Dynamical behavior of a stochastic HBV infection model with logistic hepatocyte growth

This paper is concerned with a stochastic HBV infection model with logistic growth. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HBV infection model. Then we obtain sufficient conditions for extinction of the disease. The stationary distribution shows that the disease can become persistent in vivo.     

Population dynamical behavior of Lotka-Volterra system under regime switching

In this paper, we investigate a Lotka-Volterra system under regime switching

     

Dynamics of Kolmogorov systems of competitive type under the telegraph noise

This paper studies the dynamics of Kolmogorov systems of competitive type under the telegraph noise. The telegraph noise switches at random two Kolmogorov competition-type deterministic models. The aim of this work is to describe the omega-limit set of the system and investigates properties of stationary density.     

Pulse vaccination delayed SEIRS epidemic model with saturation incidence

A delayed SEIRS epidemic model with pulse vaccination and saturation incidence rate is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution and the permanence of disease. Our results indicate that a long latent period of the disease or a proper pulse vaccination rate will lead to eradication of the disease.     

Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses

A delayed stage-structured predator–prey model with non-monotone functional responses is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age, and that immature predators do not have the ability to attack prey. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the stage-structured predator–prey model. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions to Lx = λNx. An example is given to illustrate the feasibility of our main result.     

Periodic solutions for a Lotka–Volterra mutualism system with several delays

In the present paper, a Lotka–Volterra type mutualism system with several delays is studied. Some new and interesting sufficient conditions are obtained for the global existence of positive periodic solutions of the mutualism system. Our method is based on Mawhin’s coincidence degree and novel estimation techniques for the a priori bounds of unknown solutions. Our results are different from the existing ones such as those in of Yang et al. [F. Yang, D. Jiang, A. Ying, Existence of positive solution of multidelays facultative mutualism system, J. Eng. Math. 3 (2002) 64–68] and Chen et al. [F. Chen, J. Shi, X. Chen, Periodicity in a Lotka–Volterra facultative mutualism system with several delays, J. Eng. Math. 21 (3) (2004) 403–409].     

Permanence of a delayed SIR epidemic model with density dependent birth rate

In this paper, we consider the permanence of a modified delayed SIR epidemic model with density dependent birth rate which is proposed in [M. Song, W. Ma, Asymptotic properties of a revised SIR epidemic model with density dependent birth rate and time delay, Dynamic of Continuous, Discrete and Impulsive Systems, 13 (2006) 199–208]. It is shown that global dynamic property of the modified delayed SIR epidemic model is very similar as that of the model in [W. Ma, Y. Takeuchi, T. Hara, E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J. 54 (2002) 581–591; W. Ma, M. Song, Y. Takeuchi, Global stability of an SIR epidemic model with time delay, Appl. Math. Lett. 17 (2004) 1141–1145].     

Global stability of a virus dynamics model with intracellular delay and CTL immune response

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R₀, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R₀ is more than one, and if immune response reproductive number, R⁰, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R⁰ is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Versal unfoldings of predator-prey systems with ratio-dependent functional response

In this paper we study the versal unfolding of a predator-prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.     

Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise

This work focuses on population dynamics of two species described by Kolmogorov systems of competitive type under telegraph noise that is formulated as a continuous-time Markov chain with two states. Our main effort is on establishing the existence of an invariant (or a stationary) probability measure. In addition, the convergence in total variation of the instantaneous measure to the stationary measure is demonstrated under suitable conditions. Moreover, the Ω-limit set of a model in which each species is dominant in a state of the telegraph noise is examined in detail.     

The linear and nonlinear diffusion of the competitive Lotka–Volterra model

This paper has studied the effects of linear and nonlinear diffusion of the competitive Lotka–Volterra model, and has investigated how the linear and nonlinear diffusions lead from the extinction of one species to the persistence or global asymptotic stability of all species. This research has important implications in the design of nature reserves.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

Qualitative analysis of a stochastic ratio-dependent Holling-Tanner system

This article addresses a stochastic ratio-dependent predator-prey system with Leslie-Gower and Holling type II schemes. Firstly, the existence of the global positive solution is shown by the comparison theorem of stochastic differential equations. Secondly, in the case of persistence, we prove that there exists a ergodic stationary distribution. Finally, numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.     

Well-posedness for a model of prion proliferation dynamics

The model considered consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins. We provide sufficient conditions for uniqueness of monomer-preserving weak solutions. In addition, we also prove existence of weak solutions under rather general assumptions on the involved degradation rates.     

Stochastic Lotka-Volterra system with infinite delay

This paper investigates a stochastic Lotka-Volterra system with infinite delay, whose initial data comes from an admissible Banach space C_r. We show that, under a simple hypothesis on the environmental noise, the stochastic Lotka-Volterra system with infinite delay has a unique global positive solution, and this positive solution will be asymptotic bounded. The asymptotic pathwise of the solution is also estimated by the exponential martingale inequality. Finally, two examples with their numerical simulations are provided to illustrate our result.     

Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances

In this paper, we consider a periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances. It is shown that toxic substances play an important role in the extinction of species. We obtain a set of sufficient conditions which guarantee that one of the components is driven to extinction while the other is globally attractive. The numerical simulation of an example verifies our main results.