Weak Allee effect, grazing, and S-shaped bifurcation curves

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.

Ecological systems, nonlinear boundary conditions, and Σ-shaped bifurcation curves

“…Additionally we recapitulate the subsequent boundary value problem analyzed by Poole et al [2012] for positive solutions: …”

“…For the logistic case with Dirichlet boundary conditions, Lee, Sasi, and Shivaji proved the existence of an S-shaped bifurcation curve in one dimension, as well as higher dimensions for a certain range of the grazing parameter [Lee et al 2011]. Regarding the one-dimensional weak Allee effect model with Dirichlet boundary conditions, Poole, Roberson, and Stephenson showed the existence of an S-shaped bifurcation curve, resembling Figure 1, both computationally and analytically for certain parameter ranges [Poole et al 2012]. In particular, our focus is to further examine the structure of positive solutions of (1-7) when the nonlinear boundary conditions (1-4) are satisfied for the range of the parameters where Poole et al [2012] showed the existence of an S-shaped bifurcation curve of positive solutions.…”

“…Regarding the one-dimensional weak Allee effect model with Dirichlet boundary conditions, Poole, Roberson, and Stephenson showed the existence of an S-shaped bifurcation curve, resembling Figure 1, both computationally and analytically for certain parameter ranges [Poole et al 2012]. In particular, our focus is to further examine the structure of positive solutions of (1-7) when the nonlinear boundary conditions (1-4) are satisfied for the range of the parameters where Poole et al [2012] showed the existence of an S-shaped bifurcation curve of positive solutions. Computationally, we show the existence of -shaped bifurcation curves as exemplified in Figure 2.…”

“…In the preceding cases the structure of the positive solutions of (1-7) varies. As our primary interest is the structure of positive solutions for the range of parameters where Poole et al [2012] showed the existence of S-shaped bifurcation curves, we focus on the case when c ∈ [0, c 0 ).…”

“…Grazing can be considered as a category of natural predation, for example, when an owl preys upon the surrounding rodent population. The term cu 2 /(1 + u 2 ) is commonly employed to model grazing of a population (see [Causey et al 2010;Poole et al 2012; van Nes and Scheffer 2005]). …”