![]() Yu, S.: Global stability of a modified Leslie–Gower model with Beddington–DeAngelis functional response. Li, T.T., Chen, F.D., et al.: Stability of a mutualism model in plant-pollinator system with stage-structure and the Beddington–DeAngelis functional response. 219(17), 8856–8862 (2013)Ĭhen, F.D., Xie, X.D., et al.: Partial survival and extinction of a delayed predator-prey model with stage structure. However, at present, we have difficulty to prove this conjecture, we leave this for future investigation.Ĭhen, F.D., Chen, W.L., et al.: Permanence of a stage-structured predator-prey system. Hence we have a conjecture: ConjectureĬondition ( 3.1) is not needed to ensure the positive equilibrium of system ( 1.1) to be globally attractive. ![]() 3) also shows that ( 3.1) could be dropped out. However, for the system without cannibalism (i.e., system ( 1.2)), the positive equilibrium is globally attractive without any restriction on the coefficients, and numeric simulation (Fig. It brings to our attention that inequality ( 3.1) is independent of the coefficients of the cannibalism term. Also, under some very simple conditions, we could also prove that the positive equilibrium is globally attractive. : for the system they considered, cannibalism may have both positive or negative effects on the stability of the system. Our results are also different to those of Deng et al. Compared with the Leslie–Gower predator prey system without cannibalism (i.e., system ( 1.2)), our result shows that cannibalism has no influence on the local stability property of the positive equilibrium, this is quite different to the results of Basheer et al. Our study shows that the system with cannibalism always admits a positive equilibrium and a predator free equilibrium, the predator free equilibrium is unstable, while the positive equilibrium is locally asymptotically stable. In this paper, we focus our attention on the most simple Leslie–Gower predator prey model. They showed that cannibalism in the prey cannot stabilize the unstable interior equilibrium in the ODE case, but can destabilize the stable interior equilibrium, leading to a stable limit cycle. incorporated the cannibalism to the Holling–Tanner model with ratio-dependent functional response (i.e., system ( 1.2)). , we proposed a Leslie–Gower predator prey model incorporating the nonlinear cannibalism. We end this paper with a brief discussion.īased on the traditional Leslie–Gower predator prey model and the works of Basheer et al. 4 to show the feasibility of the main results. Numeric simulations are presented in Sect. 3, we discuss the global stability of the equilibrium by using the iterative method. In the next section, we investigate the existence and local stability of the equilibrium of system ( 1.1). The rest of the paper is arranged as follows. As far as system ( 1.1) is concerned, two interesting issues are proposed: Can we obtain sufficient conditions to ensure the existence of a unique globally stable positive equilibrium? Can we give some positive answer on the influence of the cannibalism on the dynamic behaviors of the system? , based on model ( 1.2), we propose the Leslie–Gower predator prey model with prey cannibalism, i.e., system ( 1.1). Stimulated by the works of Basheer et al. showed that if system ( 1.7) has a positive equilibrium, it then is globally asymptotically stable. , by constructing some suitable Lyapunov function, Deng et al. In this case, cannibalism has an unstable effect. ![]() ![]() If the two species coexist in the stable state in the original system, then predator cannibalism may lead to the extinction of the prey species. If the predator species in the system without cannibalism is extinct, then suitable cannibalism may lead to the coexistence of both species in this case, cannibalism stabilizes the system. The authors showed that cannibalism has both positive and negative effects on the stability of the system, it depends on the dynamic behaviors of the original system. \(r_< c\), x and y are the density of the prey and predator at time t, respectively. Where H and P are the density of prey species and the predator species at time t, respectively. ![]()
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