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This paper is concerned with the dynamical behavior of a stochastic SIQR epidemic model with standard incidence which is disturbed by both white and telegraph noises. Firstly, we obtain sufficient conditions for persistence in the mean of the disease. Then we establish sufficient conditions for extinction of the disease. In addition, in the case of persistence, we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, some numerical simulations are introduced to demonstrate the analytical results.

For a nonlinear equation with several variable delays $$ \dot{x}(t)=\sum_{k=1}^m f_k(t, x(h_1(t)),\dots,x(h_l(t)))-g(t,x(t)), $$ where the functions $f_k$ increase in some variables and decrease in the others, we obtain conditions when a positive solution exists on $[0, \infty)$, as well as explore boundedness and persistence of solutions. Finally, we present sufficient conditions when a solution is unbounded. Examples include the Mackey-Glass equation with non-monotone feedback and two variable delays; its solutions can be neither persistent nor bounded, unlike the well studied case when these two delays coincide., 24 pages, published in Applied Mathematics and Computation, 2016

In this paper, a two-species stochastic non-autonomous food chain predator-prey system with jumps is proposed and studied. The asymptotic properties of the system are examined. Sufficient conditions for extinction, non-persistence in the mean, weak persistence in the mean and weak persistence of the system are established. Results show that Levy jumps are disadvantageous for the persistence of the populations. Furthermore, we also show that the solution is stochastically ultimate bounded under certain conditions. Some examples together with numerical simulations are introduced to illustrate our main results.

Prey-predator interactions are influenced by many ecological factors. In this paper, a prey-predator model with spatial diffusion under homogeneous Neumann boundary conditions is investigated to study the impact of water level on persistence of two fish populations living in an artificial lake. We obtain some important qualitative properties, including the existence of the global positive solution, the dissipation and permanence of this system. The existence of a unique globally stable periodic solution is also presented. Some numerical examples are presented to verify our results.

A prey-predator system with nonlinear incidence rate of the disease in prey is described.The dynamical behavior of the system is investigated at each equilibrium point.Sufficient conditions are derived for the uniformly strong persistence of the system.Results are obtained to extract the criteria for disease persistence and extinction. A prey-predator system with disease in prey is proposed. The proposed system is an extension of the model analyzed by Bhattacharyya and Mukhopadhyay (2011) which did not consider the density of fish population as a dynamic variable which significantly influence the dynamics of the system. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Incidence rate of the disease is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibria. Sufficient conditions are derived for the global stability of the system around coexistence equilibria. Uniform strong persistence of the system is discussed in order to ensure long-term survival of the species. The obtained results are useful to extract the criteria for disease extinction and persistence. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

In this paper, a stochastic non-autonomous food-limited Lotka-Volterra cooperative model is proposed and investigated. Sufficient conditions are formulated and verified for p-moment persistence and extinction of the system. Results demonstrate that cooperative coefficients have important effects on the convergence rate of the solution to the equilibria of the stochastic model. Some numerical simulations are introduced to support the analytical results.

This paper focuses on studying the dynamics of the stochastic Gilpin-Ayala model under regime switching with jumps. The aim is to analyze what happens under the perturbations of regime switching and jumps. Some asymptotic properties are investigated and sufficient conditions for stochastic permanence, extinction, non-persistence in the mean and weak persistence are provided. The critical value among the extinction, non-persistence in the mean and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the jumps and the stationary distribution of the Markov chain.

In this paper we consider the persistence of hyperbolic lower dimensional invariant tori with prescribed frequency for reversible system, and prove that if the frequency mapping has nonzero topological degree at some Diophantine frequency, then the hyperbolic lower invariant dimensional torus with this frequency persists under small perturbations.

