Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment

In this paper, two classes of single-species models with logistic growth and impulse dispersal (or migration) are studied: one model class describes dissymmetric impulsive bi-directional dispersal between two heterogeneous patches; and the other presents a new way of characterizing the aggregate migration of a natural population between two heterogeneous habitat patches, which alternates in direction periodically. In this theoretical study, some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness and global stability of positive periodic solutions are established by using analysis based on the theory of discrete dynamical systems. From this study, we observe that the dynamical behavior of populations with impulsive dispersal differs greatly from the behavior of models with continuous dispersal. Unlike models where the dispersal is continuous in time, in which the travel losses associated with dispersal make it difficult for such dispersal to evolve e.g., [25], [26], [28], in the present study it was relatively easy for impulsive dispersal to positively affect populations when realistic parameter values were used, and a rich variety of behaviors were possible. From our results, we found impulsive dispersal seems to more nicely model natural dispersal behavior of populations and may be more relevant to the investigation of such behavior in real ecological systems.

In this paper, we study the permanence of a periodic Ivlev-type predator–prey system where the prey disperses in patchy environment with two patches. We assume the Ivlev-type functional response within-patch dynamics and provide a sufficient condition to guarantee the predator and prey species to be permanent. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful to ecosystem control.

In this paper, the existence of positive periodic solutions for a generalized delayed population model with stocking and feedback control is established by using the continuation theorem of coincidence degree theory. Then, by constructing a Lyapunov functional, the global attractivity of a positive periodic solution for the above model is established.

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.

In this paper we consider a nonautonomous periodic dispersal system which models the diffusion of a single species between n patches connected by discrete dispersal in a periodic environment. Using Gaines and Mawhin continuation theorem of coincidence degree (cf. [R.E. Gaines, J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977]), we prove that the system has at least one positive periodic solution. With the help of an appropriately chosen Lyapunov functional it is proved that this periodic solution is globally attractive. We end the paper by some numerical simulations which illustrate the feasibility of our results.

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for the generalized two-predator and one-prey system with time delay is established. Further, by constructing a Lyapunov functional and using the result in the existence part of positive periodic solutions, the attractivity of positive periodic solutions for the above system is obtained.