The long-term dynamic behaviors of a stochastic predator-prey model with fear and group defense

This paper analyzes a realistic predator-prey model accounting for the stochastic environment, fear and group defense in modeling. Employing stochastic analysis techniques such as ergodic theory, stochastic comparison theorem and occupation measure theory, we provide a group of sharp threshold-type conditions to characterize the long-term behaviors of this stochastic predator-prey model, including persistence, extinction, and the corresponding issues of ergodicity. When the population is persistent, we additionally give its ultimate stationary distribution. Furthermore, our results demonstrate that this stationary distribution can be effectively approximated by simulating the empirical measure of one sample path, which provides a convenient way for practical simulation and prediction. Our results reveal that excessive environment noise intensities will be detrimental to the survival of the prey and predator populations, whereas small environment noise intensities promote harmonious coexistence of the prey and predator populations.