties of the classical competition model (1) [19]. We consider System (1) after the change of variables leading to System (9). This is the same as the setting $a_i=0={\tilde{a}}_i$, $i=1,2$ in (9).

The acronyms GAS and LocAS refer to the global and local asymptotic stability of a fixed point, respectively. Let $E_1^{*}=(1,0)$ and $E_2^{*}=(0,1)$ be the so-called semi-trivial equilibrium points that are fixed points of both Systems (1) and (9). It is known that $E_1^{*}$ is GAS if $c_{12}\in (0,1)$ and $c_{21}\in (1,+\infty )$. Also, $E_2^{*}$ is GAS if $c_{12}\in (1,+\infty )$ and $c_{21}\in (0,1)$. Moreover,

$$E^{*}=\left({\displaystyle \frac{1-c_{21}}{1-c_{12}{c}_{21}}},{\displaystyle \frac{1-c_{12}}{1-c_{12}{c}_{21}}}\right)$$

(10)

is an equilibrium point that is in the positive cone if either $0<c_{12},c_{21}<1$ or $c_{12},c_{21}>1$. Indeed, $E^{*}$ is GAS if $0<c_{12},c_{21}<1$. When $c_{12},c_{21}>1$$E^{*}$ becomes unstable. In this case, $E_1^{*}$ and $E_2^{*}$ are LocAS, and the unstable manifold of $E^{*}$ is the separatrix of the corresponding basins of attraction. Figure 1 displays the possible phase portrait configurations leading to each of the competition outcomes. Figure 2 shows the corresponding bifurcation diagram with $c_{12}$ and $c_{21}$ as bifurcation parameters.

The above results point to 1 as a threshold value for comparison with the competitive strength $c_{ij},i\ne j$. Let us recall that the competitive strength

$$c_{ij}=\frac{a_{ij}/r_i}{a_{jj}/r_j}$$

(11)

relates the intra-species and inter-species competition coefficient and species intrinsic growth rates (growth rates must be taken into account to avoid meaningless results [21]). From a different viewpoint, $c_{ij}$ compares the ratio of the competition effect of species j on species i over the intrinsic growth rate of species i to the carrying capacity of species j. Species j cannot drive species i to extinction if, and only if, the competitive strength $c_{ij}$ is less than 1.

We will refer throughout the manuscript to global coexistence when there is a single GAS interior equilibrium point, to coexistence when there is a LocAS interior equilibrium point, and to priority effects when $E_i^{*}$, $i=1,2$ are LocAS and there exists an interior unstable equilibrium point with a separatrix passing through it that defines their basins of attraction.

3. The Beddington–DeAngelis Competitive Response for Just One Species

We analyze in this section the case where only species 2 suffers the effect of interference when competing. We then assume that species 1 does not interfere when competing with species 2, $a_2=0$, and when encountering individuals of species 1, $\widetilde{a_1}=0$. Thus, the effect of the competition of species 2 on species 1 becomes density-dependent. That is to say, we analyze the system

$$\left\{\begin{array}{cc}\hfill u_1^{\prime }=& r_1\left(u_1-u_1^2-{\displaystyle \frac{c_{12}{u}_1{u}_2}{1+a_1{K}_1{u}_1+{\tilde{a}}_2(K_2{u}_2-1)}}\right),\hfill \\ \hfill u_2^{\prime }=& r_2(u_2-u_2^2-c_{21}{u}_1{u}_2).\hfill \end{array}\right.$$

(12)

System (12) is a particular case of System (9), so it is well behaved (see Theorem 1). We first draw conditions for the stability of the semi-trivial equilibrium points.

Theorem 3.

Consider System (12) with $a_1>0$ and ${\tilde{a}}_2>0$ (recall that $K_i>1$). Then, the following are true:

1. 

The trivial equilibrium point $E_0^{*}=(0,0)$ is unstable.

2. 

The semi-trivial equilibrium point $E_1^{*}=(1,0)$ is LocAS if $c_{21}>1$.

3. 

The semi-trivial equilibrium point $E_2^{*}=(0,1)$ is LocAS if

$$c_{12}>1+{\tilde{a}}_2(K_2-1).$$

(13)

Proof. 

This follows by direct calculations and analyzing the sign of the eigenvalues of the Jacobian matrix calculated at $E_0^{*}$ and $E_i^{*}$, $i=1,2$. □

We next focus on the existence and stability of coexistence (positive) equilibrium points. These points are found from the nullclines of System (12), that are given, respectively, by

$$u_2=f_1\left(u_1\right)=\frac{(1-u_1)(1+a_1{K}_1{u}_1-{\tilde{a}}_2)}{c_{12}+K_2{\tilde{a}}_2(u_1-1)},u_2=f_2\left(u_1\right)=1-c_{21}{u}_1.$$

(14)

Note that $u_2=f_2\left(u_1\right)$ is a straight line, as in the classical model (1), and $u_2=f_1\left(u_1\right)$ is the rational function defined in (14).

Equating $f_1\left(u_1\right)=f_2\left(u_1\right)$ yields a second-degree equation:

$${\alpha }_2{u}_1^2+{\alpha }_1{u}_1+{\alpha }_0=0$$

(15)

where

$$\begin{array}{c}{\alpha }_2=a_1{K}_1-{\tilde{a}}_2{c}_{21}{K}_2,\hfill \\ {\alpha }_1=1-{\tilde{a}}_2-a_1{K}_1-c_{12}{c}_{21}+{\tilde{a}}_2{K}_2+{\tilde{a}}_2{K}_2{c}_{21},\hfill \\ {\alpha }_0=-1+{\tilde{a}}_2+c_{12}-{\tilde{a}}_2{K}_2.\hfill \end{array}$$

(16)

It is a trivial task to solve Equation (15) and present conditions for the existence of no, one, or two solutions. However, it is not straightforward to decide whether the solutions belong to the positive cone or not. We perform such an analysis in Appendix A by combining Descartes’s rule of signs with analyzing the discriminant of the solution of Equation (15), and Theorems 4 and 5 fully describe the possible outcomes of System (12). Figure 3 displays the configurations of the nullclines and the phase planes of System (12) that are not allowed by the classical model. Figure 4 shows the possible outcomes of System (12) on the $c_{12}{c}_{21}$ plane.

