Dynamical behaviour of a two-prey and one-predator system with help and time delay

Abstract

In this paper, we have investigated a three-species food web model consisting of two preys and one predator with the assumption that in the presence of predator both teams of prey help each other. The interaction between second prey and predator is assumed as a Holling type I (Volterra) functional response because in the absence of any predation, the second prey grows unboundedly following Malthusian law. The relationship between the Canada lynx (Lynx canadensis) and the snowshoe hare (Lepus americanus) can be considered (in good agreement with our model) as an example of how interaction between a predator and its second prey can influence population dynamics. On the other hand, a modified Holling type II functional response is considered to represent the interaction between first prey and predator incorporating the Malthusian law of growth for the second prey. Also, there is no inter-specific competition among the two-prey species as the second prey has sufficient resources. Moreover, it is assumed that there may be competition among the individuals of the predator species. Next, we have discussed the positivity of the solutions of the proposed system. In this work, we have studied about various types of equilibrium points and their stability behaviour. Also, transcritical bifurcations of the planer equilibrium points and persistent of system are discussed. The effect of discrete time delay (as gestation period) is analysed to observe the switching behaviour of the delay parameter. The maximum length of delay has been determined to preserve the stability of one-periodic limit cycle. Also, the direction, period and the stability of bifurcating periodic solutions have been examined based on normal form method and centre manifold theory. Numerical simulations are also performed to verify analytical findings.