Explicit form of global solution to stochastic logistic differential equation and related topics

Abstract

This paper presents the explicit form of positive global solution to stochastically perturbed nonautonomous logistic equation $$dN(t)=N(t)\left[(a(t)-b(t)N(t))dt+\alpha(t)dw(t)+\int_{\mathbb{R}}\gamma(t,z)\tilde\nu(dt,dz)\right],\ N(0)=N_0,$$ where $w(t)$ is the standard one-dimensional Wiener process, $\tilde\nu(t,A)=\nu(t,A)-t\Pi(A)$, $\nu(t,A)$ is the Poisson measure, which is independent on $w(t)$, $E[\nu(t,A)]=t\Pi(A)$, $\Pi(A)$ is a finite measure on the Borel sets in $\mathbb{R}$. If coefficients $a(t), b(t), \alpha(t)$ and $\gamma(t,z)$ are continuous on $t$, $T$-periodic on $t$ functions, $a(t)>0, b(t)>0$ and $$\int_{0}^{T}\left[a(s)-\alpha^2(s)-\int_{\mathbb{R}}\frac{\gamma^2(s,z)}{1+\gamma(s,z)}\Pi(dz)\right]ds>0,$$ then there exists unique, positive $T$-periodic solution to equation for $E[1/N(t)]$.