The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator

Abstract

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators' gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

Economic Growth by Means of Interspecific Functional Response of Capital-Labour in Dynamical System

“…After the contrivance of theoretical ecology by Lotka [10] and Volterra [11], the models, named as Lotka-Volterra equations, played an important role in different fields of applied sciences [12][13][14][15]. The novelty in this attempt is the inclusion of Holling-type II function [16][17][18][19][20] in the system, which can effectively measure the functional response between the capital and the labour force. The existence of competitive interaction between these two factors produces a major change in the production process that yields a salient change in economic growth.…”