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Resilience in reaction-diffusion systems |
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JH van Vuuren1
Abstract
Reaction-diffusion systems with zero-flux Neumann boundaries are widely used to model various kinds of interaction in, for example, the scientific fields of ecology, biology, chemistry, medicine and industry. The physical systems within these fields are often known to be (conditionally or unconditionally) resilient with respect to shocks, disturbances or catastrophes in the immediate environment. In order to be good mathematical models of such situations the reaction-diffusion systems must have the same rilient or asymptotic behaviour as that of the physical situation. Three fundamentally different kinds of reacion terms are usually distinguished according to entry signs of the reaction Jacobian: mutualism, mixed (predator-prey) interaction and competition. The asymptotic stability (in the Poincare sense) of mutualistic systems has already been studied extensively, but the results cannot be generalized (globally) to the other two types, which are not order-preserving. A partial (local) generalization is, however, given here for these two types, involving simple Jacobian inequalities and knowledge (often prompted by the physical situation) of invariant sets in solution space. The return time of resilient systems and the approach rate of asymptotically stable solutions are also estimated.
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Affiliations
1Department of Applied Mathematics, Stellenbosch University, Private Bag X1, Matieland, 7602, Republic of South Africa, fax: +27 21 8083778, email: vuuren@sun.ac.za