Theorem 4.

Consider System (12) with $r_i>0$, $a_1>0$, and ${\tilde{a}}_2>0$. Then, the following are true:

1. 

Global coexistence. There exists a unique coexistence equilibrium point $E_3^{*}$ that is globally asymptotically stable if $(c_{12},c_{21})\in (0,1+{\tilde{a}}_2(K_2-1))\times (0,1)$.

2. 

Priority effects. Condition $(c_{12},c_{21})\in (1+{\tilde{a}}_2(K_2-1),\infty )\times (1,\infty )$ entails the existence of a unique coexistence equilibrium point $E_3^{*}$ that is unstable, while $E_1^{*}$ and $E_2^{*}$ are LocAS. Furthermore, $E_3^{*}$ is a saddle, and its stable manifold is a separatrix curve that divides the positive cone into two open regions, $R_1$ and $R_2$, such that $E_3^{*}$ is on its boundary. Any solution with initial values in $R_1$ converges to $E_1^{*}$, while any solution with initial values in $R_2$ converges to $E_2^{*}$.

Proof. 

See Appendix A. □

In what follows, two curves are key to complete the description of the outcomes of System (12). These are the curves $c_{21}={\Psi }_{\pm }\left(c_{12}\right)$, where

$${\Psi }_{\pm }\left(c_{12}\right)=\frac{\begin{array}{c}({\tilde{a}}_2(1+a_1{K}_1+{\tilde{a}}_2(-1+K_2))K_2\\ -c_{12}(-1+{\tilde{a}}_2+a_1{K}_1+{\tilde{a}}_2{K}_2)\end{array}\pm 2\sqrt{\begin{array}{c}{c}_{12}(-1+{\tilde{a}}_2+c_{12}-{\tilde{a}}_2{K}_2)\\ (a_1{K}_1{c}_{12}+{\tilde{a}}_2(-1+{\tilde{a}}_2-a_1{K}_1)K_2))\end{array}}}{{(c_{12}-{\tilde{a}}_2{K}_2)}^2}$$

(17)

These curves arise from equating to zero the discriminant of the solution of Equation (15) and solving for $c_{21}$ in the resulting equation. Given fixed values of $a_1{K}_1$, ${\tilde{a}}_2$, and $K_2$, these curves separate the regions of the $c_{12}{c}_{21}$ plane containing the real and the complex roots of Equation (15).

Lemma 1.

Consider (17). Then, the following are true:

1. 

${\Psi }_{+}\left(c_{12}\right)=0$ at $c_{12}={\displaystyle \frac{{(a_1{K}_1+1+{\tilde{a}}_2(K_2-1))}^2}{4a_1{K}_1}}=c_{12}^{*}$.

2. 

$1+{\tilde{a}}_2(K_2-1)<{\displaystyle \frac{{(a_1{K}_1+1+{\tilde{a}}_2(K_2-1))}^2}{4a_1{K}_1}}$.

3. 

${\Psi }_{\pm }\left(c_{12}\right)=1$ at $c_{12}=a_1{K}_1+1-{\tilde{a}}_2$.

4. 

$a_1{K}_1+1-{\tilde{a}}_2=1+{\tilde{a}}_2(K_2-1)$ if, and only if, $a_1{K}_1={\tilde{a}}_2{K}_2$.

Proof. 

This follows from direct calculations. □

The following result describes the outcomes of System (12) in terms of the relative size of ${\tilde{a}}_2{K}_2$ and $a_1{K}_1$. Refer to Section 4.1 for a biological interpretation.

Theorem 5.

Consider System (12) with $r_i>0$, $a_1>0$, and ${\tilde{a}}_2>0$. Then, the following are true:

1. 

Condition $a_1{K}_1={\tilde{a}}_2{K}_2$ (see the bottom-left central panel in Figure 4) entails the following:

(a) 

Species 2 wins. The equilibrium point $E_1^{*}$ is unstable, and $E_2^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in (1+{\tilde{a}}_2(K_2-1),+\infty )\times (0,1).$$

(18)

(b) 

Species 1 wins. The equilibrium point $E_2^{*}$ is unstable, and $E_1^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in (0,1+{\tilde{a}}_2(K_2-1))\times (1,+\infty ).$$

(19)

2. 

Consider $a_1{K}_1<{\tilde{a}}_2{K}_2$ (see the bottom-right panel in Figure 4). This entails the following:

(a) 

Species 2 wins. The equilibrium point $E_1^{*}$ is unstable, and $E_2^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in (1+{\tilde{a}}_2(K_2-1),+\infty )\times (0,1).$$

(20)

(b) 

Species 1 wins. The equilibrium point $E_2^{*}$ is unstable, and $E_1^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in \left\{\begin{array}{ccc}\hfill 0& <c_{12}<& 1+{\tilde{a}}_2(K_2-1),\hfill \\ \hfill max\left\{1,{\mathsf{\Psi }}_{-}\left(c_{12}\right)\right\}& <c_{21}<& +\infty \hfill \end{array}\right\}$$

(21)

(c) 

Coexistence or species 1 wins. Assume that $(c_{12},c_{21})$ are such that

$$(c_{12},c_{21})\in \left\{\begin{array}{ccc}\hfill a_1{K}_1+1-{\tilde{a}}_2& <c_{12}<& 1+{\tilde{a}}_2(K_2-1)\hfill \\ \hfill \mathcal{1}& <c_{21}<& {\Psi }_{-}\left(c_{12}\right)\hfill \end{array}\right\}.$$

(22)

Then, $E_2^{*}$ is unstable, while $E_1^{*}$ is LocAS. In addition, two coexistence equilibrium points appear in the positive cone. That closer to $E_2^{*}$ is LocAS. The other is a saddle whose stable manifold is a separatrix defining the basins of attraction of the coexistence equilibrium and $E_1^{*}$.

3. 

