International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2015): 6.391 Volume 5 Issue 7, July 2016 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Strong Coexistence (Local and Global Stability) of Two Mutually Interdependent Predator Species for Sharing a Single Prey Species Brahampal Singh 1 , Sunita Gakkhar 2 1 Department of Mathematics, J.V.Jain College, Saharanpur 247001, India 2 Department of Mathematics, Indian Institute of Technology, Roorkee 247667, India Abstract: A model consisting of two mutually interdependent predator species feeding on a single prey species is studied. Both the predator species has symbiotic interaction that is mutually beneficial. The local stability of equilibria points is analyzed. It is observed that the persistence is possible in the form of stable non zero equilibrium point or periodic limit cycle in the positive octant. It is shown that in the case of two competing predator species feeding over a single prey species, coexistence is not possible and only the fittest will survive [7], but in the case when two predator species have mutual cooperation feeding over a single prey species, then strong coexistence (local and global stability) is possible. In this paper the mathematical model comprising two mutually interdependent predator species and a single prey species shows rich dynamics numerically as well as analytically. The effect of mutually Cooperation on the two predators feeding on a common prey was investigated. Keywords: Mutual Cooperation among predator; prey predation; Food web; stability; limit cycle. 1. Introduction Many investigators have discussed three species food chains and food webs [1]-[12]. Two prey and one predator systems are shown to have complex dynamical behavior [5]-[7]. A prey and two predator system has been investigated by considering various types of interactions among the two predators. The two predators themselves may have different types of interaction between them. In the case of two competing species, coexistence is not possible and only the fittest will survive [7]. It may be interesting to see the changes in the behavior of the dynamical system as the competition is added between two of the species in food webs. The effect of implicit competition on the two predators sharing a common prey was investigated in [7]. Apart from the implicit competition, the two predators may have explicit competition between themselves [14]. Due to explicit competition, the Competitive exclusion of the weak predator is possible. In specialist and generalist prey predator models, the two predators may be in a prey- predator type of interaction [3], and due to this additional interaction a rich complex dynamics is observed. In a two species system, cooperation is found to have a destabilizing effect on the stability of equilibrium. Freedman et. al. [13] investigated a three species food web considering the mutualism between two predators sharing a prey. The effect of their cooperation is considered implicitly in the functional response of the prey species. Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions [2]. It is excluded such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two predator and one prey species in the chemostat [2]. In this paper, we are investigating a three species food web with mutualist predators. The three species food web model has all the three types of interactions among the interacting species: prey predation, implicit competition and mutualism. Consider a three species food web comprised of two mutualist predators feeding on a single prey species. There is an implicit competition between them due to sharing of prey. The dynamical equations of this food web are given as (1) Here 1 X , denotes the density of prey species while 2 X and 3 X are the densities of two predators,   X F i represent the Holling type-II functional response. The model parameters and i i i i r,K,a ,b ,d e can assume only positive values. The parameter 1  and 2  represent the coefficient of cooperation between the predators. The model (1) has 12 parameters, which are reduced to 8 by introducing the following non-dimensional variables and parameters are Paper ID: ART2016116 10.21275/ART2016116 29