On a generalized population dynamics equation with environmental noise

Abstract

We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions for the following stochastic differential equation dXt=(θXtm0+kXtm)dt+εXtm+12φ(Xt)dWt,t>0,Xt>0,m>m0⩾1,X0=x>0, with θ,k,ε∈R being constants and φ(r)=rϑ or |log(r)|ϑ(ϑ>0), and we also show that the index ϑ>0 is sharp in the sense that if ϑ=0, one can choose certain proper constants θ,k and ε such that the solution Xt will explode in a finite time almost surely.