Assume now that $a_1{K}_1>{\tilde{a}}_2{K}_2$ (see the top row of Figure 4). This entails the following:

(a) 

Species 1 wins. The equilibrium point $E_2^{*}$ is unstable, and $E_1^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in (0,1+{\tilde{a}}_2(K_2-1))\times (1,+\infty ).$$

(23)

(b) 

Species 2 wins. The equilibrium point $E_1^{*}$ is unstable, and $E_2^{*}$ is GAS to the positive cone for any

$$(c_{12},c_{21})\in \left\{\begin{array}{ccc}\hfill c_{12}^{*}& <c_{12}<& +\infty \hfill \\ \hfill 0& <c_{21}<& {\Psi }_{+}\left(c_{12}\right)\hfill \end{array}\right\}$$

(24)

where $c_{12}^{*}$ is defined in Lemma 1.

(c) 

Coexistence or species 2 wins. Assume that $(c_{12},c_{21})$ are such that

$$(c_{12},c_{21})\in \left\{\begin{array}{ccc}\hfill 1+{\tilde{a}}_2(K_2-1)& <c_{12}<& a_1{K}_1+1-\widetilde{a_2}\hfill \\ \hfill max\left\{0,{\Psi }_{+}\left(c_{21}\right)\right\}& <c_{21}<& 1\hfill \end{array}\right\}.$$

(25)

Then, $E_1^{*}$ is unstable, while $E_2^{*}$ is LocAS. In addition, two coexistence equilibrium points appear in the positive cone. That closer to $E_1^{*}$ is LocAS. The other is a saddle whose stable manifold is a separatrix defining the basins of attraction of the coexistence equilibrium and $E_2^{*}$.

Proof. 

See Appendix A. □

We leave the biological interpretation and discussion of the above results for Section 5.3.1.

4. The Beddington–DeAngelis Competitive Response for Both Species

In this section, we focus on the complete model (9). We already know that System (9) is well behaved (see Theorem 1), and we seek the existence and stability of the positive equilibrium points, which are closely related to the stability of the semi-trivial equilibrium points.

Theorem 6.

Consider System (9) with $r_i>0$, $i=1,2$. This entails the following:

1. 

The trivial equilibrium point $E_0^{*}=(0,0)$ is unstable.

2. 

Consider the semi-trivial equilibrium points $E_j^{*}$, $j=1,2$. Then, the following are true:

(a) 

$E_j^{*}$ is LocAS if

$$c_{ij}>1+{\tilde{a}}_j(K_j-1).$$

(26)

(b) 

$E_j^{*}$ is unstable if

$$c_{ij}<1+{\tilde{a}}_j(K_j-1).$$

(27)

Proof. 

This follows by accounting for the sign of the eigenvalues of the Jacobian matrix calculated at $E_0^{*}$ and $E_j^{*}$, $j=1,2$. □

As for the non-trivial equilibrium points of system (9), one needs to find the intersection points of the nullclines, given by $u_2=f_1\left(u_1\right)$ in (14) and

$$u_1=f_2\left(u_2\right)=\frac{(1-u_2)(1+a_2{K}_2{u}_2-{\tilde{a}}_1)}{c_{21}+K_1{\tilde{a}}_1(u_2-1)}.$$

(28)

Thus, the positive equilibrium points are given by the positive roots of

$$0=f_2\left(f_1\left(u_1\right)\right)=(u_1-1)(1-{\tilde{a}}_2+a_1{K}_1{u}_1)\sum _{k=0}^4{\gamma }_k{u}_1^k=0,$$

(29)

where the explicit expression of the coefficients can be found in Appendix B. The study of (29) is feasible, but the conditions enabling one outcome or another are extremely numerous and not meaningful at all.

Thus, we deal with two particular but interesting cases. The study will be completed with numerical simulations.

4.1. Considering Only Intra-Species Interference

We now let $a_i=0$ and $i=1,2$ in System (9), so that we neglect the effect of inter-species interference when competing and focus on the role of intra-species interference. Some of the results in Section 4.1 were published in [22]. We include herein the full version for the sake of completeness.

Theorem 6 deals with the semi-trivial equilibrium points. We next focus on the non-trivial equilibrium points.

Theorem 7.

Consider System (9) with $a_i=0$. Then, the following are true:

1. 

Global coexistence. There exists an equilibrium point in the non-negative cone that is GAS for any

$$(c_{12},c_{21})\in \left(0,1+{\tilde{a}}_2(K_2-1)\right)\times \left(0,1+{\tilde{a}}_1(K_1-1)\right).$$

(30)

2. 

Priority effects. There exists a saddle equilibrium point in the non-negative cone that is unstable for any

$$(c_{12},c_{21})\in \left(1+{\tilde{a}}_2(K_2-1),+\infty \right)\times \left(1+{\tilde{a}}_1(K_1-1),+\infty \right).$$

(31)

Indeed, $E_1^{*}$ and $E_2^{*}$ are locally asymptotically stable, and the stable manifold of the positive (component-wise) equilibrium defines the basins of attraction of each semi-trivial equilibrium point.

Proof. 

The non-trivial equilibrium points are the solutions to the equation resulting from equating the nullclines of System (9). Locating the equilibrium points in the positive cone follows the reasoning in Appendix A. The stability of the non-trivial equilibrium points follows from Theorem 2 and the stability conditions of the semi-trivial equilibrium points proved in Theorem 6. □

The feasible non-trivial equilibrium points are the positive roots of the polynomial

$${\alpha }_2{u}_1^2+{\alpha }_1{u}_1+{\alpha }_0=0$$

(32)

where

$${\alpha }_0=-({\tilde{a}}_1-1)({\tilde{a}}_2+c_{12}-{\tilde{a}}_2{K}_2-1)$$

(33)

$$\begin{array}{cc}\hfill {\alpha }_1=& {\tilde{a}}_1{\tilde{a}}_2-{\tilde{a}}_2-{\tilde{a}}_1+{\tilde{a}}_1{K}_1(-1+{\tilde{a}}_2+c_{12})\hfill \\ \hfill & +{\tilde{a}}_2{K}_2(1-{\tilde{a}}_1+c_{21})-{\tilde{a}}_1{\tilde{a}}_2{K}_1{K}_2+1-c_{12}{c}_{21}\hfill \end{array}$$

(34)

$${\alpha }_2={\tilde{a}}_1{K}_1(1-{\tilde{a}}_2)-{\tilde{a}}_2{K}_2(c_{21}-{\tilde{a}}_1{K}_1)$$

(35)

Solving the discriminant of Equation (32) for $c_{21}$ yields two curves, $c_{21}={\psi }_{\pm }\left(c_{12}\right)$, that separate the regions in the $c_{12}$-$c_{21}$ plane leading to no, one, or two equilibrium points.

