We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat O, governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on \(\partial O\) with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness.

中文翻译：

---

线性乘性高斯噪声扰动的随机种群动力学方程的重标度方法

我们关注一个非线性的非自治模型，该模型由一个方程式表示，该方程式描述了在空间栖息地O中扩散的年龄结构人口的动力学，受本地Lipschitz生命因子控制，并且受人口率的随机行为（可能表示移民，移民和偶然性）的控制。死亡。该模型由随机初始条件，\（\ partial O \）上的通量类型边界条件完成人口密度的随机跳跃和在零年龄给出的非局部非线性边界条件。随机影响由方程中的线性乘法高斯噪声摄动表示。主要结果证明了随机模型的适定性，其解属于路径连续函数类，并且满足有关年龄和空间的某些特殊规律。该方法基于将随机方程重新缩放为具有局部Lipschitz非线性的时间确定性随机双曲抛物线方程。通过组合半群，变分和逼近技术，证明了随机确定性方程强解的存在性和唯一性。