On the dynamics of a discrete pest-natural enemy model with Holling II response and stochastic perturbations

The survival environment of pests and natural enemies is often subject to various stochastic factors, and the introduction of white noise can better simulate such uncertainty. Compared to continuous models, discrete models are more accurate in describing the situations where the successive generations of many species in nature are non-overlapping and the data of biological samples are usually collected in discrete time. Furthermore, when natural enemies are confronted with abundant prey, the predation rate does not increase indefinitely but instead tends to saturate due to the limitation of handling capacity. The Holling II functional response captures this saturation phenomenon in the predation process, which is more consistent with the behavior of natural enemies in actual ecosystems. Based on these, a stochastic discrete pest control model with a Holling II functional response is established, incorporating white noise that affects the growth rate of pest populations and the mortality rate of natural enemy populations. In this paper, we assume that the stochastic variables in the model are independent, doubly truncated stochastic sequences following the standard normal distribution. By constructing auxiliary equations, we prove the positivity and boundedness of the solutions, derive sufficient conditions for pest extinction, and analyze the effects of white noise on population survival and extinction. Numerical simulations are conducted to verify the correctness of the theoretical results and to discuss the effects of key parameters on pest extinction. The results indicate that white noise of relatively low intensity does not alter the survival or extinction of populations, however, if the intensity of white noise affecting a population is too high, it can lead to the extinction of that population.