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On a generalized population dynamics equation with environmental noise

Rongrong Tian, Jinlong Wei, Jiang-lun Wu Orcid Logo

Statistics & Probability Letters, Volume: 168, Start page: 108944

Swansea University Author: Jiang-lun Wu Orcid Logo

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DOI (Published version): 10.1016/j.spl.2020.108944

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 f...

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Published in: Statistics & Probability Letters
ISSN: 0167-7152
Published: ELSEVIER Elsevier BV 2021
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa55191
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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 followingstochastic differential equation\begin{eqnarray*}dX_t=(\theta X_t^{m_0}+kX_t^m)dt+\varepsilon X_t^{\frac{m+1}{2}}\varphi(X_t)dW_t, \ t>0, \ X_t>0,\ m> m_0\geq 1,\ \X_0=x>0,\end{eqnarray*}with $\theta,k,\varepsilon\in \mR$ being constants and $\varphi(r)=r^{\vartheta}$ or $|\log(r)|^{\vartheta} \ (\vartheta>0)$, and we also show that the index $\vartheta>0$ is sharp in the sense that if $\vartheta=0$, one can choose certain proper constants $\theta,k$ and $\varepsilon$ such that the solution $X_t$ will explode in a finite time almost surely.
Keywords: Stochastic differential equation; Environmental noise; Explosion; Positive strong solution.
College: Faculty of Science and Engineering
Start Page: 108944

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