Vol. 6, No. 4, 2013

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Ecological systems, nonlinear boundary conditions, and $\Sigma$-shaped bifurcation curves

Kathryn Ashley, Victoria Sincavage and Jerome Goddard II

Vol. 6 (2013), No. 4, 399–430

We examine a one-dimensional reaction diffusion model with a weak Allee growth rate that appears in population dynamics. We combine grazing with a certain nonlinear boundary condition that models negative density dependent dispersal on the boundary and analyze the effects on the steady states. In particular, we study the bifurcation curve of positive steady states as the grazing parameter is varied. Our results are acquired through the adaptation of a quadrature method and Mathematica computations. Specifically, we computationally ascertain the existence of Σ-shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter.

nonlinear boundary conditions, weak Allee effect, positive solutions
Mathematical Subject Classification 2010
Primary: 34B08, 34B18
Received: 23 July 2012
Revised: 4 April 2013
Accepted: 10 April 2013
Published: 8 October 2013

Communicated by John Baxley
Kathryn Ashley
Department of Mathematical Sciences
Clemson University
Clemson, SC 29634
United States
Victoria Sincavage
Department of Mathematical Sciences
Clemson University
Clemson, SC 29634
United States
Jerome Goddard II
Department of Mathematics
Auburn University Montgomery
Montgomery, AL 36124-4023
United States