A predator–prey model with generic birth and death rates for the predator and Beddington–DeAngelis functional response

Abstract

We study a predator–prey model with a Beddington–DeAngelis response function and generic birth and death rates for the predator. The mathematical analysis of the model includes existence and uniqueness of positive solutions, their uniform boundedness, existence and global stability of equilibrium points. Numerical simulation confirms the theoretical results.

Introduction

Since the Lotka–Volterra model was published  [17], [25], a lot of work has been devoted to studying the interactions between predator and prey populations (see, e.g.,  [5], [18] and the references therein). A very general model, called Gause-type model (see, e.g.,  [14] and the references therein) is described by the equations

$$\frac{dN}{dt}=rN(1-\frac{N}{K})-PF(N,P)\text{,}$$$$\frac{dP}{dt}=\chi PF(N,P)-dP\text{.}$$

Here, $N\left(t\right)$ and $P\left(t\right)$ are the densities of prey and predator populations at time $t$; $K$ is the maximum carrying prey capacity of the environment in the absence of predators; $\chi$ and $d$ are positive constants, $d$ means the death rate of the predator. The function $F(N,P)$ describes the feeding rate of prey consumption by predators and is also called a response function. Recent theoretical studies show that the particular form of $F(N,P)$ leads to specific properties of the model solutions like stability, existence and uniqueness of limit cycles, persistence, etc. Some of the most studied functional responses are presented below:

$$F(N,P)=F\left(N\right)=\frac{aN}{1+bN}\text{Holling type II}$$$$F(N,P)=\frac{aN}{bN+P^m}\text{Hassell–Varley}$$$$F(N,P)=\frac{aN}{1+bN+cP}\text{Beddington–DeAngelis .}$$

All constants $a$, $b$, $c$, and $m$ in the above response functions are positive.

The model (1) with $F(N,P)=F\left(N\right)$ is known as the Rosenzweig–MacArthur model  [21] and is one of the most studied models in the literature (cf.  [22]).

In a very recent paper, A. J. Terry, 2014  [23] proposes the following generalization of the Rosenzweig–MacArthur model

$$\frac{dN}{dt}=rN(1-\frac{N}{K})-PF\left(N\right)$$$$\frac{dP}{dt}=P[ℬ\left(F\left(N\right)\right)-\mathcal{D}\left(F\left(N\right)\right)]$$

under the following conditions:

The function $ℬ\left(F\left(N\right)\right)=ℬ\left(F\right)$ is continuously differentiable in $N$ and $F$, for $N\ge 0$ and $F\ge 0$. For $F\ge 0$, $0\le ℬ\left(F\right)\le cF$ holds for some positive constant $c$. Also, $dℬ/dF\ge 0$ is fulfilled. Moreover, $dℬ/dF>0$ either for $F\in [F_1,\infty )$ where $F_1$ is a nonnegative constant, or for $F\in [F_1,F_2]$ where $F_1$ and $F_2$ are constants with $0\le F_1<F_2$.

The function $\mathcal{D}\left(F\left(N\right)\right)=\mathcal{D}\left(F\right)$ is continuously differentiable in $N$ and $F$, for $N\ge 0$ and $F\ge 0$. For $F\ge 0$, $0<d_m\le \mathcal{D}\left(F\right)\le d_M$ where $d_m$ and $d_M$ are constants. Also, for $F\ge 0,d\mathcal{D}/dF\le 0$.

The reason for introducing the generic functions $ℬ$ and $\mathcal{D}$ is based on several observations.

For sufficiently small values of the predator functional response, the predator reproduction will be zero rather than linearly increasing w.r.t. the predator functional response. During times of low prey density a cessation of predator’s breeding may occur. On the other hand, the reproduction rate of predators can achieve a plateau level if their prey consumption rate becomes sufficiently high. There will always be a limit to the rate at which an individual predator can reproduce.

The predator death rate depends on the predator functional response. The predator needs to consume prey at some minimal rate to avoid death by hunger. If the predator has a sufficiently high prey consumption rate, then further increases of this rate may have little impact on its short-term chance of death.

More details can be found in  [23].

Although the ideas underlying the aforementioned birth and death rates of the predator have been mentioned a few times in the literature (see, e.g.,  [1], [2]), to the best of our knowledge they have been utilized for the first time in Terry’s paper  [23]. We believe this kind of models are more biologically realistic and, therefore, deserve further attention.

As can be seen in (2), a Holling type II functional response is used in Terry’s model. It is discussed in  [22] that in many cases the Beddington–DeAngelis functional response  [3], [6] is to be preferred, since it accommodates interference among predators and gives a better fit to experimental data. The classical Beddington–DeAngelis model has the form:

$$\frac{dN}{dt}=rN(1-\frac{N}{K})-\frac{aNP}{1+bN+cP}\text{,}$$$$\frac{dP}{dt}=\frac{\chi aNP}{1+bN+cP}-dP\text{.}$$

A detailed study of (3) can be found in  [9], [12], [13]. We shall mention here just one important feature of the Beddington–DeAngelis model. It does not necessarily exhibit the “paradox of enrichment”  [20] (although it might), which always exists in the classical models with prey-dependent (in the sense, defined in  [1]) functional response. For more details on the conditions under which this paradox occurs in predator–prey models, see  [9], [10]. The paradox of enrichment means that increasing the prey carrying capacity of the environment in a stable predator–prey system could lead to the destabilization of the system. As discussed, e.g., in  [4], however, this is not always in line with field observations.

