Positive Periodic Solutions for Neutral Delay Ratio-Dependent Predator-Prey Model with Holling-Tanner Functional Response

By using a continuation theorem based on coincidence degree theory, we establish some easily verifiable criteria for the existence of positive periodic solutions for neutral delay ratio-dependent predator-prey model with Holling-Tanner functional response , . 1. Introduction The dynamic relationship between the predator and the prey has long been and will continue to be one of the dominant themes in population dynamics due to its universal existence and importance in nature [1]. In order to precisely describe the real ecological interactions between species such as mite and spider mite, lynx and hare, and sparrow and sparrow hawk, described by Tanner [2] and Wollkind et al. [3], May [4] developed the Holling-Tanner prey-predator model In system (1.1), and stand for prey and predator density at time . , , , , , are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half-capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predators biomass, respectively. Nowadays attention have been paid by many authors to Holling-Tanner predator-prey model (see [5–7]). Recently, there is a growing explicit biological and physiological evidence [8–10] that in many situations, especially when predators have to search for food (and, therefore, have to share or compete for food), a more suitable general predator-prey theory should be based on the so-called ratio-dependent theory, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance and so should be the so-called ratio-dependent functional response. This is strongly supported by numerous field and laboratory experiments and observations [11, 12]. Generally, a ratio-dependent Holling-Tanner predator-prey model takes the form of Liang and Pan [13] obtained results for the global stability of the positive equilibrium of (1.2). However, time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing [14], Gopalsamy [15], Kuang [16], and MacDonald [17] for general delayed biological systems. Time delay due to gestation is a common example, because generally the consumption of prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, more realistic models of population interactions should take into account the effect of time delays. Recently, Saha and Chakrabarti [18] considered the following delayed ratio-dependent Holling-Tanner