捕食者具有传染病的捕食系统的全局稳定性

Global stability of a prey-predator system with epidemic in predator

疾病可以在不同的种群之间传播。研究疾病在相互作用种群之间的传播规律,是种群生态学与传染病动力学的一种结合。通过假设捕食者和食饵均是密度制约、捕食者具有传染病、染病的捕食者不能捕食、染病的捕食者可以恢复但具有暂时的免疫力,建立了一类食饵-捕食系统的SIS传染病模型,利用比较定理研究了解的有界性,利用特征根法和Hurwitz判据分析了系统的无病平衡点和地方病平衡点的局部稳定性,通过构造Lyapunov函数,讨论了无病平衡点和地方病平衡点的全局渐近稳定性,从而得到了疾病流行与否的阈值R,并证明当R≤1时无病平衡点全局渐近稳定,从而疾病消除;当R>1时,地方病平衡点全局渐近稳定,从而疾病流行。

Disease can be spread between different populations. The study of disease spread rules in the interacted populations is a kind of combination between population ecology and epidemic dynamics. By assuming that predators and preys are of density-dependence, the predator has infectious disease, the infected predator couldn＇t prey, and they can recover and has temporary immunity, an SIS epidemic model of a predator-prey system is established. Boundedness of solution is studied by using the comparison theorem. Local stabilities of the disease-free equilibrium and the endemic equilibrium are analyzed by using the characteristic root method and Hurwitz criterion. By constructing the Lyapunov function the global stabilities of the disease-free equilibrium and the endemic equilibrium are discussed, then the threshold value of determining spread of disease or not is obtained. It is proved that the disease-free equilibrium is global asymptotical stable when R ≤1, so that the disease is eliminated, while the endemic equilibrium is globally asymptotical stable for the case of R〉1, and therefore disease is spread.