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## A reaction-diffusion model for inter-species competition and intra-species cooperation

Rasheed, Shaker M. (2013) A reaction-diffusion model for inter-species competition and intra-species cooperation. PhD thesis, University of Nottingham.

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## Abstract

This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range $0< \lambda \ll 1$ (fast diffusion and slow reaction), and in a few cases we assume $\lambda=O(1)$. We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, $(u_0,w_0)$ to an extinction state $(0,0)$ when $0< \lambda \ll 1$. This type can be found when the reaction-diffusion system satisfy the symmetry property and $\lambda=1$.

Item Type: Thesis (University of Nottingham only) (PhD) Billingham, J.Coombes, S. Q Science > QA Mathematics > QA299 AnalysisQ Science > QH Natural history. Biology > QH471 Reproduction. Life UK Campuses > Faculty of Science > School of Mathematical Sciences 13545 EP, Services 05 Mar 2014 13:22 18 Dec 2017 07:58 https://eprints.nottingham.ac.uk/id/eprint/13545

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