Said Kouachi

A remark on “Study of a Leslie–Gower-type tritrophic population model” [Chaos, Solitons and Fractals 14 (2002) 1275–1293]

In Aziz-Alaoui (2002) a three species ODE model, based on a modified Leslie–Gower scheme is investigated. It is shown that under certain restrictions on the parameter space, the model has bounded solutions for all positive initial conditions, which eventually enter an invariant attracting set. We show that this is not true. To the contrary, solutions to the model can blow up in finite time, even under the restrictions derived in Aziz-Alaoui (2002), if the initial data is large enough. We also prove similar results for the spatially extended system. We validate all of our results via numerical simulations.

A multi-temperature kinetic Ising model and the eigenvalues of some perturbed Jacobi matrices

In this paper we analyze the eigenvalues of some perturbed Jacobi matrices. The results contain as particular cases the known spectra of several classes of tridiagonal matrices studied recently. As a motivation, we discuss a three and a four-temperature kinetic Ising model that can be analyzed using some perturbed Jacobi matrices. The analytical results can also be used for the associated reaction-diffusion systems to solve for the particle density.

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]

In the current manuscript we comment on (Misra and Babu, Model Earth Syst Environ 2(1):1–11, 2016), where two novel five-species ODE models are proposed and analyzed, in order to investigate the population dynamics of a three-species food chain, in a polluted environment. It is shown in Misra and Babu (Model Earth Syst Environ 2(1):1–11, 2016) that under certain restrictions on the parameters, the models have bounded solutions for all positive initial conditions. Furthermore, a globally attracting set is explicitly constructed for initial conditions in

On “small” data blow-up in a three species food chain model

\mathbb {R}^{5}_{+}

Global existence for a strongly coupled reaction diffusion system

R+5. We prove these results are not true. To the contrary, solutions to these models can blow-up in finite time, even under the parametric restrictions derived in Misra and Babu (Model Earth Syst Environ 2(1):1–11, 2016), for sufficiently large initial conditions. We provide both analytical proofs and numerics to confirm our results.