Lemma 2.

Consider the curves $c_{21}={\psi }_{\pm }\left(c_{12}\right)$ defined above. This entails the following:

1. 

${\psi }_{+}\left(c_{12}\right)=1+{\tilde{a}}_1(K_1-1)\iff c_{12}=(1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}$.

2. 

${\psi }_{\pm }(1+{\tilde{a}}_2(K_2-1))=(1+{\tilde{a}}_2(K_2-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}$.

3. 

If

$${\displaystyle \frac{{\tilde{a}}_2{K}_2}{{\tilde{a}}_1{K}_1}}={\displaystyle \frac{1-{\tilde{a}}_2}{1-{\tilde{a}}_1}},$$

(36)

then

$${\psi }_{-}(1+{\tilde{a}}_2(K_2-1))={\psi }_{+}(1+{\tilde{a}}_2(K_2-1))=1+{\tilde{a}}_1(K_1-1).$$

4. 

If

$${\displaystyle \frac{{\tilde{a}}_2{K}_2}{{\tilde{a}}_1{K}_1}}<{\displaystyle \frac{1-{\tilde{a}}_2}{1-{\tilde{a}}_1}},$$

(37)

then

$$(1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}>1+{\tilde{a}}_2(K_2-1).$$

5. 

If

$${\displaystyle \frac{{\tilde{a}}_2{K}_2}{{\tilde{a}}_1{K}_1}}>{\displaystyle \frac{1-{\tilde{a}}_2}{1-{\tilde{a}}_1}},$$

(38)

then

$$(1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}<1+{\tilde{a}}_2(K_2-1).$$

Proof. 

This follows from direct calculations. □

Theorem 8.

Consider System (9) with $a_i=0$. Then, consider the following:

1. 

Assume now that condition (36) holds. This entails the following (see the bottom-left panel in Figure 5):

(a) 

Species 1 wins. $E_1^{*}$ is GAS whenever

$$(c_{12},c_{21})\in {\mathcal{R}}_1:=\left\{\begin{array}{ccc}\hfill 0& <c_{12}<& 1+{\tilde{a}}_2(K_2-1),\hfill \\ \hfill 1+{\tilde{a}}_1(K_1-1)& <c_{21}<& +\infty \hfill \end{array}\right\}$$

(39)

(b) 

Species 2 wins. $E_2^{*}$ is GAS whenever

$$(c_{12},c_{21})\in {\mathcal{R}}_2:=\left\{\begin{array}{ccc}\hfill 1+{\tilde{a}}_2(K_2-1)& <c_{12}<& +\infty ,\hfill \\ \hfill 0& <c_{21}<& 1+{\tilde{a}}_1(K_1-1)\hfill \end{array}\right\}$$

(40)

2. 

Instead, if condition (37) holds, this entails the following (see the panels in the first row in Figure 5):

(a) 

Species 1 wins. $E_1^{*}$ is GAS if $(c_{12},c_{21})\in {\mathcal{R}}_1$, as defined in (39).

(b) 

Now, two different outcomes may occur in ${\mathcal{R}}_2$:

i. 

Species 2 wins. $E_2^{*}$ is GAS whenever $(c_{12},c_{21})\in {\mathcal{A}}_1\cup {\mathcal{A}}_2$, where

$$\begin{array}{c}{\mathcal{A}}_1:=\left\{\begin{array}{ccc}\hfill 1+{\tilde{a}}_2(K_2-1)& <c_{12}<& (1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}\hfill \\ \hfill 0& <c_{21}<& {\psi }_{+}\left(c_{12}\right)\hfill \end{array}\right\}\\ \\ {\mathcal{A}}_2:=\left((1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}},+\infty \right)\times (0,1+{\tilde{a}}_1(K_1-1))\end{array}$$

(41)

ii. 

Conditional coexistence or species 2 wins. On the contrary, if $(c_{12},c_{21})\in {\mathcal{A}}_3$, where

$$\begin{array}{c}{\mathcal{A}}_3:=\left\{\begin{array}{ccc}\hfill 1+{\tilde{a}}_2(K_2-1)& <c_{12}<& (1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}\hfill \\ \hfill {\psi }_{+}\left(c_{12}\right)& <c_{21}<& 1+{\tilde{a}}_1(K_1-1)\hfill \end{array}\right\}\end{array}$$

(42)

then $E_2^{*}$ is locally asymptotically stable and $E_1^{*}$ is unstable. In addition, there exist two equilibrium points in the positive cone, one locally asymptotically stable and one unstable. The latter is a saddle equilibrium point whose stable manifold separates the basins of attraction of $E_2^{*}$ and the positive (coexistence) equilibrium point.

3. 

Finally, if condition (38) holds, this entails the following (see the bottom-right panel in Figure 5):

(a) 

Species 2 wins. $E_2^{*}$ is GAS if $(c_{12},c_{21})\in {\mathcal{R}}_2$, as defined in (40).

(b) 

Two different outcomes may occur in ${\mathcal{R}}_1$:

i. 

Species 1 wins. $E_1^{*}$ is GAS whenever $(c_{12},c_{21})\in {\mathcal{A}}_4\cup {\mathcal{A}}_5$, where

$$\begin{array}{c}{\mathcal{A}}_4:=\left\{\begin{array}{ccc}\hfill (1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}& <c_{12}<& 1+{\tilde{a}}_2(K_2-1)\hfill \\ \hfill {\psi }_{-}\left(c_{12}\right)& <c_{21}<& +\infty \hfill \end{array}\right\}\\ \\ {\mathcal{A}}_5:=\left(0,(1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}\right)\times (1+{\tilde{a}}_1(K_1-1),+\infty )\end{array}$$

(43)

ii. 

