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Linear and nonlinear stability in a diffusional ecotoxicological model with time delays

Linear and nonlinear stability in a diffusional ecotoxicological model with time delays
Linear and nonlinear stability in a diffusional ecotoxicological model with time delays
We propose a reaction-diffusion extension of a two species ecotoxicological model with time-delays proposed by Chattopadhyay et al (1997). Each species has the capacity to produce a substance toxic to its competitor, and a distributed time-delay is incorporated to model lags in the production of toxin. Additionally, nonlocal spatial effects are present because of the combination of delay and diffusion. The stability of the various uniform equilibria of the model are studied by using linearised analysis, on an infinite spatial domain. It is shown that simple exponentially decaying delay kernels cannot destabilise the coexistence equilibrium state. In the case of a finite spatial domain, with purely temporal delays, a nonlinear convergence result is proved using ideas of Lyapunov functionals together with invariant set theory. The result is also applicable to the purely temporal system studied by other investigators and, in fact, extends their results.
1078-0947
575-590
Schley, D.
3d807658-2cfd-40e6-90ba-5032f88bb54b
Gourley, S.A.
1aa9c89e-dfca-43f6-ad4d-4ecbe379094c
Schley, D.
3d807658-2cfd-40e6-90ba-5032f88bb54b
Gourley, S.A.
1aa9c89e-dfca-43f6-ad4d-4ecbe379094c

Schley, D. and Gourley, S.A. (2002) Linear and nonlinear stability in a diffusional ecotoxicological model with time delays. Discrete and Continuous Dynamical Systems, 2 (4), 575-590.

Record type: Article

Abstract

We propose a reaction-diffusion extension of a two species ecotoxicological model with time-delays proposed by Chattopadhyay et al (1997). Each species has the capacity to produce a substance toxic to its competitor, and a distributed time-delay is incorporated to model lags in the production of toxin. Additionally, nonlocal spatial effects are present because of the combination of delay and diffusion. The stability of the various uniform equilibria of the model are studied by using linearised analysis, on an infinite spatial domain. It is shown that simple exponentially decaying delay kernels cannot destabilise the coexistence equilibrium state. In the case of a finite spatial domain, with purely temporal delays, a nonlinear convergence result is proved using ideas of Lyapunov functionals together with invariant set theory. The result is also applicable to the purely temporal system studied by other investigators and, in fact, extends their results.

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Published date: 2002

Identifiers

Local EPrints ID: 29283
URI: http://eprints.soton.ac.uk/id/eprint/29283
ISSN: 1078-0947
PURE UUID: 30dd6c44-d640-4cc6-ab6e-bedf8ba4f4e8

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Date deposited: 06 Feb 2007
Last modified: 08 Jan 2022 03:51

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Contributors

Author: D. Schley
Author: S.A. Gourley

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