Development and analysis of a malaria transmission mathematical model with seasonal mosquito life-history traits

In this paper, we develop and analyze a malaria model with seasonality of mosquito life-history traits: periodic-mosquitoes per capita birth rate, -mosquitoes death rate, -probability of mosquito to human disease transmission, -probability of human to mosquito disease transmission, and -mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation system. We provide a global analysis of the model depending on two threshold parameters and (with ). When , then the disease-free stationary state is locally asymptotically stable. In the presence of the human disease-induced mortality, the global stability of the disease-free stationary state is guarantied when . On the contrary, if , the disease persists in the host population in the long term and the model admits at least one positive periodic solution. Moreover, by a numerical simulation, we show that a sub-critical (backward) bifurcation is possible at . Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malaria-epidemic region in Mpumalanga province in South Africa.     

一类食饵采取鹰鸽对策的阶段结构捕食模型性态分析

建立和研究了一类食饵采取鹰鸽对策的阶段结构捕食模型.当收益大于损失时,成年食饵种群全都采取鹰策略,证明了系统的一致持续生存性,利用Lyapunov函数证明了正平衡点的全局稳定性.当收益小于损失时,成年食饵种群采取鹰鸽混合策略,讨论了边界平衡点的存在性,并分析了fold分支.通过极限系统理论和杜拉克判据研究了边界平衡点的全局稳定性.     

Dynamics of an epidemic model with relapse and delay

In this paper, we consider a new epidemiological model with delay and relapse phenomena. Firstly, a basic reproduction number $R_0$ is identified, which serves as a threshold parameter for the stability of the equilibria of the model. Then, beginning with the delay-free model, the global asymptotic stability of the equilibria is obtained through the construction of suitable Lyapunov functions. For the delay model, the stability of the positive equilibrium and the existence of the local Hopf bifurcation are discussed. Furthermore, the application of the normal form theory and center manifold theorem is used to determine the direction and stability of these Hopf bifurcations. Finally, we shed light on corresponding biological implications from a numerical perspective. It turns out that time delay affects the stability of the positive equilibrium, leading to the occurrence of periodic oscillations and disease recurrence.     

Persistence in a discrete-time,stage-structured epidemic model

Discrete-time SI and SIR epidemic models, formulated by Emmert and Allen [J. Differ. Equ. Appl., 10 (2004), pp. 1177–1199] for the spread of a fungal disease in a structured amphibian host population, are analysed. Criteria for persistence of the population as well as for persistence of the disease are established. Global stability results for host extinction and for the disease-free equilibrium are presented.     

Uniform persistence and flows near a closed positively invariant set

In this paper, the behavior of a continuous flow in the vicinity of a closed positively invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equivalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider varity of ecological models.     

A conjecture by G. Ladas

In this paper, a sufficient condition for boundedness and persistence of the solu1 /of the following delay difference equation is obtained. A conjecture by G. Ladas is proved here xn-1=A/x^pn B/x^qn-1, n=0,1,...where A, B, p, q, x-1, xo∈(0,∞).     

一类具有非单调型功能反应的捕食-食饵扩散系统的持久性和全局吸引性

利用微分方程的比较原理和重合度理论中的延拓定理,及Lyapunov函数和Barbalat引理,研究了一类具有双密度制约和非单调型功能性反应的捕食-食饵扩散系统的持久性和全局吸引性,获得了周期系统存在唯一全局渐近稳定正周期解的充分条件.     

Some Results on Sinai Diffusion

The diffusion of a particle in a one-dimensional random force field (Sinai diffusion) is studied using the replica method. This method, which maps the problem onto a quantum problem, is shown to be a simple and direct way to calculate the long-time diffusive behavior. Results for the distribution of the local Green's function, the particle distribution, and persistence are obtained.     

一类食饵捕食者模型的渐近性态

研究了一类带周期系数的阶段结构的食饵-捕食者模型的渐近性态。利用比较原理和泛函数微分方程定理建立了种群一致持续生存的条件，并得到了正周期解的存在性。     

具有Beddington-DeAngelis功能反应的捕食模型

研究了一类具有Beddington-DeAngelis 功能反应的渐近周期捕食模型,得到了该系统一致强持久的充分条件.