Coexistence or species 1 wins. On the contrary, if

$${\mathcal{A}}_6:=\left\{\begin{array}{ccc}\hfill (1+{\tilde{a}}_1(K_1-1)){\displaystyle \frac{1-{\tilde{a}}_1}{1-{\tilde{a}}_2}}& <c_{12}<& 1+{\tilde{a}}_2(K_2-1)\hfill \\ \hfill 1+{\tilde{a}}_1(K_1-1)& <c_{21}<& {\psi }_{-}\left(c_{12}\right)\hfill \end{array}\right\}$$

(44)

then $E_1^{*}$ is locally asymptotically stable and $E_2^{*}$ unstable. In addition, there exist two equilibrium points in the positive cone, one locally asymptotically stable and one unstable. The latter is a saddle equilibrium point whose stable manifold separates the basins of attraction of $E_1^{*}$ and the positive (coexistence) equilibrium point.

Proof. 

When equating the nullclines of System (9) with $a_i=0$ for $i=1,2$, we obtain a second-degree equation for $u_1$. The solution of such an equation is the $u_1$ component of the equilibrium points of System (9). Setting the discriminant of the solution of that equation equal to zero, the curves ${\Psi }_{+}$ and ${\Psi }_{-}$ are obtained. These curves bound the regions on the $c_{21}{c}_{12}$ plane where there are two, one, or no equilibrium points (that is, the algebraic equation has either real or complex solutions).

The signs of the coordinates of the equilibrium points are determined by using Descartes’ rule of signs. The number of equilibrium points inside the non-negative cone, in addition to the stability of the semi-trivial equilibrium points (Theorem 2), yields the stability of the non-trivial equilibrium points. □

4.2. Symmetric Competition

Symmetric competition may take place, for instance, when individuals of different species display similar phenotypic traits [23]. We introduce this feature in System (9) by setting

$$c_{ij}:=c_{12}=c_{21},a:=a_1=a_2,K:=K_1=K_2,\tilde{a}:={\tilde{a}}_1={\tilde{a}}_2.$$

(45)

Assumption (45) is a strong one. We are interested in the qualitative outcomes of the model. Note that the stability of hyperbolic equilibrium points is robust under small (enough) perturbations in the parameters of the system. Thus, the results achieved also hold for coefficients close to the perfect symmetry settings stated above.

A first consequence of the symmetry assumptions (45) is that there exists either one equilibrium or three equilibrium points in the positive cone. In the former case, there is either coexistence or priority effects. In the latter case, the outcomes are either global coexistence through two positive stable equilibrium points or tri-stability (coexistence or priority effects).

The symmetry conditions (45) simplify the fixed-point Equation (29) so that a computer algebra system (we used the commercial software Mathematica [24]) can find the explicit expression of the solutions:

$$E_3^{*}={\alpha }_3-\sqrt{{\beta }_3},E_4^{*}={\alpha }_3+\sqrt{{\beta }_3},$$

(46)

$$E_5^{*}={\alpha }_5-\sqrt{{\beta }_5},E_6^{*}={\alpha }_5+\sqrt{{\beta }_5},$$

(47)

where

$${\alpha }_3={\displaystyle \frac{-1+\tilde{a}(1+K)+aK-c_{12}}{2(aK+\tilde{a}K)}},$$

$${\beta }_3={\displaystyle \frac{{(1-\tilde{a}(1+K)-aK+c_{12})}^2-4(-1+\tilde{a})(aK+\tilde{a}K)}{4{(aK+\tilde{a}K)}^2}},$$

and

$${\alpha }_5={\displaystyle \frac{\tilde{a}+aK+c_{12}-\tilde{a}K-1}{2aK}},$$

$${\beta }_5={\displaystyle \frac{\begin{array}{c}(a-\tilde{a}){(\tilde{a}(1-K)+aK+c_{12}-1)}^2-\\ 4a(aK+c_{12}-\tilde{a}K)(\tilde{a}(1-K)+c_{12}-1)\end{array}}{4a^2(a-\tilde{a})K^2}},$$

Note that $\tilde{a}\in [0,1)$ implies ${\beta }_3>0$ so that $E_3^{*}$ and $E_4^{*}$ always exist. Indeed, both lie on the line $u_1=u_2$: $E_3^{*}$ on the positive semi-line, and $E_4^{*}$ on the negative side. $E_5^{*}$ and $E_6^{*}$ can be either complex or real and, in the latter case, belong to any quadrant of the real plane. Thus, we seek conditions that attribute these equilibrium points to the positive cone and describe their stability. The system depends on a, $\tilde{a}$, K, and $c_{ij}$. To control the sign of ${\beta }_5$, we solve ${\beta }_5=0$ in $c_{ij}$, which yields

$$c_{ij}=\mathsf{\Phi }(\tilde{a},a,K)$$

(48)

where

$$\mathsf{\Phi }(\tilde{a},a,K):=\frac{\begin{array}{c}a+\tilde{a}+{\tilde{a}}^2(K-1)\\ -a^{2}K+a\tilde{a}(2K-1)\end{array}+\sqrt{\begin{array}{c}4a^2({(\tilde{a}-1)}^2\\ -(\tilde{a}-1)(a+\tilde{a})K+a^2{K}^2\left)\right)\end{array}}}{3a+\tilde{a}}$$

It will turn out that comparing $1+\tilde{a}(K-1)$, $c_{ij}$, and $\mathsf{\Phi }$ yields a full description of the dynamics of the symmetric competition model. Consider the parameter plane $\mathsf{\Pi }$ with $c_{ij}$ and $1+\tilde{a}(K-1)$ in the horizontal and vertical axes, respectively. We next show that the curve $\mathsf{\Phi }$ as defined in (48) and the bisector $c_{ij}=1+\tilde{a}(K-1)$ divides $\mathsf{\Pi }$ into four regions (for $\tilde{a}>0$, $a>0$). For this purpose, we need the following lemma.

Lemma 3.

Consider Φ as defined in (48) along with $\tilde{a},a\in [0,1]$, $K>1$. Then, the following are true.

1. 

