Competitive Lotka-Volterra Population Dynamics with Jumps

TL;DR

This paper studies stochastic Lotka-Volterra models with jumps, proving existence, boundedness, and explicit solutions, and analyzing population extinction and Lyapunov exponents in a stochastic setting.

Contribution

It provides the first explicit solutions for 1D Lotka-Volterra models with jumps and analyzes stability and extinction in multi-dimensional stochastic populations.

Findings

Unique global positive solutions established

Boundedness of moments and Lyapunov exponents analyzed

Explicit solutions and extinction criteria derived

Abstract

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) We discuss the uniform boundedness of $p$th moment with $p>0$ and reveal the sample Lyapunov exponents; (c) Using a variation-of-constants formula for a class of SDEs with jumps, we provide explicit solution for 1-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our $n$-dimensional model.