In this paper, we investigate the following non-local Cauchy problem of nonlinear population evolution equation with random periodic migration perturbation,{(p(r,t))/(t)+(p(r,t))/(r)=-μ(r)p(r,t)+f(t,p(r,t)),0<r<rm,t0p(r,0)=p0(r)+g(p(r,t0)),T>t0>0 p(0,t)=β(t)integral from n=r1 to r2 k(r)h(r)p(r,t)dr, t≥0.

It was widely used in many fields such as Schrodinger equation, heat equation and nonlocal abstract Cauchy problem.

The upper and lower solution method of nonlocal problem for the first order ordinary differential equation;

The nonlocal problems of semilinear parabolic equations and systems are studied.

The singularly perturbed nonlocal problem (when 0<ε<<1) often Occurs in the model problem of many physical phenomena.

Constructive method for the Cauchy problem of parabolic equations and its mechanization;

Cauchy problem for a class of coupled hyperbolic systems of conservation laws;

The generalization of the sabstitution-iteration method for high-dimension wave equation s Cauchy problem;

The propagation of singularities of solutions to the Cauchy problems of a linear thermoelastic system for microstretch materials in one space variable is studied by using an argument of frequencies analysis.