Elsevier

Applied Mathematics and Computation

Volume 273, 15 January 2016, Pages 54-67
Applied Mathematics and Computation

A three species dynamical system involving prey–predation, competition and commensalism

https://doi.org/10.1016/j.amc.2015.09.036Get rights and content

Abstract

In this paper, a three species dynamical system is explored. The system consisting of two logistically growing competing species and the third species acts as a predator as well as host. It is predating over second species with Holling type II functional response, while first species is benefited from the third species. In addition, the prey species move into a refuge to avoid high predation. The essential mathematical features of the proposed model are studied in terms of boundedness, persistence, local stability and bifurcation. The existence of transcritical bifurcations have been established about two axial points. It has been observed that survival of all three species may be possible due to commensalism. Numerical simulations have been performed to show the Hopf bifurcation about interior equilibrium point. The existence of period-2 solution is observed. Further, the bifurcations of codimension-2 have also been investigated.

Introduction

The study of ecological models has created much interest among authors. Most of the two species models consider only one type of interaction at a time: prey–predation, competition or mutualism. The prey–predator models with Holling types I, II, III and IV are discussed in [1], [2], [3], [4], [5], [6]. The Leslie–Gower, ratio-dependent and Beddington–DeAngelis functional responses are also extensively investigated in [7], [8], [9], [10], [11], [12]. To avoid high risk of predation, prey species defend themselves by taking refuge. The term refuge can be generally defined to include any strategy that decreases predation risk. Many authors have studied refuge of prey species in [13], [14], [15]. Lot of work is available on two species competing systems but limited work is available on mutualism [16], [17], [18], [19]. The stability, extinction, co-existence, bifurcations and limit cycles are the typical behaviors investigated for such nonlinear two dimensional systems.

These models are extended for three species food chain and food-web systems [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. Food-web systems have been studied with various combinations of prey–predation and competition in [30], [31]. These three species systems exhibit rich dynamical behaviors as compared to their counter parts in two species systems. The existence of chaos, strange attractors and quasi-periodic solutions may be possible in food-webs and food chains. Few food-webs are also investigated considering mutualism in [32], [33]. Commensalism is a form of mutualism between individuals of two species in which one is benefited and other is unaffected. The commensal may obtain food, shelter or support from the host species. Kumar and Pattabhiramacharyulu [34] have studied the three species system in combination with prey–predation and commensalism. One or more species taking refuge to avoid predation are considered by many authors in food-web models in [23], [25], [35].

In this paper, a three species food-web system consisting of prey–predation, competition and commensalism is considered. The refuge is also incorporated in the prey species.

Section snippets

Model formulation

Consider a three species food-web system, whose interactions are shown in schematic diagram (Fig. 1). Let X and Y are the two logistically growing competing species with intrinsic growth rates and carrying capacities ri and Ki(i=1,2), respectively. The species Z is predating over Y with Holling type II response. The species X is assumed to be benefited from the presence of Z, as such X is commensal of Z. The proportion p of species Y is refuged from predation. The coefficients αij(i,j=1,2;ij)

Analysis

In the absence of species z, the system (2.2) reduces to well-studied Lotka–Volterra competition model. In the absence of y, species x will be logistically growing while z will go to extinction in absence of food.

Numerical simulation

In this section, some numerical results are given to discuss the dynamical aspects of system (2.2) and to verify analytical findings. Consider the following set of parameters: w1=0.14,w2=0.32009,w3=0.5,p=0.4,r=0.5,δ=0.04,β12=0.4andβ21=0.1The positive equilibrium point E* is found as (0.84044, 0.41514, 0.16238). The three eigenvalues 0.849670 and 0.019512±(0.16238)i have negative real parts. Accordingly, the interior point E* is locally asymptotically stable. Further, the phase plane diagram,

Discussion

In this paper, we have considered a three species system in which two competing species are logistically growing. One of the competitors is being predated by third species with Holling type II functional response while the other competitor is in commensal relationship with the third species acting as its host. The prey species take refuge to avoid predation. The stability analysis has been carried out for all possible feasible equilibrium points. Also, sufficient conditions for persistence of

Acknowledgment

The authors are thankful to reviewers for their valuable suggestions. The second author (K. Gupta) would like to thank “University Grants Commission (UGC)” for providing Junior Research Fellowship through grant no. 6405-11-044.

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