Abstract
Under consideration is construction of a model of age-structured population reflecting random oscillations of the death and birth rate functions. We arrive at an Ito-type difference equation in a Hilbert space of functions which can not be transformed into a proper Ito equation via passing to the limit procedure due to the properties of the operator coefficients. We suggest overcoming the obstacle by building the model in a space of Hilbert space valued generalized random variables where it has the form of an operator-differential equation with multiplicative noise. The result on existence and uniqueness of the solution to the obtained equation is stated.
Highlights
IntroductionA well known model of an age-structured population dynamics is the famous McKendrick–von
A well known model of an age-structured population dynamics is the famous McKendrick–vonFoerster equation ∂u(x, t) ∂t + ∂u(x, t) ∂x = −m(x)u(x, t), (0.1)x2 where u(x, t) is density of the population at age x at time t (so, that u(s, t)ds is the number of x1 individuals with the age belonging to [x1; x2] at the time t) and m(x) is the death rate
We show that the obstacle connected with non-differentiability of the Brownian sheet can be overcome with the help of the concept of a cylindrical random variable on a Hilbert space
Summary
A well known model of an age-structured population dynamics is the famous McKendrick–von. The aim of our work is clarification of this question in order consistent with the desired properties of the noisy influence on the population Since both of the rates are described by functions m and b of age x ∈ [0; 1], it seems natural to model these oscillations by appropriate random processes taking values in spaces of functions of x and to build a model having form of a stochastic equation in such a space. We show that the random fluctuations of the death rate can be modeled by increments of a cylindrical Wiener process. We arrive at a model having form of an operator-differential equation in (S)−ρ(H) and formulate the existence and uniqueness result for the Cauchy problem for this equation
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