${\displaystyle \frac{d\mathsf{\Phi }(\tilde{a},a,K)}{d\tilde{a}}}>0$, so that Φ is strictly increasing.

2. 

$\mathsf{\Phi }(\tilde{a},0,K)=1+\tilde{a}(K-1)$.

3. 

Thus, ${\left.\mathsf{\Phi }(\tilde{a},0,K)\right|}_{\tilde{a}=0}=1$.

4. 

$\mathsf{\Phi }(0,a,K)>1$ for any $a\in (0,1)$.

5. 

$\mathsf{\Phi }(1,a,K)<K$ for any $a\in (0,1)$.

6. 

$\mathsf{\Phi }(\tilde{a},a,K)=1+\tilde{a}(K-1)\iff \tilde{a}=a$. That is, Φ crosses the bisector at $\tilde{a}=a$, which entails that the crossing point is $\left(1+a(K-1),1+a(K-1)\right)$.

Proof. 

This follows from direct calculations. □

Lemma 3 implies that the curve $c_{ij}=\mathsf{\Phi }(\tilde{a},a,K)$ and the bisector $c_{ij}=1+\tilde{a}(K-1)$ are the same at $a=0$. Furthermore, for $a>0$, the curve $\mathsf{\Phi }$ increases (in $\tilde{a}$) faster than the bisector, and they cross at $\tilde{a}=a$, thus dividing the square $[0,K]\times [1,K]$ into the above-mentioned four regions ${\mathsf{\Omega }}_i$, $i=1,2,3,4$. See Figure 6.

Theorem 9.

Consider System (9) along with the symmetry conditions (45). Consider fixed values of K and a. Then, the following are true:

1. 

The equilibrium point $E_3^{*}$ is always in the positive cone and on the bisector of the positive cone.

2. 

Global coexistence. For any $\tilde{a}$ such that $\left(c_{ij},1+\tilde{a}(K-1)\right)\in {\mathsf{\Omega }}_1$, the semi-trivial equilibrium points $E_1^{*}$ and $E_2^{*}$ are unstable. The equilibrium $E_3^{*}$ is a global attractor to the positive cone.

3. 

Priority effects. For any $\tilde{a}$ such that $\left(c_{ij},1+\tilde{a}(K-1)\right)\in {\mathsf{\Omega }}_2$, the semi-trivial equilibrium points $E_i$ for $i=1,2$ are asymptotically stable and $E_3^{*}$ is unstable (a saddle). The stable manifold of $E_3^{*}$ determines the basins of attraction of the semi-trivial equilibrium points.

4. 

Global bi-coexistence. For any $\tilde{a}$ such that $\left(c_{ij},1+\tilde{a}(K-1)\right)\in {\mathsf{\Omega }}_3$, the semi-trivial equilibrium points $E_1^{*}$ and $E_2^{*}$ are unstable. There exist positive equilibrium points $E_3^{*}$, $E_5^{*}$, and $E_6^{*}$ in the positive cone. $E_3^{*}$ is unstable (a saddle). The latter two are asymptotically stable and located symmetrically with respect to the bisector. The stable manifold of $E_3^{*}$ determines the basins of attraction of $E_5^{*}$ and $E_6^{*}$. See the bottom-left panel in Figure 7 and the right panel in Figure 8.

5. 

Coexistence or priority effects. For any $\tilde{a}$ such that $\left(c_{ij},1+\tilde{a}(K-1)\right)\in {\mathsf{\Omega }}_4$, the semi-trivial equilibrium points $E_1^{*}$, $E_2^{*}$, and $E_3^{*}$ are locally asymptotically stable. There exist another two equilibrium points $E_5^{*}$ and $E_6^{*}$ in the positive cone, which are unstable (saddles) and located symmetrically with respect to the bisector. The stable manifolds of $E_5^{*}$ and $E_6^{*}$ determine the basins of attraction of $E_1^{*}$, $E_2^{*}$, and $E_3^{*}$. See the bottom-right panel in Figure 7 and the left panel in Figure 8.

Proof. 

Statement 1 follows from the fact that $E_3^{*}={\alpha }_3+\sqrt{{\beta }_3}$, as defined in (46), and $|{\alpha }_3|<|{\beta }_3^2|$.

As for statements 2 to 5, we recall that the stability of the semi-trivial equilibrium points is related to the balance between $c_{ij}$ and $1+{\tilde{a}}_i(K_i-1)$; see Theorem 6. Thus, for $\left(c_{ij},1+\tilde{a}(K-1)\right)$ such that $c_{ij}<1+\tilde{a}(K-1)$ (that is, in ${\mathsf{\Omega }}_1\cup {\mathsf{\Omega }}_2$), there is global species coexistence, and, on the contrary, priority effects are possible if $c_{ij}>1+\tilde{a}(K-1)$ (that is, in ${\mathsf{\Omega }}_3\cup {\mathsf{\Omega }}_4$).

Indeed, note that the graph of $\mathsf{\Phi }$ describes the combination of parameters such that $E_3^{*}$, $E_5^{*}$, and $E_6^{*}$ collide, entailing a pitchfork bifurcation. That is to say that $E_3^{*}$ gains/loses local stability as $E_5^{*}$ and $E_6^{*}$ lose/gain local stability. Direct calculations (meaning aided by Mathematica [24]) showed that $E_5^{*}$ and $E_6^{*}$ exist and belong to the positive cone for values of $\left(c_{ij},1+\tilde{a}(K-1)\right)\in {\mathsf{\Omega }}_3\cup {\mathsf{\Omega }}_4$. Thus, outside these regions (i.e., in ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$), $E_3^{*}$ is the only equilibrium point in the positive cone. With the system being competitive (sensu [20]), the stability of the semi-trivial equilibrium points yields statements 2 and 3.

Also, by applying the topological index (see Theorem 4.2, statement 2(a) in [3]) along with a symmetry argument, the stability of the semi-trivial equilibrium points again yields statements 4 and 5. □

4.3. Numerical Simulations

Figure 8 displays two relevant bifurcation diagrams and phase portraits computed for the full model and not found under the settings presented up to now. The left panel displays simultaneously the coexistence or species 1 (or 2) wins regions along with the conditional coexistence (dark gray) region. Also, the right panel in Figure 8 displays together the coexistence or species 2 wins region and the global bi-coexistence region (dark green, tiny region close to (1.5, 1.5) within the global coexistence region).

