Original articlesA predator–prey model with generic birth and death rates for the predator and Beddington–DeAngelis functional response
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 Here, and are the densities of prey and predator populations at time ; is the maximum carrying prey capacity of the environment in the absence of predators; and are positive constants, means the death rate of the predator. The function 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 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: All constants , , , and in the above response functions are positive.
The model (1) with 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 under the following conditions:
The function is continuously differentiable in and , for and . For , holds for some positive constant . Also, is fulfilled. Moreover, either for where is a nonnegative constant, or for where and are constants with .
The function is continuously differentiable in and , for and . For , where and are constants. Also, for .
The reason for introducing the generic functions and 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: 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) we obtain the following model: within
We assume that the functions and 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 the positive cone in , i.e.
(B) Conditions for:
- (i)
is continuously differentiable w.r.t. and ;
- (ii)
and for some constant ;
- (iii)
there exist non-negative constants ( possibly equal to ) such that, for if , and if .
(D) Conditions for
- (i)
is continuously differentiable w.r.t. and ;
- (ii)
there exist constants and such that for and ;
- (iii)
there exist non-negative constants ( possibly equal to ) such that, for , if and if .
Remark 1 The new condition in (B) (compared with [23]) is (iii). The latter captures the idea of a threshold (in our notation, ), under which no reproduction can occur, and of a certain amount of intake () 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 and , the function is strictly monotone for and, thus, it includes as a particular case the classical (linear w.r.t. ) 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 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.
Section snippets
Basic properties of the model
Proposition 1 The positive cone is a positively invariant set for (4).
Proof We can rewrite the first equation in (4) as Therefore, and it is obvious that implies for every . Similarly, it can be shown that is also nonnegative, if is nonnegative. Obviously, if and then (due to uniqueness of the solution) and for all . This means that the coordinate axes and
Equilibrium points of the model
First, let us note that we are interested only in nonnegative equilibrium points. No other equilibria have any biological relevance. The equilibrium points of (4) are solutions of the algebraic system
Obviously, if in (6), then , and if , then either , or . Therefore, and are always equilibrium points of (4). If a third equilibrium exists, its coordinates should be strictly positive and
Local stability of the equilibria
Proposition 4 The equilibrium point is a saddle point for all positive values of the model parameters.
Proof For the Jacobian matrix of (4), evaluated at we have Condition(D)(ii) implies that is a saddle point. Its stable manifold is the positive -axis. □
Proposition 5 Let be defined as in (12). If holds true (i.e. no internal equilibrium exists), then is a locally asymptotically stable equilibrium point. If is valid, then is a saddle point.
Proof
Global behavior of the solutions
Theorem 1 If (or equivalently ) then is a globally asymptotically stable equilibrium point of (4).
Proof From the first equation in (4) it is obvious that for every there exists such that for every . Therefore, taking into account conditions (B) and (D), the following inequalities hold true for : Similarly, it can be shown that for , is also fulfilled.
Numerical simulation and discussion
In this section, we shall show some numerical examples that illustrate the behavior of the solutions. For the particular expressions of and , in the numerical examples we shall follow [23]. Define and where all the parameters are positive and, furthermore, , and .
As discussed in [23], these functions are constructed such that they capture several
Conclusions
Motivated by the suggestions and the results presented in the paper of A. J. Terry [23], we consider a predator–prey model with generic birth and death rate functions for the predator and Beddington–DeAngelis response function. The latter is known to give a better fit to experimental data in many cases and under certain conditions it solves the “paradox of enrichment”. We investigate the model solutions and show that they exhibit the typical behavior of predator–prey models like extinction of
Acknowledgments
The work of the first author has been partially supported by the Sofia University “St. Kl. Ohridski” under Contract No. 075/2015. The work of the second author has been partially supported by the Sofia University “St. Kl. Ohridski” under Contract No. 08/26.03.2015.
The authors are grateful to the anonymous referees for their valuable advices and comments.
References (26)
- et al.
Coupling in predator–prey dynamics: ratio-dependence
J. Theor. Biol.
(1989) - et al.
Persistence in models of three interacting predator–prey populations
Math. Biosci.
(1984) A detailed study of the Beddington–DeAngelis predator–prey model
Math. Biosci.
(2011)Global analysis of the predator–prey system with Beddington–DeAngelis functional response
J. Math. Anal. Appl.
(2003)Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response
J. Math. Anal. Appl.
(2004)- et al.
Uniqueness of limit cycles in Gause-type models of predator–prey systems
Math. Biosci.
(1988) - et al.
Stability and bifurcation in a delayed predator–prey system with Beddington–DeAngelis functional response
J. Math. Anal. Appl.
(2004) - et al.
On the dynamical behavior of three species food web model
Chaos Solitons Fractals
(2007) A predator prey model with generic birth and death rates for the predator
Math. Biosci.
(2014)- et al.
Dynamical analysis in a delayed predator–prey model with delays
Discrete Dyn. Nat. Soc.
(2012)
How Species Interact: Altering the Standard View on Trophic Ecology
Mutual interference between parasites or predators and its effect on searching efficiency
J. Anim. Ecol.
The origins and evolution of predator–prey theory
Ecology
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