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

Abstract

This paper aims to study the dynamical behaviours of a prey-predator system, where both the prey and predator show herd behaviours. Positivity, boundedness, stability of equilibrium points, et cetera, are discussed in deterministic environment. To incorporate the effect of fluctuating environment, we have perturbed the birth rate of prey species and death rate of predator species by Gaussian white noises. Then the resulting model is cultured by the method of statistical linearization to study the stability and non-equilibrium fluctuation of the populations in stochastic environment. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically. ARTICLE HISTORY

Deterministic and stochastic analysis of a predator–prey model with Allee effect and herd behaviour

“…These definitely underpin the utility and appeal of the white-noise-idealization in applications to natural systems. 50 Recently, some researchers have utilized white noises to investigate the impact of fluctuating environment (see Maiti et al 49 and references therein). From this viewpoint, we formulate the stochastic version of the system (2.2) (using white noises) to study the effect of environmental fluctuations.…”

“…Recently, many researchers have tried to manifest the sensibility of stochastic models (see Dimentberg, 40 Samanta, 41,42 Maiti and Samanta, 43,44 Maiti et al, 45 Bandyopadhyay and Chakrabarti, 46 Bandyopadhyay and Chattopadhyay, 47 and references cited therein). Very recently, stochastic analysis of predator–prey systems with herd behaviour has been done by Bera et al 48 and Maiti et al 49 Modelling and analysis of the effect of a fluctuating environment on a predator–prey system with Allee effect and herd behaviour would undoubtedly be a captivating area of research. But this area has so far remained unexplored.…”

Dynamical study of the theta-logistic predator-prey model incorporating gregarious behavior of prey

“…In our existing literature, we also observe the study of monotonic and non-monotonic functional response ( [11], [16]), sigmoid functional response [29], Ivlev-type functional response [38], ratio-dependent functional response ( [34], [30]). In the square root functional response ( [25], [27], [7], [4], [6], [31], [12], [1], [40], [28]), the predator interacts with prey along the outer corridor of the prey herd. In this paper, we generalize the square-root functional response as x b where 0 < b ≤ 1.…”

Prey herd behavior modeled by a generic non-differentiable functional response

“…In recent years, a type of antipredator behavior (APB) called prey herd behavior has received the attention of modelers [1,43,60]. A prey species exhibits herd behavior when the individuals demonstrate collective social conduct to avoid predation and when each realizes a typical reaction equal to that effectuated by the majority of the other members [40].…”

“…The drivers of prey herd behavior are several and varied, and this behavior works as protection against predators (predators might hesitate to attack a large group of animals) [40]. Generally, the strongest individuals are on the border, and the weakest are concentrated in the middle of the group.…”