Figure 8. Bifurcation diagrams, with $c_{12}$ and $c_{21}$ being the bifurcation parameters, and phase portraits with several orbits (in red). The color code is as in Figure 2 and Figure 6. Furthermore, the left panel shows in dark gray the coexistence or one species extinction scenario that is illustrated by the corresponding phase portrait. The right panel shows in dark green the bi-coexistence scenario along with the corresponding phase portrait. Parameter values of the phase portraits: left, $r_1=1$, $c_{12}=1.6$, $c_1=0.5$, ${\tilde{a}}_2=0.52$$K_1=2.2$; right, $r_2=1$, $c_{21}=1.35$, $c_2=0.5$, ${\tilde{a}}_1=0.6$, $K_2=1.6$.

Figure 8. Bifurcation diagrams, with $c_{12}$ and $c_{21}$ being the bifurcation parameters, and phase portraits with several orbits (in red). The color code is as in Figure 2 and Figure 6. Furthermore, the left panel shows in dark gray the coexistence or one species extinction scenario that is illustrated by the corresponding phase portrait. The right panel shows in dark green the bi-coexistence scenario along with the corresponding phase portrait. Parameter values of the phase portraits: left, $r_1=1$, $c_{12}=1.6$, $c_1=0.5$, ${\tilde{a}}_2=0.52$$K_1=2.2$; right, $r_2=1$, $c_{21}=1.35$, $c_2=0.5$, ${\tilde{a}}_1=0.6$, $K_2=1.6$.

5. Disentangling Interference: Discussion

In the following, we discuss three ideas that gather the main results obtained from the study of System (9): (i) the impact of interference when competing on global coexistence, (ii) the competition outcomes yielded by intra-/inter-species interference, and (iii) the trade-off between intra-/inter-species interference.

Global species coexistence in the classical competition model (1) as well as recent expansions [3,11,12,13,14] is driven by the so-called competitive strengths $c_{ij}$, $i\ne j$ defined by (11). Thus, we discuss our results in the $c_{12}{c}_{21}$ plane.

5.1. Homospecific Interference Enhances Global Coexistence

The main finding is that considering homospecific interference enlarges the chance of global coexistence in the competitive strength parameter space, from the square region

$$(c_{12},c_{21})\in (0,1)\times (0,1)$$

(49)

found in the classical model [6,19] as well as in [3,13,14,25] to the rectangular region

$$(c_{12},c_{21})\in \left(0,1+{\tilde{a}}_2(K_2-1)\right)\times \left(0,1+{\tilde{a}}_1(K_1-1)\right).$$

(50)

See, for instance, Figure 8. This feature is exclusively due to considering homospecific interference, as shown in Section 4.1.

This fact is closely related to the usual interpretation of the classical competition model (1): coexistence depends on the balance between intra-/inter-species competition pressure. If intra-species competition pressure is stronger ($c_{ij}<1$), a species that survives alone would not be much affected by a competitor [9,26] and cannot be removed.

In line with this, both inter- and intra-species interference diverts efforts in the inter-species competition term. Interestingly, considering just inter-species interference (as in [3]) leaves the global coexistence region the same as in the classical model.

5.2. Competition Outcomes Induced by Interference

Remarkably, there are other competition outcomes apart from those allowed by the classical model (global coexistence, one species extinction, or priority effects):

• Coexistence or one of the species wins; see Section 3, Figure 3.
• Global bi-coexistence; see item 4 in Theorem 9 and Figure 6. This is also possible under non-symmetric settings; see Figure 8.
• Coexistence or priority effects; see item 5 in Theorem 9 and Figure 6. This may also happen under non-symmetric settings; see Figure 8.

Dynamically, the above-mentioned scenarios appear in previous models, such as [3,13] or [14] (only the latter exhibits all of them). However, the conditions (and the biological reasons) leading to one or another are rather different, as discussed in the following Section 5.3.

Interestingly, bi-stable global coexistence is not allowed when considering only inter-species interference [3] under symmetric competition or asymmetric competition settings (also in the general case). Instead, it is allowed under symmetric competition settings when the group defense strategy operates as in [14], but not when herd behavior (a form of group defense) is active [13].

5.3. The Trade-Off between Intra-Species and Inter-Species Interference

Interference when competing is reflected in the denominator of the species’ interaction term (9), so that the interaction coefficient becomes density-dependent. This feature relaxes the strength of species interactions. However, intra-species and inter-species interference lead to different outcomes, which we discuss next under different settings.

5.3.1. Beddington–DeAngelis Competitive Response in Species 1

In Section 5.1, we pointed out that intra-species interference makes extinction depend in the first instance on the value of $c_{ij}$ compared to $1+{\tilde{a}}_j(K_j-1)$ (note that in this section we assume that ${\tilde{a}}_1=0=a_2$). However, the situation is not so simple, and $a_1{K}_1$ and ${\tilde{a}}_2{K}_2$ play a key role.

• Assume that

  $$1+{\tilde{a}}_2(K_2-1)>c_{12}\mathrm{and}0<c_{21}<1.$$

  (51)
  In the absence of inter-species interference, $E_2^{*}$ is globally asymptotically stable. However, if $a_1{K}_1>{\tilde{a}}_2{K}_2>0$ (the effect of inter-species interference is stronger than the effect of intra-species interference), a new region appears where species can either coexist or species 2 wins depending on the initial values; see the top-row panels in Figure 4. An equivalent but smaller region was found in the Holling II competition model [3] (see the last paragraph before Section 4.2 for further details). Interestingly, $0<a_1{K}_1\le {\tilde{a}}_2{K}_2$ has no effect when Condition (51) holds.
• Consider now that

  $$0<c_{12}<1+{\tilde{a}}_2(K_2-1)\mathrm{and}1<c_{21}.$$

  (52)
  In the absence of inter-species interference, $E_1^{*}$ is globally asymptotically stable. However, if $0<a_1{K}_1<{\tilde{a}}_2{K}_2$ (the effect of inter-species interference is weaker than the effect of intra-species interference), a new region appears where there is either coexistence or species 1 wins; see the bottom-right panel in Figure 4. This region does not exist in either the Holling II competition model [3] or the group defense [14] or herd behavior [13] competition models. As with the previous item, $a_1{K}_1\ge {\tilde{a}}_2{K}_2>0$ has no effect when Condition (52) holds.
• In the intermediate case $0<a_1{K}_1={\tilde{a}}_2{K}_2$, the regions where either coexistence occurs or one species excludes the other one vanish; see the bottom-left panel in Figure 4.
• Furthermore, as ${\tilde{a}}_2$ increases, the species 1 wins and the global coexistence regions expand, while the priority effects and species 2 wins regions contract.

