Vol. 5, No. 2, 2012

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Weak Allee effect, grazing, and S-shaped bifurcation curves

Emily Poole, Bonnie Roberson and Brittany Stephenson

Vol. 5 (2012), No. 2, 133–158
Abstract

We study a one-dimensional reaction-diffusion model arising in population dynamics where the growth rate is a weak Allee type. In particular, we consider the effects of grazing on the steady states and discuss the complete evolution of the bifurcation curve of positive solutions as the grazing parameter varies. We obtain our results via the quadrature method and Mathematica computations. We establish that the bifurcation curve is S-shaped for certain ranges of the grazing parameter. We also prove this occurrence of an S-shaped bifurcation curve analytically.

Keywords
ordinary differential equations, nonlinear boundary value problems, grazing, Allee effect, population dynamics
Mathematical Subject Classification 2010
Primary: 34B15
Milestones
Received: 8 February 2011
Revised: 26 September 2011
Accepted: 27 September 2011
Published: 27 January 2013

Communicated by John Baxley
Authors
Emily Poole
Department of Mathematics and Statistics
University of Arkansas
Fayatteville, AR 72701
United States
Bonnie Roberson
Department of Mathematics and Statistics
Center for Computational Sciences
Mississippi State University
Mississippi State, MS 39762
United States
Brittany Stephenson
Department of Mathematics and Statistics
Center for Computational Sciences
Mississippi State University
Mississippi State, MS 39762
United States