IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 _______________________________________________________________________________________ Volume: 05 Issue: 09 | Sep-2016, Available @ http://ijret.esatjournals.org 233 A RATIO-DEPENDENT PREDATOR-PREY MODEL WITH STRONG ALLEE EFFECT IN THE PREY AND AN ALTERNATIVE FOOD SOURCE FOR THE PREDATOR Alakes Maiti 1 *, Rita Paul 2 , Shariful Alam 3 1 Department of Mathematics, Vidyasagar Evening College, Kolkata – 700006, India, alakeshmaity@hotmail.com *1 Corresponding Author 2 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah - 711103, India 3 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah - 711103, India, salam50in@yahoo.co.in Abstract This paper aims to study the dynamical behaviours of a ratio-dependent predator-prey model, where the prey is subject to strong Allee effect, and the predator has an alternative food source. Positivity and boundedness of the system are discussed. Some extinction criteria are derived. Stability analysis of the equilibrium points is presented including some global results. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically. Keywords: Predator-Prey, Allee Effect, Extinction, Stability. --------------------------------------------------------------------***---------------------------------------------------------------------- 1. INTRODUCTION Nowadays the study of predator-prey relationship has become an extremely important part of ecology. In this relationship, one species serves as a food for the other. It is true that the preys always try to develop the methods of evasion to avoid being eaten. However, it is certainly not true that a predator-prey relationship is always harmful for the preys, it might be beneficial to both. Further, such a relationship often plays an important role to keep ecological balance in nature. Mathematical modelling of predator-prey interaction was started in 1920s. Interestingly, the first predator-prey model in the history of theoretical ecology was developed independently by Alfred James Lotka (a US physical chemist) and Vito Volterra (an Italian mathematician) [30, 48]. Subsequently, this model has been used as a machine to introduce numerous mathematical and practical concepts in theoretical ecology. Many refinements of the Lotka-Volterra model have also been made to overcome the shortcomings of the model and to get better insights of predator-prey interactions. In the last five or six decades, a number of predator-prey models are developed and systematically cultured in literature. However, urge for incorporating many parameters of real systems had been felt day by day. If we summarize the basic considerations behind the modelling of predator-prey systems, it would be evident that the most crucial elements of predator-prey models are the choices of growth function of the prey and functional response of the predator. It has long been recognized that the famous logistic growth function has the capability of describing individual population growth. The function is introduced in 1838 by the Belgian mathematician Pierre Francois Verhulst [47] and later it is rediscovered in 1920 by American biologists Reymon Pearl and Lowell Reed [39]. If X ( T) denotes the population density at time T,then the logistic growth equation is given by 1 , dX X rX dT K         (1.1) where r is the intrinsic per capita growth rate and K is the carrying capacity of the environment. The logic behind this is very simple. As the resources (e.g., space, food, and essential nutrients) are limited, every population grows into a saturated phase from which it cannot grow further; the ecological habitat of the population can carry just so much of it and no more[83]. This suggests that the per capita growth rate is a decreasing function of the size of the population, and reaches zero as the population achieved a size K (in the saturated phase). Further, any population reaching a size that is above this value will experience a negative growth rate. The term −rX 2 /K may also be regarded as the loss due to intraspecific competition. Although logistic growth function became extremely popular, but, in real life situations, researchers found many evidences where the populations show a reverse trend in low population density [18, 16, 36, 12, 19, 42]. This phenomenon of positive density dependence of population growth at low densities is known as the Allee effect [45, 19].