Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response
© Yu; licensee Springer 2014
Received: 21 October 2013
Accepted: 24 February 2014
Published: 13 March 2014
A predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response is studied. The local stability of the equilibria and the permanence of the system are investigated. By applying the fluctuation lemma, qualitative analysis and Lyapunov direct method, respectively, three sufficient conditions on the global asymptotic stability of a positive equilibrium are obtained.
MSC:34D23, 92D25, 34D20, 34D40.
where , stand for the population (the density) of the prey and the predator at time t, respectively. The parameters and are the intrinsic growth rates of the prey and the predator, respectively. measures the strength of competition among individuals of species x. The value is the carrying capacity of the prey in the absence of predation. The predator consumes the prey according to the functional response and carries capacity . The parameter is a measure of the food quantity that the prey provides converted to predator birth. The term is the Leslie-Gower term which measures the loss in the predator population due to rarity (per capita ) of its favorite food. Leslie model is a predator-prey model where the carrying capacity of the predator is proportional to the number of prey, stressing the fact that there are upper limits to the rates of increase in both prey x and predator y, which are not recognized in the Lotka-Volterra model. These upper limits can be approached under favorable conditions: for the predators, when the number of prey per predator is large; for the prey, when the number of predators (and perhaps the number of prey also) is small . For more details of the model, one can see [4–9] and the references cited therein. Holling  suggested three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. When , the functional response is called Holling-type II.
where , , , have the same meaning as in models (1.1). is the maximum value of the per capita reduction rate of x due to y, (respectively, ) measures the extent to which the environment provides protection to prey x (respectively, to the predator y). The authors studied the boundedness and global stability of positive equilibrium of system (1.2). Since then, system (1.2) and its non-autonomous versions have been studied by incorporating delay and impulses, harvesting and so on (see, for example, [12–29]). In , we studied the structure, linearized stability and the global asymptotic stability of equilibria of (1.2). Nindjin et al.  incorporated time delay to system (1.2) and studied the global stability and persistence of the delayed system by using the Lyapunov functional. Yafia et al.  and  further studied the occurrence of Hopf bifurcation at equilibrium by using the time delay as a parameter of bifurcation. Nindjin and Aziz-Alaoui  studied uniform persistence and global stability of three Leslie-Gower-type species food chain system. Gakkhar and Singh  studied the dynamic behaviors of a modified Leslie-Gower predator-prey system with seasonally varying parameters. Guo and Song , Song and Li  further considered the influence of impulsive effect. Zhu and Wang  obtained sufficient conditions for the existence and global attractivity of positive periodic solutions of system (1.2) with periodic coefficients. Liu and Wang  considered a stochastic predator-prey system with modified Leslie-Gower and Holling-type II schemes with Lévy jumps. The results showed that the Lévy jumps can change the properties of the population systems significantly. Kar and Ghorai  obtained local stability, global stability, influence of harvesting and bifurcation of a delayed predator-prey system in the presence of harvesting. Two stage-structured predator-prey models with modified Leslie-Gower and Holling-type II schemes were studied in [23–25]. Gupta and Chandra  discussed the permanence, stability and bifurcation (saddle-node bifurcation, transcritical, Hopf-Andronov and Bogdanov-Takens) of a modified Leslie-Gower prey-predator model with Michaelis-Menten type prey harvesting. Ji et al. [27, 28] showed there was a stationary distribution of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property. Lian and Xu  discussed the Hopf bifurcation of a predator-prey system with Holling type IV functional response and time delay.
As we all know, the functional response can be classified into two types: prey-dependent and predator-dependent. Prey-dependent depends on prey density only, while predator-dependent means that the functional response is a function of both the preys and the predators densities. Recently, the prey-dependent functional responses have been challenged by several ecologists. There is a growing explicit biological and physiological evidence [30–32] that in many situations, especially when the predator has to search for food (and therefore has to shave or compete for food), a more suitable general predator-prey theory should be predator-dependent. This is supported by numerous fields and laboratory experiments and observations [33, 34]. Starting from this argument and the traditional prey-dependent-only model, Arditi and Ginzburg  first proposed the ratio-dependent predator-prey model. Many authors have observed that the ratio-dependent models can exhibit much richer, more complicated and more reasonable or acceptable dynamics, but it has somewhat singular behavior at low densities which has been the source of controversy . For the ratio-dependent predator-prey models, one can refer to [36–39].
