Analysis of a Single Species Model with Dissymmetric Bidirectional Impulsive Diffusion and Dispersal Delay

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in the real natural ecosystem, impulsive diffusion provides a more suitable manner to model the actual dispersal (or migration) behavior for many ecological species. In addition, the species not only requires some time to disperse or migrate among the patches but also has some possibility of loss during dispersal. In view of these facts, a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay is formulated. Criteria on the permanence and extinction of species are established. Furthermore, the realistic conditions for the existence, uniqueness, and the global stability of the positive periodic solution are obtained. Finally, numerical simulations and discussion are presented to illustrate our theoretical results

Permanence of a single-species dispersal system and predator survival

AbstractThis paper considers permanence of a single-species dispersal periodic system with the possibility of the loss for the species during their dispersion among patches. The condition obtained for permanence generalizes the known condition on the system without loss for the species in the process of movement. Next, we add predators into every patch and consider the survival possibility of the predator. It is shown that the total amount of the predators can remain positive, if the single-species (prey) dispersal system has a positive periodic solution and the quantity of prey in each patch is enough for survival of the predator

Traveling wavefronts of a prey–predator diffusion system with stage-structure and harvesting

AbstractFrom a biological point of view, we consider a prey–predator-type free diffusion fishery model with stage-structure and harvesting. First, we study the stability of the nonnegative constant equilibria. In particular, the effect of harvesting on the stability of equilibria is discussed and supported with numerical simulation. Then, employing the upper and lower solution method, we show that when the wave speed is large enough there exists a traveling wavefront connecting the zero solution to the positive equilibrium of the system. Numerical simulation is also carried out to illustrate the main result

The effect of diffusion on the time varying logistic population growth

AbstractIn this paper, we consider the effect of diffusion on the species that live in changing patches environment. Different from the former studies [1–4], we pay attention to the more important situation in conservation biology that species live in a weak patches environment, in the sense that species in some of the isolated patches will be extinct without the contribution from other patches. We obtain an interesting result: the identical species can persist for some diffusion rates, and can also vanish for another set of restriction on diffusion rates

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at time t are estimated by rit+σitB˙i(t),  i=1,2, respectively. Viewing the different intensities of the noises σi(t), i=1,2 as two parameters at time t, we conclude that there exists a global positive solution and the pth moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system

On a comparison method to reaction-diffusion systems and its applications to chemotaxis

In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations