On a generalized population dynamics equation with environmental noise

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 dX(t) = (theta X-t(m0) + kX(t)(m))dt + epsilon X-t m+1/2 phi(X-t)dW(t), t > 0, X-t > 0, m > m(0) >= 1, X-0 = x > 0, with theta, k, epsilon is an element of R being constants and phi(r) = r nu or vertical bar log(r)vertical bar nu (nu > 0), and we also show that the index nu > 0 is sharp in the sense that if nu = 0, one can choose certain proper constants theta, k and epsilon such that the solution Xt will explode in a finite time almost surely. (C) 2020 Elsevier B.V. All rights reserved.