Abstract
. The present investigation deals with the disease in the prey population having significant role in curbing the dynamical behaviour of the system of prey-predator interactions from both ecological and mathematical point of view. The predator-prey model introduced by Cosner et al. [1] has been wisely modified in the present work based on the biological point of considerations. Here one introduces the disease which may spread among the prey species only. Following the formulation of the model, all the equilibria are systematically analyzed and the existence of a Hopf bifurcation at the interior equilibrium has been duly carried out through their graphical representations with appropriate discussion in order to validate the applicability of the system under consideration
Highlights
Around 1800, the British Economist Malthus formulated a single species model [2] and subsequently modified by Verhulst [3]
The deterministic prey dependent predator-prey model exhibits the “paradox of enrichment”, formulated by Hariston et al [13] and Rosenzweig [14] and the “biological control paradox” which was taken by Luck [15]
We introduce the following facts: (i) In the presence of disease, the prey population X is divided among susceptibles S and infected I individuals
Summary
Around 1800, the British Economist Malthus formulated a single species model [2] and subsequently modified by Verhulst [3]. Prey dependent predator-prey models have been studied extensively in several investigations (cf [6,7,8,9,10,11,12]). The simplest models contain a bilinear mass action term, quadratic in both the interacting populations, called Holling type I. This term appears due to the fact that an individual can in principle interact with the whole other population, the product of the two populations is the obvious outcome. We consider the fact that in general a single individual can feed only until the stomach is full, a saturation function indicate the intake of food The latter can be modeled using the concept of the “law of diminishing returns” or technically speaking Michaelis–Menten or Holling type II term. The standard approach has been used to establish the local stability, Hopf bifurcations and limit cycles
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