Finite Time Blowup in a Realistic Food-Chain Model

We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos. 1. Introduction Interaction networks in natural ecosystems can be visualized as consisting of simple units known as food chains or food webs that are made up of a number of species linked by trophic interaction [1]. A food chain model essentially comprises of the predator-prey relationship between interacting species in a given ecosystem [2]. In their seminal work [3], Hastings and Powell for the first time demonstrated that the evolution of species participating in a tritrophic relationship might be chaotic [3]. This led to a great deal of research activity in analyzing the dynamical behavior of food chain models. Upadhyay and Rai [4] provided a new example of a chaotic population system in a simple three-species food chain with Holling type II functional response. This model is different from the Hastings and Powell model, in that it considers a generalist top predator, one that can switch its food source, in the absence of its favorite prey. Letellier and Aziz-Alaoui [5] and Aziz-Alaoui [6] revisited the Upadhyay and Rai model and found that the chaotic dynamics is observed via sequences of period-doubling bifurcation of limit cycles which however suddenly break down and reverse giving rise to a sequence of period-halving bifurcation leading to limit cycles. Upadhyay in [7] next proposed a three-species food-chain model, by incorporating mutual interference, in the original model [4], thus generalizing the models in [3, 4, 8]. Parshad and Upadhyay [9] considered the diffusive form of the model proposed in [7]. Under certain restrictions on the parameter space, they proved