Solvability of reaction–diffusion model with variable exponents

In this article, we study the existence and uniqueness of a weak solution of a degenerate reaction–diffusion parabolic system with variable exponents. This model describes the spread of epidemic diseases with a nonlinear diffusion operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Weak-renormalized solutions for predator–prey system

In this article, we consider the predator–prey system with Dirichlet boundary conditions which is used in the modelling of ecology. Under the assumptions of no growth conditions and integrable data, we prove the existence of weak-renormalized solutions to the predator–prey system.     

Optimal Control Problem for Cancer Invasion Reaction–Diffusion System

Abstract

An optimal control problem constrained by a reaction–diffusion mathematical model which incorporates the cancer invasion and its treatment is considered. The state equations consisting of three unknown variables namely tumor cell density, normal cell density, and drug concentration. The main goal of the considered optimal control problem is to minimize the density of cancer cells and decreasing the side effects of treatment. Moreover, existence of a weak solution of brain tumor reaction–diffusion system and the corresponding adjoint system of optimal control problem is also investigated. Further, existence of minimizer for the optimal control problem is established and also the first-order optimality conditions are derived.     

Global existence and blow up of solutions of quasilinear chemotaxis system

In this paper, we study the global existence of solution for the quasilinear chemotaxis system with Dirichlet boundary conditions, and further we show that the blow up properties of the solution depend only on the first eigenvalue. Copyright © 2014 John Wiley & Sons, Ltd.     

A time-fractional competition ecological model with cross-diffusion

This paper is concerned with some mathematical and numerical aspects of a Lotka-Volterra competition time-fractional reaction-diffusion system with cross-diffusion effects. First, we study the existence of weak solutions of the model following the well-known Faedo-Galerkin approximation method and convergence arguments. We demonstrate the convergence of approximate solutions to actual solutions using the energy estimates. Next, the Galerkin finite element scheme is proposed for the considered model. Further, various numerical simulations are performed to show that the fractional-order derivative plays a significant role on the morphological changes of the considered competition model.     

Existence and Uniqueness of Solutions of Predator-Prey Type Model with Mixed Boundary Conditions

In this work we consider the predator-prey model in ℝ³ with mixed boundary conditions on the Lipschitz boundary and prove the existence of solutions by Schauder’s fixed point theorem and uniqueness of solutions by Gronwall’s lemma.