In the Discussion section of  [23] the author suggests that the consequences of changing the predator functional response to different forms, and specifically to a Beddington–DeAngelis form, could be explored.

Taking into account the aforementioned reasons, we modify the model (2) by using the Beddington–DeAngelis functional response, i.e. we consider a generalization of (3) in the form of (2).

After the rescaling in (3)

$$P≔P\text{,}N≔\frac{N}{K}\text{,}a≔\frac{a}{bK}\text{,}b≔\frac{1}{bK}\text{,}c≔\frac{c}{bK}$$

we obtain the following model:

$$\frac{dN}{dt}=rN(1-N)-P\mathcal{A}(N,P)\text{,}$$$$\frac{dP}{dt}=P[ℬ\left(\mathcal{A}(N,P)\right)-\mathcal{D}\left(\mathcal{A}(N,P)\right)]$$

within

$$\mathcal{A}(N,P)=\frac{aN}{b+N+cP}\text{.}$$

We assume that the functions $ℬ$ and $\mathcal{D}$ satisfy conditions that capture the underlying ideas of Terry’s model (2). We modify some of the original assumptions that Terry made, however, for reasons, explained in Remark 1 below.

Denote by ${\mathbb{R}}_{+}^2$ the positive cone in ${\mathbb{R}}^2$, i.e.

$${\mathbb{R}}_{+}^2=\{(N,P):N\ge 0,P\ge 0\}\text{.}$$

(B) Conditions for$ℬ$:

• $ℬ=ℬ\left(\mathcal{A}\right)=ℬ\left(\mathcal{A}(N,P)\right)$ is continuously differentiable w.r.t. $\mathcal{A}\ge 0$ and $(N,P)\in {\mathbb{R}}_{+}^2$;

• $ℬ\left(0\right)=0$ and $0\le ℬ\left(\mathcal{A}\right)\le C\mathcal{A}$ for some constant $C>0$;

• there exist non-negative constants $A_1<A_2$ ($A_2$ possibly equal to $+\infty$) such that, for $(N,P)\in {\mathbb{R}}_{+}^2,{ℬ}^{\prime }=dℬ/d\mathcal{A}=0$ if $\mathcal{A}\in [0,A_1]\cup [A_2,+\infty )$, and ${ℬ}^{\prime }>0$ if $\mathcal{A}\in (A_1,A_2)$.

(D) Conditions for$\mathcal{D}$

• $\mathcal{D}=\mathcal{D}\left(\mathcal{A}\right)=\mathcal{D}\left(\mathcal{A}(N,P)\right)$ is continuously differentiable w.r.t. $\mathcal{A}\ge 0$ and $(N,P)\in {\mathbb{R}}_{+}^2$;

• there exist constants $D_1$ and $D_2$ such that $0<D_1\le \mathcal{D}\left(\mathcal{A}\right)\le D_2$ for $\mathcal{A}\ge 0$ and $(N,P)\in {\mathbb{R}}_{+}^2$;

• there exist non-negative constants ${\theta }_1<{\theta }_2$ (${\theta }_2$ possibly equal to $+\infty$) such that, for $(N,P)\in {\mathbb{R}}_{+}^2$, ${\mathcal{D}}^{\prime }=d\mathcal{D}/d\mathcal{A}=0$ if $\mathcal{A}\in [0,{\theta }_1]\cup [{\theta }_2,+\infty )$ and ${\mathcal{D}}^{\prime }<0$ if $\mathcal{A}\in ({\theta }_1,{\theta }_2)$.

Remark 1

The new condition in (B) (compared with  [23]) is (iii). The latter captures the idea of a threshold (in our notation, $A_1$), under which no reproduction can occur, and of a certain amount of intake ($A_2$) after which the reproduction rate does not increase. This assumption does not allow certain degenerate behavior of the system (4), leading to structural instability (see  [23]). Let us also note that if $A_1=0$ and $A_2=+\infty$, the function $ℬ\left(\mathcal{A}\right)$ is strictly monotone for $\mathcal{A}\in [0,+\infty )$ and, thus, it includes as a particular case the classical (linear w.r.t. $\mathcal{A}$) birth rate function.

Similar comments can be made on condition (D) (iii).

Let us note that from conditions (B)(ii) and (D)(ii) it follows that $\mathcal{D}\left(0\right)>ℬ\left(0\right)=0$ is fulfilled. The latter inequality has a clear biological meaning — if there is no food, the predator mortality will be higher than the reproduction rate.

The paper is structured in the following way. Section  2 studies the basic properties of the model like invariance of the positive quadrant, boundedness of the solutions, etc. In Section  3, the equilibrium points of the model are found. Conditions for the existence of an internal equilibrium are derived. In Section  4, we study the local stability of the equilibrium points and in Section  5 the global behavior of the solutions. It is shown that there exist three possibilities for the solutions — extinction of the predator, globally stable internal equilibrium, or a periodic solution. Numerical simulation is included in Section  6 to demonstrate the theoretical results.