Vol. 11, No. 2, 2018

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The behavior of a population interaction-diffusion equation in its subcritical regime

Mitchell G. Davis, David J. Wollkind, Richard A. Cangelosi and Bonni J. Kealy-Dichone

Vol. 11 (2018), No. 2, 297–309

A model interaction-diffusion equation for population density originally analyzed through terms of third-order in its supercritical parameter range is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further, there also exists a region of metastability allowing for the possibility of population outbreaks. These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-groundwater model system, as opposed to periodic patterns and dense homogeneous distributions occurring in its supercritical regime.

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interaction-diffusion, Stuart–Watson method, subcritical bifurcation analysis
Mathematical Subject Classification 2010
Primary: 34D20, 35Q56, 92D40
Received: 15 October 2016
Revised: 27 January 2017
Accepted: 4 February 2017
Published: 17 September 2017

Communicated by Martin J. Bohner
Mitchell G. Davis
Department of Mathematics
Washington State University
Pullman, WA
United States
David J. Wollkind
Department of Mathematics
Washington State University
Pullman, WA
United States
Richard A. Cangelosi
Department of Mathematics
Gonzaga University
Spokane, WA
United States
Bonni J. Kealy-Dichone
Department of Mathematics
Gonzaga University
Spokane, WA
United States