Li and Lim (2011) [8] proposed a three-dimensional system of ordinary differential equations modeling the role of Tax expression in the persistence of Human T-lymphotropic virus Type I and presented a mathematical analysis and its immunovirological explanation. The previous paper shows that three-dimensional system is a cooperative system if a sharp condition is satisfied. By means of the theory of cooperative dynamical systems, they studied the global dynamics. If the sharp condition is not satisfied, they leave it an open problem. The present paper is to provide a complete classification of global dynamics for this ODE model. In particular, our global analysis establishes that the dynamics of the HTLV-I infection are completely determined by a basic reproductive number, which completely solves the open problem above.

In this paper we consider a non-autonomous ratio-dependent predator–prey system driven by Levy noise. Firstly, we show the existence of global positive solution and stochastic boundedness. Secondly, the conditions of persistent in mean and extinction are established and we also give the asymptotic properties of the solution. Finally, we simulate the model to illustrate our main analytical results.

This paper characterizes the qualitative dynamics of a stochastic SIRS epidemic model. The study shows that the dynamics of the model are determined by a certain threshold quantity R S expressed in terms of the model parameters and the intensity of the noise. If R S 1 , the disease will be eliminated from the community; whereas an epidemic occurs if R S > 1 . Our results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.

A two-species ratio-dependent predator–prey diffusion model with variable time delays is investigated. By constructing some suitable Lyapunov functionals and utilizing some analysis techniques, we obtain the permanence of this system. Our results generalize and improve the results of [10] , [11] .

We study the travelling fronts of KdV-Burgers-Kuramoto equation from geometric singular perturbation point of view. Motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations, we prove the persistence of these waves for sufficiently small dissipation. The result of numerical investigation also establishes our analysis.

The persistence of some stage structured predator–prey model with finite and infinite delay was studied in this paper. By analyzing right-hand functions and using the comparison theorem, some sufficient conditions for the persistence of populations with finite and infinite delay were derived.

In this paper, an nonlinear asymptotically periodic multispecies competition predator-prey system with general functional response is studied. And we obtain sufficient conditions for the uniformly strong persistence of the system. In particular, our result extend the result in [F.Y. Wei, K. Wang, Uniform persistence of asymptotically periodic multispecies competition predator-prey systems with Holling III type of functional response, Appl. Math. Comput., in press].

This paper discusses a discrete Lotka-Volterra competition system with feedback controls. We first obtain the persistence of the system. Assuming that the coefficients in the system are periodic, we obtain the existence of a periodic solution. Moreover, under some additional conditions, this periodic solution is globally stable.

A ratio-dependent predator-prey model with stage structure for prey is investigated. First, sufficient conditions are derived for the uniform persistence and impermanence of the model. Next, by constructing appropriate Lyapunov functions, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Numerical simulations are presented to illustrate the validity of our main results.

A delayed predator-prey model with stage structure for the predator is proposed and investigated. Sufficient conditions are derived for persistence, the local and global asymptotic stability of a positive equilibrium of the model. Numerical simulations are presented to illustrate the feasibility of our main results.

In this paper, we consider a nonautonomous Lotka-Volterra competitive system with nonlinear discrete dispersion and a finite number of discrete delays. The system, which is consisted of two Lotka-Volterra patches, has two competitors: one can disperse between two patches, but the other is confined to one patch and cannot disperse. Our main focus is persistence, i.e. the survival in two patches of the population, and the existence of a periodic solution. We also establish conditions under which the system admits a positive periodic solution which attracts all solutions.

A spherical reaction-diffusion unit-world model is analytically approached. This model serves as a paradigm in investigations of persistence and spatial range of non-polar chemicals. The presented framework allows to handle any combination of rate constants. For a huge class of chemicals, a closed-form approximation is derived. An exact expression for the persistence is presented. Although degradation and diffusion processes are linked in a multi-compartment system, the persistence is proven to be independent of any diffusion rate, geometry, and dimension. A close relation of time-domain @d-like releases and steady-state situations is established.

We comment on an error made in a recently published paper [Junliang Lu et al., Applied Mathematics and Computation 191 (2) (2007) 347–352]; an explanation as to why this is so is given. Moreover, we also give some questions to be discussed.