Persistence and global stability in discrete models of Lotka–Volterra type

AbstractIn this paper, we establish new sufficient conditions for global asymptotic stability of the positive equilibrium in the following discrete models of Lotka–Volterra type:{Ni(p+1)=Ni(p)exp{ci−aiNi(p)−∑j=1naijNj(p−kij)},p⩾0,1⩽i⩽n,Ni(p)=Nip⩾0,p⩽0,andNi0>0,1⩽i⩽n, where each Nip for p⩽0, each ci, ai and aij are finite and{ai>0,ai+aii>0,1⩽i⩽n,andkij⩾0,1⩽i,j⩽n. Applying the former results [Y. Muroya, Persistence and global stability for discrete models of nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 273 (2002) 492–511] on sufficient conditions for the persistence of nonautonomous discrete Lotka–Volterra systems, we first obtain conditions for the persistence of the above autonomous system, and extending a similar technique to use a nonnegative Lyapunov-like function offered by Y. Saito, T. Hara and W. Ma [Y. Saito, T. Hara, W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka–Volterra system with two delays, J. Math. Anal. Appl. 236 (1999) 534–556] for n=2 to the above system for n⩾2, we establish new conditions for global asymptotic stability of the positive equilibrium. In some special cases that kij=kjj, 1⩽i,j⩽n, and ∑j=1najiajk=0, i≠k, these conditions become ai>∑j=1naji2, 1⩽i⩽n, and improve the well-known stability conditions ai>∑j=1n|aji|, 1⩽i⩽n, obtained by K. Gopalsamy [K. Gopalsamy, Global asymptotic stability in Volterra's population systems, J. Math. Biol. 19 (1984) 157–168]

Global stability for a discrete SIS epidemic model with immigration of infectives

Abstract. In this paper, we propose a discrete-time SIS epidemic model which is derived from continuous-time SIS epidemic models with immigration of infectives by the backward Euler method. For the discretized model, by applying new Lyapunov function techniques, we establish the global asymptotic stability of the disease-free equilibrium for R 0 ≤ 1 and the endemic equilibrium for R 0 > 1, where R 0 is the basic reproduction number of the continuous-time model. This is just a discrete analogue of continuous SIS epidemic model with immigration of infectives

Global analysis of a multi-group SIR epidemic model with nonlinear incidence rates and distributed moving delays between patches

In this paper, applying Lyapunov functional approach, we establish sufficient conditions under which each equilibrium is globally asymptotically stable for a class of multi-group SIR epidemic models. The incidence rate is given by nonlinear incidence rates and distributed delays incorporating not only an exchange of individuals between patches through migration but also cross patch infection between different groups. We show that nonlinear incidence rates and distributed delays have no influence on the global stability, but patch structure has. Moreover, the present results generalize known results on the global stability of a heroin model with two delays considered in the recent literatures. We also offer new techniques to prove the boundedness of the solutions, the existence of the endemic equilibrium and permanence of the model