Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

一类具有无穷时滞和功能反应的捕食扩散系统的持久性与稳定性

研究一类非自治的具有HollingⅡ类功能性反应且包含时变时滞与多个无穷时滞的两种群n斑块捕食扩散系统的持久性与稳定性.利用比较原理,结合构造Lyapunov泛函的方法,得到了保证该系统永久持续生存和任意正解全局渐近稳定的充分性条件.     

A cannibalistic eco-epidemiological model with disease in predator population

In this present article, we propose and analyze a cannibalistic predator–prey model with disease in the predator population. We consider two important factors for the dynamics of predator population. The first one is governed through cannibalistic interaction, and the second one is governed through the disease in the predator population via cannibalism. The local stability analysis of the model system around the biologically feasible equilibria are investigated. We perform global dynamics of the model using Lyapunov functions. We analyze and compare the community structure of the system in terms of ecological and disease basic reproduction numbers. The existence of Hopf bifurcation around the interior steady state is investigated. We also derive the sufficient conditions for the permanence and impermanence of the system. The study reveals that the cannibalism acts as a self-regulatory mechanism and controls the disease transmission among the predators by stabilizing the predator–prey oscillations.     

On almost periodic processes in impulsive competitive systems with delay and impulsive perturbations

In the present paper we investigate the existence of almost periodic processes of ecological systems which are presented with the general impulsive nonautonomous Lotka–Volterra system of integro-differential equations with infinite delay. The impulses are at fixed moments of time, and by using the techniques of piecewise continuous Lyapunov’s functions, new sufficient conditions for the global exponential stability of the unique positive almost periodic solutions of these systems are given.     

Analysis of a nonautonomous eco‐epidemic diffusive model with disease in the prey

This paper deals with the behavior of positive solutions to a nonautonomous reaction‐diffusion system with homogeneous Neumann boundary conditions, which describes a two‐species predator‐prey system in which there is an infectious disease in prey. The sufficient condition on the permanence of the prey and the predator is established by combining the comparison principle with the results related to the corresponding ODE system. Some sufficient conditions for the spreading and vanishing of the disease are obtained. The global attractivity is also discussed by constructing a Lyapunov functional. Our results show that the disease is spreading if the transmission rate is suitably large, while if the transmission rate is small, the disease must be vanishing.     

Analysis of a nonautonomous dynamical model of diseases through droplet infection and direct contact

In this paper, we have considered a nonautonomous dynamical model of diseases that spread by droplet infection and also through direct contact (with a lower risk) with varying total population size and distributed time delay to become infectious. It is assumed that there is a time lag due to incubation period of pathogens, i.e. the development of an infection from the time the pathogen enters the body until signs or symptoms first appear. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.     

Periodic solutions of a nonautonomous predator–prey system with stage structure and time delays

A nonautonomous Lotka–Volterra type predator–prey model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators do not feed on prey and do not have the ability to reproduce. By some comparison arguments we first discuss the permanence of the model. By using the continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions to the model. By means of a suitable Lyapunov functional, sufficient conditions are obtained for the uniqueness and global stability of the positive periodic solutions to the model.     

Permanence and Global Attractivity of a Discrete Semi-Ratio-Dependent Predator-Prey System with Holling Ⅳ Type Functional Response

In this paper,we investigate a discrete semi-ratio dependent predator-prey system with Holling IV type functional response.For general nonautonomous case,sufficient conditions which ensure the permanence and the global stability of the system are obtained.Meanwhile,we discuss the existence of the positive periodic solution and global stability of the system.     

带有Holling IV功能反应的离散半比率依赖捕食-食饵系统的持久性和全局吸引性

In this paper, we investigate a discrete semi-ratio dependent predator-prey system with Holling IV type functional response. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained. Meanwhile, we discuss the existence of the positive periodic solution and global stability of the system.     

Permanence and global attractivity for discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls

A discrete nonautonomous two-species Lotka-Volterra competitive system with delays and feedback controls is proposed and investigated. By using the method of discrete Lyapunov functionals, new sufficient conditions on the permanence of species and global attractivity of the system are established. Particularly, an interesting fact is found in our results, that is, the feedback controls are harmless to the permanence of species for the considered system.     

The asymptotic behavior of a nonautonomous eco-epidemic model with disease in the prey

A nonautonomous eco-epidemic model with disease in the prey is formulated and studied. Some sufficient and necessary conditions on the permanence and extinction of the infective prey are established by introducing the new research method. Some sufficient conditions on the global attractivity of the model are presented by constructing a Lyapunov function. Finally, an example is given to show that the periodic model is global attractivity if the infective prey is permanent.     

