Electronic Journal of Differential Equations (Oct 2013)
S-shaped bifurcation curves for logistic growth and weak Allee effect growth models with grazing on an interior patch
Abstract
We study the positive solutions to the steady state reaction diffusion equations with Dirichlet boundary conditions of the form $$displaylines{ -u''= cases{ lambda[u - frac{1}{K}u^2 - c frac{u^2}{1+u^2}], & $x in (L,1-L)$,cr lambda[u - frac{1}{K}u^2], & $x in (0,L)cup(1-L,1)$, } cr u(0)=0, quad u(1)=0 }$$ and displaylines{ -u''= cases{ lambda[u(u+1)(b-u) - c frac{u^2}{1+u^2}], & $x in (L,1-L)$, cr lambda[u(u+1)(b-u)], & $x in (0,L)cup(1-L,1)$, } cr u(0)=0,quad u(1)=0. }$$ Here, $lambda, b, c, K, L$ are positive constants with 0<L<1/2. These types of steady state equations occur in population dynamics; the first model describes logistic growth with grazing, and the second model describes weak Allee effect with grazing. In both cases, u is the population density, $1/lambda$ is the diffusion coefficient, and c is the maximum grazing rate. These models correspond to the case of symmetric grazing on an interior region. Our goal is to study the existence of positive solutions. Previous studies when the grazing was throughout the domain resulted in S-shaped bifurcation curves for certain parameter ranges. Here, we show that such S-shaped bifurcations occur even if the grazing is confined to the interior. We discuss the results via a modified quadrature method and Mathematica computations.
