Almost periodic solution of non-autonomous Lotka-Volterra predator-prey dispersal system with delays

This paper studies a non-autonomous Lotka-Volterra almost periodic predator-prey dispersal system with discrete and continuous time delays which consists of n-patches, the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. By using comparison theorem and delay differential equation basic theory, we prove the system is uniformly persistent under some appropriate conditions. Further, by constructing suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. By using almost periodic functional hull theory, we show that the almost periodic system has a unique globally asymptotical stable strictly positive almost periodic solution. The conditions for the permanence, global stability of system and the existence, uniqueness of positive almost periodic solution depend on delays, so, time delays are "profitless". Finally, conclusions and two particular cases are given. These results are basically an extension of the known results for non-autonomous Lotka-Volterra systems.     

扩散与时滞有关的单种群植物模型的全局稳定性

研究了具有周期系数及扩散与时滞有关的单种群植物模型,得到了该系统存在全局渐近稳定的正周期解的充分条件,其中该条件依赖时滞与扩散     

非自治阶段结构合作系统的持久性与周期解

本文研究一类非自治阶段结构的合作系统,得到系统的最终有界性,对应周期系统正周期解的存在性,唯一性以及全局渐近稳定性的充分条件。     

Periodic solutions to nonautonomous difference equations

A technique is presented for determining when periodic solutions to nonautonomous periodic difference equations exist. Under certain constraints, stable periodic solutions can be guaranteed to exist, and this is used to compare the analogous behavior of a nonautonomous periodic hyperbolic difference equation to that of the nonautonomous periodic Pearl-Verhulst logistic differential equation.     

具有经济阈值的植物疾病模型

研究了具有经济阈值和人文控制策略的植物疾病模型．根据某一参数的三种情况分析了唯一的正的周期解的存在性,并利用定性理论给出了在该参数某种范围下周期解全局稳定的充分条件,同时得到在其它两种情况下周期解的不稳定性．文章所得结论推广了综合疾病管理中植物疾病模型的经典结论．     

Periodic difference equations, population biology and the Cushing-Henson conjectures

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of Cushing and Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We give an equality linking the average population with the growth rates and carrying capacities (in the 2-periodic case) which shows that out-of-phase oscillations in these quantities always have a deleterious effect on the average population. We give an example where in-phase oscillations cause the opposite to occur.     

Eigensolutions of nonlinear wave equations in one dimension

A class of nonlinear equations describing the steady propagation of a disturbance on the infinite interval in one dimensional space are shown under certain conditions to admit solution with a unique velocity of propagation. The class of equations describe both initial and final homogeneous steady states which are asymptotically stable with respect to uniform perturbations, in contrast to the Fisher equation, which does not.     

A Malaria Transmission Model with Temperature-Dependent Incubation Period

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio \(R_0\) and establish a threshold type result on the global dynamics in terms of \(R_0\), that is, the unique disease-free periodic solution is globally asymptotically stable if \(R_0<1\); and the model system admits a unique positive periodic solution which is globally asymptotically stable if \(R_0>1\). Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

Erratum to: Mathematical properties of pump-leak models of cell volume control and electrolyte balance

Homeostatic control of cell volume and intracellular electrolyte content is a fundamental problem in physiology and is central to the functioning of epithelial systems. These physiological processes are modeled using pump-leak models, a system of differential algebraic equations that describes the balance of ions and water flowing across the cell membrane. Despite their widespread use, very little is known about their mathematical properties. Here, we establish analytical results on the existence and stability of steady states for a general class of pump-leak models. We treat two cases. When the ion channel currents have a linear current-voltage relationship, we show that there is at most one steady state, and that the steady state is globally asymptotically stable. If there are no steady states, the cell volume tends to infinity with time. When minimal assumptions are placed on the properties of ion channel currents, we show that there is an asymptotically stable steady state so long as the pump current is not too large. The key analytical tool is a free energy relation satisfied by a general class of pump-leak models, which can be used as a Lyapunov function to study stability.     

具有扩散的非自治两种群Lotka-Volterra模型的概周期问题

研究非自治两种群竞争系统,其中一种群可以在两个斑块之间扩散。而另一种群在一个斑块中,不能扩散。本文结合运用Liapunov函数,得到该系统唯一存在全局渐近稳定的正概周期解的条件．     

具有周期系数和连续时滞的扩散模型的周期解

本文讨论了具有周期系数和连续时滞的竞争扩散模型,得到了保证其存在唯一周期解及全局渐近稳定的充分条件．     

Dynamics of an HIV-1 therapy model of fighting a virus with another virus

In this paper, we rigorously analyse an ordinary differential equation system that models fighting the HIV-1 virus with a genetically modified virus. We show that when the basic reproduction ratio ?(0)<1, then the infection-free equilibrium E (0) is globally asymptotically stable; when ?(0)>1, E (0) loses its stability and there is the single-infection equilibrium E (s). If ?(0)∈(1, 1+δ) where δ is a positive constant explicitly depending on system parameters, then the single-infection equilibrium E (s) that is globally asymptotically stable, while when ?(0)>1+δ, E (s) becomes unstable and the double-infection equilibrium E (d) comes into existence. When ?(0) is slightly larger than 1+δ, E (d) is stable and it loses its stability via Hopf bifurcation when ?(0) is further increased in some ways. Through a numerical example and by applying a normal form theory, we demonstrate how to determine the bifurcation direction and stability, as well as the estimates of the amplitudes and the periods of the bifurcated periodic solutions. We also perform numerical simulations which agree with the theoretical results. The approaches we use here are a combination of analysis of characteristic equations, fluctuation lemma, Lyapunov function and normal form theory.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

三种群食饵系统的平稳振荡

本文研究了一类三种群周期食饵系统,并给出了其存在平稳振荡的充分条件。     

Periodic solutions of some ecological models

A result is given on the existence of an asymptotically stable periodic solution of a class of systems of periodic ordinary differential equations. The result is applied to a lake eutrophication model with seasonal effects, and some suggestions are made for the solution of such models.     

稀疏效应下周期系数捕食-被捕食系统的全局渐近稳定性

研究一类稀疏效应下周期系数捕食-被捕食系统,得到了该系统存在唯一全局渐近稳定的正周期解的充分条件．     

A global stability criterion for simple control loops

Summary A method for studying biological control loops has been developed, which suffices to prove global stability for the Goodwin equations when the Hill coefficient is equal to 1. This holds for arbitrary reaction constants, even if time delays are included in the system. For a generalized class of repfessible systems, including the Goodwin Equations for >1, the method gives a sufficient condition for global stability, in terms of solutions of an algebraic equation in a single variable. When the criterion is not satisfied, the same equation gives bounds on any possible limit cycles. The method also shows that inducible systems with a unique equilibrium are globally stable. The system of equations studied allows each reaction rate equation to be non-linear, and to include a time delay.