Mathematical study of a Leslie–Gower-type tritrophic population model in a polluted environment

Abstract

In this paper, the effect of toxicant on the dynamics of Leslie-Gower tritrophic food chain population model is studied. Two models have been studied to visualize the dynamical behaviour of prey and predator populations under toxicant stress. All the feasible equilibria of the systems are obtained and using stability theory the conditions are derived for the survival or extinction of species. From the analysis of the models it is observed that the effect of toxicant on predator population increases the equilibrium density of prey population. Finally, we support our analytical findings with numerical simulations.

A comment on “Mathematical study of a Leslie-Gower type tritrophic population model in a polluted environment” [Modeling in Earth Systems and Environment 2 (2016) 1–11]

“…This is shown via theorem 2.3. • Similar results are proved via corollary 1, for the first model system proposed in Misra and Babu (2016). • We demonstrate our results via numerical simulations.…”

“…In this regard the work of the authors in Misra and Babu (2016) is very interesting, and such investigations are highly desireable. Therein, two five-species ordinary differential equation (ODE) models are proposed and analyzed in an attempt to understand the effect of toxicity/ pollution on a three species food chain, modeled via a combination of a modified Leslie-Gower type scheme and Holling type II functional response (Upadhyay and Rai 1997;Aziz-Alaoui 2002).…”

“…However, one must take extreme care in deriving global existence results for such systems, given recent results that show finite time blow-up in such models (Parshad et al 2013;Parshad 2015;Parshad and Basheer 2016). The purpose of this short communication is to remark on certain results in Misra and Babu (2016). The authors therein derive global boundedness results for solutions to the ODE model systems proposed, under certain restrictions on the parameters.…”

“…We note, these results are incorrect. In particular there is no bound on the solutions to the proposed models, even under the parametric restrictions imposed in Misra and Babu (2016). Our primary contributions in the current manuscript are:…”

“…• We consider the second model proposed in Misra and Babu (2016), and show solutions to this model system are not bounded. In fact the solutions can blow-up in finite time, for sufficiently large initial data.…”