Dynamics of a Stochastic Predator–Prey Model with Smith Growth Rate and Cooperative Defense

• DOI: 10.3390/math12121796
• Publication Date: 2024-06-10T12:49:03Z
• Authors: Qiuyue Zhao; Xinglong Niu
• Keywords: extinction; 0103 physical sciences; QA1-939; cooperative defense; stochastic perturbation; Smith growth rate; 01 natural sciences; Mathematics; stationary distribution

The random changes in the environment play a crucial role in the sustainability of ecosystems. Usually, the construction of stochastic models does not take into account the non-linear growth of intrinsic growth rate. In addition, prey only considers the collective response of the population when encountering predators and ignores the role of individual prey. To address this issue, we contemplate the dynamics of a stochastic prey–predator model with Smith growth rate and cooperative defense. The population density of prey is measured by mass, and the growth limitations are based on the proportion of unused available resources. Additionally, the grazing pattern of the predator incorporates cooperative characteristics into the functional response. We carry out existence and uniqueness analysis for the global positive solution. Then, we construct sufficient conditions for the existence of an ergodic stationary distribution of positive solutions for investigating whether prey and predator populations continue to survive. Numerical examples indicate that the Smith growth rate, cooperative defense and environmental disturbance play crucial roles in the coexistence of interacting populations.