Weak Allee effect, grazing, and S-shaped bifurcation curves (Q1950924)

The authors give a survey on a one-dimensional reaction-diffusion model arising in population dynamics with a weak Allee type's growth rate \[ \begin{aligned} -u''(x)&=\lambda f(u(x)), \;\;x\in(0,1),\\ u(0)&=u(1)=0, \end{aligned} \] where \(f:\;[0,\infty)\rightarrow (0,\infty)\) is a \(C^1\) function and \(\lambda\) is a nonnegative parameter. They consider the effects of grazing on the steady states and discuss the complete evolution of bifurcation curve of positive solutions as the grazing parameter varies. Especially, they obtain that the bifurcation curve is S-shaped for certain ranges of the grazing parameter by using quadrature method and \texttt{Mathematica}-computations.