**Travelling waves in near-degenerate bistable competition models.** / Alzahrani, E. O.; Davidson, F. A.; Dodds, N.

Research output: Contribution to journal › Article

Alzahrani, EO, Davidson, FA & Dodds, N 2010, 'Travelling waves in near-degenerate bistable competition models' *Mathematical Modelling of Natural Phenomena*, vol 5, no. 5, pp. 13-35.

Alzahrani, E. O., Davidson, F. A., & Dodds, N. (2010). Travelling waves in near-degenerate bistable competition models. *Mathematical Modelling of Natural Phenomena*, 5(5), 13-35doi: 10.1051/mmnp/20105502

Alzahrani EO, Davidson FA, Dodds N. Travelling waves in near-degenerate bistable competition models. Mathematical Modelling of Natural Phenomena. 2010;5(5):13-35.

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title = "Travelling waves in near-degenerate bistable competition models",

author = "Alzahrani, {E. O.} and Davidson, {F. A.} and N. Dodds",

year = "2010",

volume = "5",

number = "5",

pages = "13--35",

journal = "Mathematical Modelling of Natural Phenomena",

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TY - JOUR

T1 - Travelling waves in near-degenerate bistable competition models

A1 - Alzahrani,E. O.

A1 - Davidson,F. A.

A1 - Dodds,N.

AU - Alzahrani,E. O.

AU - Davidson,F. A.

AU - Dodds,N.

PY - 2010

Y1 - 2010

N2 - We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

AB - We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

KW - Competition

KW - Reaction-diffusion

KW - Free energy

KW - Bistable

KW - Travelling waves

U2 - 10.1051/mmnp/20105502

DO - 10.1051/mmnp/20105502

M1 - Article

JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

SN - 0973-5348

IS - 5

VL - 5

SP - 13

EP - 35

ER -