Explicit numerical approximation for logistic models with regime switching in finite and infinite horizons

TL;DR

This paper introduces a positivity-preserving explicit numerical scheme for stochastic logistic models with regime switching, ensuring accurate approximation of solutions and long-term behaviors without additional restrictions.

Contribution

The paper develops a novel explicit truncated Euler-Maruyama scheme that preserves positivity and converges strongly with a rate of 1/2, applicable to models with regime switching.

Findings

The scheme achieves strong convergence with rate 1/2.

It accurately captures long-term dynamical properties like stability and extinction.

Simulations confirm theoretical convergence and validity.

Abstract

The stochastic logistic model with regime switching is an important model in the ecosystem. While analytic solution to this model is positive, current numerical methods are unable to preserve such boundaries in the approximation. So, proposing appropriate numerical method for solving this model which preserves positivity and dynamical behaviors of the model's solution is very important. In this paper, we present a positivity preserving truncated Euler-Maruyama scheme for this model, which taking advantages of being explicit and easily implementable. Without additional restriction conditions, strong convergence of the numerical algorithm is studied, and 1/2 order convergence rate is obtained. In the particular case of this model without switching the first order strong convergence rate is obtained. Furthermore, the approximation of long-time dynamical properties is realized, including the stochastic permanence, extinctive and stability in distribution. Some simulations and examples are provided to confirm the theoretical results and demonstrate the validity of the approach.