We prove existence and stability of entropy solutions for a predator-prey system consisting of an hyperbolic equation for predators and a parabolic-hyperbolic equation for preys. The preys' equation, which represents the evolution of a population of voles as in [2], depends on time, t, age, a, and on a 2-dimensional space variable x, and it is supplemented by a nonlocal boundary condition at $a=0$. The drift term in the predators' equation depends nonlocally on the density of preys and the two equations are also coupled via classical source terms of Lotka-Volterra type, as in [4]. We establish existence of solutions by applying the vanishing viscosity method, and we prove stability by a doubling of variables type argument.

我们证明了一个由捕食者的双曲方程和被捕食者的抛物-双曲方程组成的捕食者-被捕食系统的熵解的存在性和稳定性。代表[2]中的田鼠种群进化的猎物方程取决于时间t，年龄，a和二维空间变量x，并且由一个非局部边界条件补充。$\mathrm{一种}=0$。捕食者方程中的漂移项非局部地取决于猎物的密度，并且这两个方程还通过Lotka-Volterra类型的经典源项进行耦合，如[4]所示。我们采用消失粘度法建立了解的存在性，并通过将变量类型参数加倍来证明稳定性。