A study of logistic growth models influenced by the exterior matrix hostility and grazing in an interior patch

Abstract

We will analyze the symmetric positive solutions to the two-point steady state reaction-diffusion equation: \begin{equation*} \begin{split} -&u^{\prime \prime}= \begin{cases} \lambda\left[ u-\dfrac{1}{K}u^2-\dfrac{cu^2}{1+u^2}\right];& x\in [L,1-L] ,\\ \lambda \left[u-\dfrac{1}{K}u^2\right];& x\in (0,L)\cup(1-L,1), \end{cases} \\ -&u^{\prime}(0) + \sqrt{\lambda}\gamma u(0) = 0,\\ &u^{\prime}(1) + \sqrt{\lambda}\gamma u(1) = 0,\\ \end{split} \end{equation*} where $\lambda$, $c$, $K$, and $\gamma$ are positive parameters and the parameter $L\in(0,\frac{1}{2})$. The steady state reaction-diffusion equation above occurs in ecological systems and population dynamics. The above model exhibits logistic growth in the one-dimensional habitat $\Omega_0=(0,1)$, where grazing (type of predation) is occurring on the subregion $[L,1-L]$. In this model, $u$ is the population density and $c$ is the maximum grazing rate. $\lambda$ is a parameter which influences the equation as well as the boundary conditions, and $\gamma$ represents the hostility factor of the surrounding matrix. Previous studies have shown the occurrence of S-shaped bifurcation curves for positive solutions for certain parameter ranges when the boundary condition is Dirichlet ($\gamma \longrightarrow \infty$). Here we discuss the occurrence of S-shaped bifurcation curves for certain parameter ranges, when $\gamma$ is finite, and their evolutions as $\gamma$ and $L$ vary