An Ecological Model with Grazing and Constant Yield Harvesting

Abstract

We study positive solutions to the steady state reaction diffusion equation with Dirichlet boundary condition of the form: \begin{equation} \left\{ \begin{aligned} -\Delta u &= au-bu^2-c \dfrac{u^p}{1+u^p}-K, \quad x \in \Omega \\u &= 0, \quad x \in\partial\Omega. \end{aligned} \right. \end{equation} Here $\Delta u=div \big(\nabla u\big)$ is the Laplacian of u, $a, b, c, p, K$ are positive constants with $p\geq2$ and $\Omega$ is a smooth bounded region with $\partial\Omega$ in $C^2$. This model describes the steady states of a logistic growth model with grazing and constant yield harvesting. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. We study the existence of positive solutions to this model. We prove our results by the method of sub-super solutions.