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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
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Abstract |
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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 -shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter. |
Keywords
nonlinear boundary conditions, weak Allee effect, positive
solutions
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Mathematical Subject Classification 2010
Primary: 34B08, 34B18
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Milestones
Received: 23 July 2012
Revised: 4 April 2013
Accepted: 10 April 2013
Published: 8 October 2013
Communicated by John Baxley
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Authors |
| Kathryn Ashley | |
| Department of Mathematical
Sciences Clemson University Clemson, SC 29634 United States |
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| Victoria Sincavage | |
| Department of Mathematical
Sciences Clemson University Clemson, SC 29634 United States |
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| Jerome Goddard II | |
| Department of Mathematics Auburn University Montgomery Montgomery, AL 36124-4023 United States |
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