Analysis of a stochastic two-patch model with nonlinear birth rate and dispersal driven by the Ornstein–Uhlenbeck process

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Abstract

This paper proposes a novel stochastic two-patch model that incorporates a Beverton–Holt-type nonlinear birth rate and dispersal driven by an Ornstein–Uhlenbeck process. We first establish the existence and uniqueness of a global positive solution, thereby ensuring the biological well-posedness of the model. By constructing appropriate Lyapunov functions and applying Itô’s formula, we further demonstrate the existence of a stationary distribution, which characterizes the long-term statistical behavior of the system. Sufficient conditions for population extinction are then derived, providing criteria for predicting when the population is destined to disappear. We perform numerical simulations using the Euler–Maruyama method to validate the main theoretical results. Finally, as an empirical application, the stochastic model achieves a good fit to zooplankton data, confirming its practical validity and offering important insights for mathematical ecology, as well as for species conservation, population regulation, and resource management in fragmented habitats.

Keywords

stochastic two-patch model / Ornstein-Uhlenbeck process / stationary distribution / extinction

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