Dynamical analysis of a prey-predator model incorporating a prey refuge with variable carrying capacity

Highlights

• A prey-predator model incorporating prey refuge with variable carrying capacity and Holling type-II functional response is proposed and analyzed.

• The model includes a case of increasing carrying capacity as well as a decreasing carrying capacity case.

• Sufficient conditions are derived to ensure the existence and local stability of the equilibrium points of the proposed model. The occurrence of transcritical bifurcation as well as Hopf bifurcation are investigated.

• The effect of some model parameter related to the prey refuge and the variable carrying capacity on the prey-predator dynamics has been examined.

• Numerical simulations are presented to demonstrate the theoretical results and to illustrate the effect of these parameters on the model dynamics.

Abstract

A prey-predator model incorporating prey refuge with variable carrying capacity and Holling type-II functional response is proposed and analyzed. The model includes a case of increasing carrying capacity as well as a decreasing carrying capacity case. Sufficient conditions are derived to ensure the existence and local stability of the equilibrium points of the proposed model. Moreover, the occurrence of transcritical bifurcation as well as Hopf bifurcation are investigated. The effect of some model parameter related to the prey refuge and the variable carrying capacity on the prey-predator dynamics has been examined. Numerical simulations are presented to demonstrate the theoretical results and to illustrate the effect of these parameters on the model dynamics. Moreover, a comparison with the constant carrying case has been presented.

Introduction

The study of prey refuge impact on the prey-predator systems has received a lot of attention from researchers in theoretical ecology. It has been observed that prey refuge is one of the ecological factors which has a significant potential to change the dynamics of such systems, see for example Chakraborty et al. (2017); Jana and Ray (2016); Kar (2005); Ma et al. (2013); Mukherjee (2016); Samanta et al. (2016); Sundari and Valliathal (2018). Different forms of refuges can be used by prey to reduce predation rate such as spatial or temporal refuge, behavioral or physical refuge, prey aggregation or reducing search activity by prey, for more details see Chakraborty et al. (2017); Jana and Ray (2016). In general, prey refuge plays an important role in stabilizing the dynamics of prey-predator interactions and in preventing the extinction of the prey species, as the prey protects itself from being attacked by predators. Several authors have studied the effect of a prey refuge on stability of prey-predator dynamics, see for exampleChakraborty and Bairagi (2019); González-Olivares and Ramos-Jiliberto (2003); Huang et al. (2006); Ma et al. (2013); Mukherjee (2016); Samanta et al. (2016); Sih (1987); Sundari and Valliathal (2018).

Moreover, changes in physical and biological processes of the environment may result changes in the carrying capacity for a specie which in turn affects the dynamics of prey-predator systems. There are positive changes which lead to an extension and increase in the carrying capacity such as food production and new resources using developing technologies Safuan (2015); Safuan et al. (2013). Evidently, changes in technology modify the carrying capacity of a specie, for example irrigation and fertilization in agriculture will raise the resources which can support an increasing population whereas cropping decreases amount of landMeyer and Ausubel (1999); Waggoner (1995). However, food depletion, pollution, disease, starvation, hunting can be considered as negative factors that results in loss of habitat and hence decline the carrying capacity. Thus, carrying capacity is one of the most significant concept as it controls how quick and up to what level a population can grow Ganguli et al. (2017). In most of the literature, carrying capacity is assumed to be constant. However, in a dynamically changing environment, it is reasonable to consider the carrying capacity as a model variable (i.e., a function of time) Al-Moqbali et al. (2018); Ang and Safuan (2019); Ganguli et al. (2017); Meyer and Ausubel (1999). In addition, in reality, the resources are finite and limited, so there must be some bounds within the growth or the decline of carrying capacity. These assumptions provide more realistic understanding about prey-predator dynamics and can be described by carrying capacity with saturation which was used to model enrichment in an inland sea by a nutrienta Ikeda and Yokoi (1980) and to describe the adoption of new technologies in carrying capacity of the developed world Meyer and Ausubel (1999). Furthermore, in Al-Moqbali et al. (2018), the authors studied prey-predator models with variable carrying capacity. They have assumed that the carrying capacity is modeled by a logistic equation that increases between an initial value $k_0>k_1$ (a lower bound for the carrying capacity) and a final value $k_1+k_2$ (an upper bound for the carrying capacity). Then, they investigated the effect of variable carrying capacity by analyzing the stability and carrying out the numerical simulations. They observed that for low growth rate of variable carrying capacity, the periodicity in the solutions of the systems decreases its magnitude. Also, the solutions reach the stable equilibrium faster compared to the constant carrying capacity case.

The main objective of this paper is to investigate the effects of incorporating both prey refuge and variable carrying capacity in prey- predator dynamics. We assume that the carrying capacity changes over time and obeys logistic growth or decay model depending on its initial and limiting values and we also assume that a constant proportion of the prey takes refuge, which provides them protection from predator.

The rest of this paper is organized as follows: in the next section, we present the proposed mathematical model for the prey-predator interactions with varying carrying capacity and constant prey refuge. In section 3, we present the mathematical analysis of the proposed model: local stability of equilibrium points and transcritical and Hopf bifurcations. Numerical simulations are carried out in section 4 to illustrate the theoretical results and to demonstrate the effect of some model parameters on prey-predator dynamics, namely those related to the varying carrying capacity and the prey refuge. Finally, in Section 5, a brief conclusion is presented.