Chaos, Solitons and Fractals 104 (2017) 350–362 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Degree of prey refuges: Control the competition among prey and foraging ability of predator Debaldev Jana a , Aniket Banerjee b , G.P. Samanta b,∗ a Department of Mathematics & SRM Research Institute, SRM University, Kattankulathur 603 203, Tamil Nadu, India b Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India a r t i c l e i n f o Article history: Received 27 April 2017 Revised 29 July 2017 Accepted 29 August 2017 Keywords: Prey-predator model Refuge Interspecific competition Hopf bifurcation a b s t r a c t Every population should exploit a specially variable and diverse environment so as to increase their Dar- winian fitness. Dynamics of any local population depends upon attributes of the local habitat. Although, use of refuge habitat by prey population can reduce their risk of predation, refuge use may also involve cost such as increased interspecific and intraspecific competition within the refuge patch. Surveys in the Sunderban mangrove ecosystem show that two detritivorous prey fishes Liza parsia and Liza tade coexist in nature by using refuges with the presence of the predator fish population Lates calcarifer. In view of such observations in mind, a three component model conceiving of two competing prey and one predator is considered in the present study with the inclusion of Holling type-II response function incorporating a fraction of prey refuge. The geographic position of these refuge patches tend to determine the popu- lation of prey residing in these patches which ultimately leads to the interspecific competition inclusion between prey. Here, we have differentiated the geographic position of the refuge patch into five differ- ent cases, for example, disjoint refuge patch (no competition between refuge prey population), partially overlapping refuge patch (competition between non-refuge and partially refuge prey population), only one prey refuge patch (competition between one prey population entirely and non-refuge prey popula- tion of the other) and common refuge patch (competition between both refuge and non-refuge prey in and out of the common patch). Equilibrium abundance of each population and the stability criterion are absolutely motivated by the interspecific competition strategies by both prey due to their patch selection. Mathematical results and numerical results support these hypothesis. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Niche [2] in ecology is defined as the behavior of a group of species and how they respond to the distribution of re- sources and competitors, for example abundance in resources and scarcity of predator etc [21], but in return to this instinct these species alter these factors itself, for example abundance of resources increase the growth rate or increase in predator decreases growth rate and increases the protection strategies etc. [1,8,11,12,13,15,16,18,22,23,24,27,34–36,38,40,42,45]. The type and number of variables comprising the dimensions of an en- vironmental niche vary from one species to another and the relative importance of particular environmental variables for a species may vary according to the geographic and biotic con- texts ([1,34,35,38,42,45]). Ecological niche depends on ecological biogeography which depends on the space [15], habitat physical ∗ Corresponding author. E-mail addresses: g_p_samanta@yahoo.co.uk, gpsamanta@math.becs.ac.in, gpsamanta@math.iiests.ac.in (G.P. Samanta). [20,24,25] and behavioral [20,24,25] complexity and so many constrains [15,17] where the ecological communities belong to. In a niche one species changes or alters the dynamics of another species in order to maintain comfortable fitness [24,25]. A perfect niche can be achieved when there is no competition for resources that is fundamental niche where a species can grow in full vol- ume but it is not achieved in real environment where there is competition and it is known as realized niche and species changes there dynamics according to the competition it faces. Temporal partitioning is achieved due to competition for seasonal change. Reproductive asynchrony would be two competing species of frog offsetting their breeding periods. By doing this the first species’ tadpoles will have graduated to a different food resource by the time the tadpoles of the second species are hatching [26]. Spatial partitioning is obtained when species occupy the same resource at different geographical spaces in order to ignore competition like two competing species of monkey using the same species of fruit trees, but in different forest [37]. Third is morphological partitioning where the species evolve morphologically to absorb http://dx.doi.org/10.1016/j.chaos.2017.08.031 0960-0779/© 2017 Elsevier Ltd. All rights reserved.