Deterministic and Stochastic Analysis of a Nonlinear Prey-Predator System

Abstract

The paper deals first with the deterministic analysis of stability and bifurcation of a non-linear model prey-predator system and then with the critical analysis of nonequilibrium fluctuation and stability of the model system under random perturbation including the comparative study of both deterministic and stochastic criteria of stability of the system.

Influence of Gestation Delay and Predator’s Interference in Predator-Prey Interaction under Stochastic Environment

“…Results show that predator's interference plays a significant role in changing the stochastic stability of the system. To study the effect of environmental fluctuation on the time-delayed predator-prey system (27), I have superimposed Gaussian white noises on (2) and then studied nonequilibrium fluctuation and stability of the resulting stochastic model (27) by using Fourier transform technique. Following the criteria of stability in the stochastic environment [30], it is seen that the environmental noises have a destabilizing effect on the system when ℎ ≪ Ω ( = 1, 2).…”

“…In spite of some shortcomings, Gaussian white noise has been proved extremely useful to model rapidly fluctuating phenomena [24,25]. The basic mechanism and factors of population growth like the resources and vital rates birth, death, and so forth change nondeterministically due to random environment and they are the main parameters subject to coupling of the system with its environment [25][26][27][28][29][30][31][32][33][34][35]. Here we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the growth coefficients of the prey and predator since these are the main parameters subject to coupling of a prey-predator pair with its environment [34,35].…”

“…Let me now assume that the system size expansion is valid such that the correlations ( = 1, 2) given by (17) decrease with the increase of the population size and they are assumed to be of the order of the inverse of the population size [24,27,28]:…”

Deterministic and stochastic analysis of a prey–predator model with herd behaviour in both

“…To compute the unknown coefficients, we use the similar techniques used by Valsakumar et al (1983), Bandyopadhyay & Chakrabarti (2003), and Maiti & Samanta (2006):…”

Allee effects in a predator--prey system with a saturated recovery function and harvesting

“…the maximum number of prey that can be eaten by a predator in each time unit and m is the half capturing saturation constant, i.e. the number of prey necessary to achieve one-half of the maximum rate c. Many species models with such functional responses are extensively studied in ecological literature [8,9,10]. In [11], a modified Holling-Tanner predator-prey model with time delay is considered.…”