Meng Fan

Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system

With the help of differential equations with piecewise constant arguments, we first propose a discrete analogue of continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modeling the dynamics of the prey and the predator having nonoverlapping generations. Then, easily verifiable sufficient criteria are established for the existence of positive periodic solutions. The approach is based on the coincidence degree and the related continuation theorem as well as some priori estimates.

Optimal harvesting policy for single population with periodic coefficients.

In this paper, we examine the exploitation of single population modeled by time-dependent Logistic equation with periodic coefficients. First, it is shown that the time-dependent periodic Logistic equation has a unique positive periodic solution, which is globally asymptotically stable for positive solutions, and we obtain its explicit representation. Further, we choose the maximum annual-sustainable yield as the management objective, and investigate the optimal harvesting policies for constant harvest and periodic harvest. The optimal harvest effort that maximizes the annual-sustainable yield, the corresponding optimal population level, the corresponding harvesting time-spectrum, and the maximum annual-sustainable yield are determined, and their explicit expressions are obtained in terms of the intrinsic growth rate and the carrying capacity of the considered population. Our interesting and brief results generalize the classical results of Clark for a population described by the autonomous logistic equation in renewable resources management.

Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments

In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka-Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka-Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized.

Global stability of an SEIS epidemic model with recruitment and a varying total population size.

This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number R(0). If R(0)<or=1, the disease-free equilibrium is globally stable and the disease dies out. If R(0)>1, a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.

Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays

The principle aim of this paper is to explore the existence of periodic solutions with strictly positive components of generalized ecological competition systems governed by impulsive differential equation with infinite delays. Easily verifiable sufficient criteria are established. The approach is based on the coincidence degree theory and its related continuation theorem as well as some a priori estimates. Applications to some famous competition models, which have been widely studied in the literature, are presented also.

Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses ✩

Abstract In this paper, we investigate the dynamics of a class of the so-called semi-ratio-dependent predator–prey interaction models with functional responses based on systems of nonautonomous differential equations with time-dependent parameters. The functional responses are classified into five types and typical examples of each type are provided. Then we establish sufficient criteria for the boundedness of solutions, the permanence of system, and the existence, uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution. Some conclusive discussion is presented at the end of this paper.

Dynamics of a non-autonomous ratio-dependent predator{prey system

We investigate a non-autonomous ratio-dependent predator{prey system, whose autonomous versions have been analysed by several authors. For the general non-autonomous case, we address such properties as positive invariance, permanence, non-persistence and the globally asymptotic stability for the system. For the periodic and almost-periodic cases, we obtain conditions for existence, uniqueness and stability of a positive periodic solution, and a positive almost-periodic solution, respectively.

Periodic solution of single population models on time scales

By using the calculus on time scales, we study and establish criterion for the existence of periodic solutions of some scalar dynamical equations on time scales. The existence of periodic solutions for some concrete well-known single population models is obtained.

Positive Periodic Solutions of a Periodic Integro-differential Competition System with Infinite Delays

By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence and globally asymptotic stability of a unique strictly positive (componentwise) periodic solution of a periodic integro-differential competition system with infinite delays. Some new criterions are established.

Necessary and sufficient criteria for the existence of exponential dichotomy on time scales

Necessary and sufficient criteria are established for the existence of exponential dichotomies for linear dynamic equations on time scales and are applied to derive some perturbation theorems on the roughness of exponential dichotomy.