Abstract
This paper explores the classification of parameter spaces for reaction-diffusion systems of two chemical species on stationary rectangular domains. The dynamics of the system are explored both in the absence and presence of diffusion. The parameter space is fully classified in terms of the types and stability of the uniform steady state. In the absence of diffusion the results on the classification of parameter space are supported by simulations of the corresponding vector-field and some trajectories of the phase-plane around the uniform steady state. In the presence of diffusion, the main findings are the quantitative analysis relating the domain-size with the reaction and diffusion rates and their corresponding influence on the dynamics of the reaction-diffusion system when perturbed in the neighbourhood of the uniform steady state. Theoretical predictions are supported by numerical simulations both in the presence as well as in the absence of diffusion. Conditions on the domain size with respect to the diffusion and reaction rates are related to the types of diffusion-driven instabilities namely Turing, Hopf and Transcritical types of bifurcations. The first condition is a lower bound on the area of a rectangular domain in terms of the diffusion and reaction rates, which is necessary for Hopf and Transcritical bifurcation to occur. The second condition is an upper bound on the area of domain in terms of reaction-diffusion rates that restricts the diffusion-driven instability to Turing type behaviour, whilst forbidding the existence of Hopf and Transcritical bifurcation. Theoretical findings are verified by the finite element solution of the coupled system on a two dimensional rectangular domain.
Highlights
IntroductionReaction-diffusion systems (RDSs) attract a significant degree of attention from researchers in applied mathematics [1–9], mathematical and computational biology [10–14], chemical engineering [15–17] and so forth
From a research perspective the study of Reaction-diffusion systems (RDSs) is conducted through different types of approaches, one of which focuses on the local behaviour of the dynamics of the RDS near a uniform steady state, which in turn relates to the subject of stability analysis [6,8,12,20,26,27] of RDSs using the stability matrix
In the absence of diffusion, theoretical results on the dynamics of the system were supported by use of the phase-plane analysis, where in each case the numerical solution of the system was observed to be in agreement with the theoretically predicted behaviour
Summary
Reaction-diffusion systems (RDSs) attract a significant degree of attention from researchers in applied mathematics [1–9], mathematical and computational biology [10–14], chemical engineering [15–17] and so forth. Comparing their work to the present study, our results are robust in the sense that we explicitly relate domain size to the reaction-diffusion rates Using this relationship, the parameter space is classified for different types of bifurcations. In the current work the existing knowledge on the conditions for diffusion-driven instability in the literature is extended using a series of analytical and numerical techniques, to obtain new insights on the combined effects of diffusion and reaction rates, and in turn relating these to domain size of the evolution of the pattern. The main findings of the present work, which relate the domain size to the diffusion and reaction rates, are presented in the form of theorems with rigorous mathematical proofs and these theoretical results are supported computationally by finite element numerical solutions corresponding to the activator-depleted RDS on fixed rectangular domains.
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