Optimal harvesting for a logistic model with grazing

Abstract

We consider semi-linear elliptic equations of the following form: \begin{equation*} \left\{ \begin{aligned} -Δu &= λ[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:λf_h(u), \quad && x \in Ω, \frac{\partial u}{\partial η}&+qu = 0, \quad && x\in\partialΩ, \end{aligned} \right. \end{equation*} where, $h\in U=\{h\in L^2(\Omega ):0\le h(x)\le H\}.$ We prove the existence and uniqueness of the positive solution for large $\lambda .$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)={\int }_{\Omega }h(x)u_h(x)\mathrm{d}\mathrm{x}-{\int }_{\mathrm{\Omega }}({\mathrm{B}}_1+{\mathrm{B}}_2\mathrm{h}(\mathrm{x}))\mathrm{h}(\mathrm{x})\mathrm{d}\mathrm{x}$ over $U$, where $u_h$ is the unique positive solution of the above problem associated with $h$, $B_1>0$ is the cost per unit effort when the level of effort is low and $B_2>0$ represents the rate at which the cost rises as more labor is employed. Finally, we provide a unique optimality system.