Dynamics of a stochastic Holling IV predator-prey model with infinite delay

Deterministic Holling-IV predator-prey model is extensively applied in biological and ecological modeling. However, most real processes undergo small noise perturbations from environmental factors or intrinsic uncertainties, and it is shown that such small perturbations could significantly influence the dynamics of these processes. Building upon the work of Han et al. Commun. Nonlinear Sci. Numer. Simul. 128:107596, 2004, this study further analyzes the dynamical behavior of a stochastic Holling-IV predator-prey model with infinite delay. Firstly, we establish the existence and uniqueness of a global positive solution. Secondly, we formulate a new lemma to derive three critical values and build the sufficient conditions for exponential extinction and persistence of species within ecosystems. Thirdly, we provide an approximate expression for the probability density function of the model. The validity of the approximation is verified using the Kolmogorov-Smirnov test. Unknown parameters within the density function are estimated via maximum likelihood estimation. Finally, we carry out numerical simulations to explore the effects of linear white noise, infinite delay and critical parameters on the dynamical behavior of the model.