Permanence and global attractivity of a discrete semi-ratio dependent predator-prey system with Holling II type functional response

In this paper, we propose a discrete semi-ratio dependent predator-prey system with Holling II type functional response. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.     

一类具有脉冲效应的非线性竞争系统的持久及全局吸引性(英文)

通过利用常微分方程比较定理以及构造恰当Lyapounov泛函,研究了一类具有脉冲效应的广义非自治n种群Gilpin-Ayala竞争系统,给出了使系统正解持久以及全局吸引的充分性条件,所得结论改进并推广了一些现有结果.     

Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses

In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. We show the existence of a bounded positive invariant and attracting set. The possibility of existence and uniqueness of positive equilibrium are considered. The asymptotic behavior of the positive equilibrium and the existence of Hopf-bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. The existence and non-existence of periodic solutions are established under suitable conditions. The permanence conditions are also established. We obtained sufficient conditions to ensure the global stability of the unique positive equilibrium, by using suitable Lyapunov functions, LaSalle invariance principle and Dulac’s criterion. We obtained also sufficient conditions for the global stability of the prey-extinction equilibrium when the unique positive equilibrium is not feasible. Finally, numerical simulations are presented to illustrate the analytical results.     

Analysis of a nonautonomous Gompertz population growth model with dispersal in a polluted environment

In this paper, the dynamic behavior of a nonautonomous system with mixed functional response is studied. The population has a history that takes them through two stages, immature and mature. The effects of diffusion on population growth in a polluted patch environment are discussed. Some sufficient conditions on permanence and extinction of population are obtained. Under some appropriate conditions, the asymptotically stability of the periodic solution is obtained. Moreover, a stochastic model is proposed and the conditions for the existence of a global positive solution are discussed.     

Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics

We investigate a system of nonlinear differential equations with distributed delays, arising from a model of hematopoietic stem cell dynamics. We state uniqueness of a global solution under a classical Lipschitz condition. Sufficient conditions for the global stability of the population are obtained, through the analysis of the asymptotic behavior of the trivial steady state and using a Lyapunov function. Finally, we give sufficient conditions for the unbounded proliferation of a given cell generation.     

Global dynamics of one class of nonlinear nonautonomous systems with time-varying delays

A class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques. The results for the nonautonomous systems with time-varying delays are novel, e.g., the global stability of differential equations with nonlinear (casual) Volterra operators is considered for the first time in the literature. Criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying/verifying in practice. We illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous Nicholson-type models.     

Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics

We investigate a system of two nonlinear age-structured partial differential equations describing the dynamics of proliferating and quiescent hematopoietic stem cell (HSC) populations. The method of characteristics reduces the age-structured model to a system of coupled delay differential and renewal difference equations with continuous time and distributed delay. By constructing a Lyapunov–Krasovskii functional, we give a necessary and sufficient condition for the global asymptotic stability of the trivial steady state, which describes the population dying out. We also give sufficient conditions for the existence of unbounded solutions, which describe the uncontrolled proliferation of HSC population. This study may be helpful in understanding the behavior of hematopoietic cells in some hematological disorders.     

ALMOST PERIODIC SOLUTION OF A NONAUTONOMOUS MODIFIED LESLIE-GOWER PREDATOR-PREY MODEL WITH NONMONOTONIC FUNCTIONAL RESPONSE AND A PREY REFUGE

A nonautonomous modified Leslie-Gower predator-prey model with nonmonotonic functional response and a prey refuge is proposed and studied in this paper. Sufficient conditions which guarantee the permanence, extinction of the prey species and the global stability of the system are obtained, respectively. Also, by constructing a suitable Lyapunov function, some sufficient conditions are obtained for the existence of a unique globally attractive positive almost periodic solution of this model. Our results indicate that the prey refuge has positive effect on the coexistence of the species. Examples together with their numeric simulation show the feasibility of our main results.     

EFFECT OF TOXIC SUBSTANCE ON A NONAUTONOMOUS COMPETITIVE SYSTEM

In this paper, we consider a nonautonomous competitive system which is also affected by toxic substances. Some averaged conditions for the permanence of this system are obtained. Our result shows that under some suitable assumption on the coefficients of the system, the toxic has no influence on the permanence of the system. Also, by using a suitable Lyapunov function, sufficient conditions which guarantee the attractivity of any two positive solutions of the system are obtained.