Note that at ${\tilde{a}}_2=0$, System (12) reduces to the Holling II competition model with interference on species 1 addressed in [3]. Seen from left to right, top to bottom, the panels in Figure 4 display how the competition outcomes evolve from the above-mentioned Holling II competition model (${\tilde{a}}_2=0$) to the Beddington–DeAngelis competition model with mild intra-species interference ($a_1{K}_1>\widetilde{a_2}{K}_2$), and finally to the case of strong intra-species interference ($a_1{K}_1<\widetilde{a_2}{K}_2$).

5.3.2. Beddington–DeAngelis Competitive Response in Both Species

When considering intra-species and inter-species interference in both species without further constraints, we could not derive a reasonable expression. Numerical simulations revealed that an appropriate balance between each intra-/inter-species interference allows the existence of conditional coexistence in favor of both species simultaneously (see the left panel in Figure 8). However, under symmetric competition assumptions (see (45)), we obtained a full description. Section 4.2 gathers the Theorem-like conditions, and Section 5.3.3 offers a biological interpretation of the results.

5.3.3. Insights on the Symmetric Competition Case

Assuming exact symmetric competition conditions (45) reduces the possible competition outcomes to global coexistence, global bi-coexistence, conditional coexistence or priority effects, or just priority effects; see Section 5.2 for a description of each scenario. Let us recall that in this case the coefficients of the model are given by (45).

Coexistence for the competitive strength ranges in the square (50) takes the form of a globally asymptotically stable equilibrium point. However, global coexistence is also possible in the form of two locally asymptotically stable equilibrium points, whose basins of attraction are separated by the stable manifold of a third (unstable) equilibrium point. Therefore, the species will coexist, but the stable distribution of individuals among species will depend on the initial number of individuals. Also, a sudden (externally forced) change in the number of individuals of each species may lead the community toward a different (coexistence) state.

This feature was previously observed in the context of group defense [14], but surprisingly not in competition models with herd behavior [11,13], which is another type of defense strategy.

As in [14], the global bi-coexistence region is also located inside the coexistence region.

Interestingly, System (9) with symmetric competition conditions (45) allows for coexistence or priority effects, precluding those scenarios where one of the species excludes the other one regardless of the initial value.

Figure 7 displays the configuration of the above-mentioned regions. The regions enabling global coexistence (left, in light green) and priority effects (right, in light gray) are separated by two other regions. One of them leads to global bi-coexistence (central-bottom region, in dark green), and the other leads to coexistence or priority effects (central-top region, in dark gray). We next interpret the bifurcation diagram in Figure 7 in terms of $\tilde{a}$, a, K, and $c_{ij}$ (see (45)).

Note that the horizontal axis is straight $c_{ij}$, but the vertical axis is $1+\tilde{a}(K-1)$; we assume that K is fixed and consider $\tilde{a}$ as a bifurcation parameter, which is the reason why the vertical axis starts at 1 (corresponding to $\tilde{a}=0$):

• For an a value close to 0, only a low level of intra-species interference $\tilde{a}$ allows for a global bi-stable coexistence transition region between global coexistence and priority effects. Instead, the bi-stable region is replaced by conditional coexistence for large values of $\tilde{a}$. The bi-stable region is smaller than the conditional coexistence one (see the top-right panel in Figure 7).
• As a increases, the global bi-stable coexistence transition region increases, while the conditional coexistence region decreases (compare the top-right panel to the bottom-left panel in Figure 7).
• For an a value close to 1, the transition bi-coexistence region is wider, and the lower the $\tilde{a}$ value, the larger the bi-coexistence region and the smaller the coexistence or exclusion region.
• In the extreme cases, for $a=0$, there is no transition from coexistence to priority effects (see top-left panel in Figure 7). This configuration was observed in the classical competition model under symmetric competition (see Figure 8 in [3]) and in the competition model for sessile individuals [25].

Note that the conditional coexistence or priority effects scenario has also been found under the herd behavior strategy [13]. This model replaces the classical interaction term $c_{ij}{u}_i{u}_j$ by $c_{ij}\sqrt{u_i}\sqrt{u_j}$, which entails a reduction in the inter-species competitive pressure, which becomes too weak to enable the global bi-coexistence scenario. Also, the stronger group defense strategy analyzed in [14] yields all the competition outcomes obtained in this work and a few more.

Additionally, as expected, small changes in the parameter values close to the boundaries of the different regions of the bifurcation diagrams may drastically change the fate of the community. For instance, see the squared area in Figure 8 (as well as Section 5 in [14]) for other related effects.

6. Final Conclusions and Future Work

We summarize next the main ideas drawn from this research and propose further research:

• The classical rule that sets $c_{ij}=1$ as a threshold value for the persistence or exclusion of each species does not hold when considering intra-species interference.
• Indeed, intra-species interference was found to be the only factor responsible for the expansion of the global coexistence region in the competitive strength’s parameter space.
• The competition outcomes obtained when considering group defense and herd behaviour also arise when, instead, intra- and inter-species interference is considered.

We hope that the results found in this paper stimulate investigation. On the one hand, the idea of species interference (regardless of the type—Holling II or Beddington–DeAngelis) is based on the idea of “time spent in”. An alternative approach is that of introducing delays in some terms of the system [27]. On the other hand, ecologists are aware of how important interference is [28] in species competition, and the system presented here constitutes a robust modelization tool ready to be tested with ecological hypotheses.