Beddington-DeAngelis functional response was first proposed by Beddington and DeAngelis [40, 41]. Predator-prey model with Beddington-DeAngelis functional response has rich dynamical features, which can describe the species and the ecological systems more reasonably. Beddington-DeAngelis functional response is similar to the well-known Holling type II functional response but has an extra term cy in the denominator modeling mutual interference among predators and has some of the same qualitative features as the ratio-dependent form but avoids some of the singular behaviors of ratio-dependent models at low densities.
with initial conditions and . The parameters , p, α, a, b, c, , β and k are positive constants and have the same meaning as in model (1.2).
It is easy to see that both the first quadrant and the positive first quadrant are invariant for system (1.3). As a result, solutions to (1.3) with are all positive solutions.
The rest of this paper is organized as follows. In Section 2, we discuss the structure of nonnegative equilibria to (1.3) and their local stability, which motivates us to study permanence and global stability of (1.3) respectively in Section 3 and Section 4.
2 Nonnegative equilibria and their local stability
respectively. As a direct consequence of (2.1), we have the following result.
Both and are unstable.
is locally asymptotically stable if , while it is unstable if .
Hence, we have the following result.
Hence, the following proposition follows from (2.1).
holds, then the positive equilibrium is locally asymptotically stable.
Proposition 2.1 and Proposition 2.3 naturally motivate us to seek sufficient conditions on the global stability of and the unique positive equilibrium to (1.3). To achieve it, we need the bounds for positive solutions.
3 Boundedness and permanence
The following result can be proved by slightly modifying the proof of Lemma 3.2 of Chen  and it will play an important role in finding the bounds for positive solutions to (1.3).
where and .
Thus by letting . □
This gives by letting . □
Combing Proposition 3.1 with Proposition 3.2 gives the permanence of (1.3).
Theorem 3.1 Suppose that (H2) holds, then (1.3) is permanent.
4 Global asymptotic stability
The goal of this section is to establish sufficient conditions on the global asymptotic stability of equilibrium to (1.3). The first two results are proved by employing the fluctuation lemma, which is cited below for the convenience of the readers. See Hirsch et al.  or Tineo  for more details on the fluctuation lemma.
Lemma 4.1 (Fluctuation lemma)
and as ;
and as .
Then is globally asymptotically stable for system (1.3).
Proposition 3.1 again tells us that is bounded and . By Lemma 4.1, there exist sequences , such that , , , and , as .
It follows from (4.3) and (4.6) that is globally asymptotically stable. □
where is defined in Proposition 3.2. Then system (1.3) has a unique positive equilibrium which is globally asymptotically stable.
Proof Note that (H2) implies (H0), thus (1.3) has a unique positive equilibrium according to Proposition 2.2. Let be any positive solution of (1.3). By the results in Section 3, (>0) and .
That is, is a positive equilibrium of (1.3). This completes the proof as the positive equilibrium is unique and so the unique equilibrium point is globally asymptotically stable. □
The following results in this section are proved by qualitative method and applying the Lyapunov direct method with the Lyapunov function.
Theorem 4.3 Assume that (H0) and (H1) hold, then (1.3) has a unique positive equilibrium which is globally asymptotically stable.
where is the vector field of (1.3). By the positivity of x, y, it is easy to obtain that if (H1) holds. Then, by the Dulac criteria, (1.3) admits no limit cycles or separatrix cycles. Proposition 2.3 shows that is locally asymptotically stable when (H1) holds. On the other hand, (1.3) admits only four equilibria () and . Also, Proposition 2.1 shows that () are all unstable when holds. So, according to Proposition 3.1 and the Poincaré-Bendixson theorem, is globally asymptotically stable. □
hold, then (1.3) has a unique positive equilibrium which is globally asymptotically stable.
According to (4.13) and (4.14), strictly for all except the positive equilibrium , where . Thus, satisfies Lyapunov’s asymptotic stability theorem, and the positive equilibrium of system (1.3) is globally asymptotically stable. This ends the proof of Theorem 4.4. □
In this paper, we consider a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. We discuss the structure of nonnegative equilibria and their local stability. Also, the permanence of the system is investigated. By applying the fluctuation lemma, qualitative analysis and Lyapunov direct method, respectively, three sufficient conditions on the global asymptotic stability of a positive equilibrium are obtained. Compare Theorem 4.2 with Theorem 4.3. Since (H2) contains (H0), what will happen when (H0) and (H4) hold? This is a further problem, which can be studied in the future.
The author would like to thank the two anonymous referees for their constructive suggestions on improving the presentation of the paper. This research is supported by the Foundation of Fujian Education Bureau (JA13365).
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