We consider semi-linear elliptic equations of the following form:

\begin{equation*}

\left\{

\begin{aligned}

-\Delta u &= \lambda[u-\dfrac{u^2}{K}-c \dfrac{u^2}{1+u^2}-h(x) u]=:\lambda f_h(u) &&\textcolor{black}{\rm{in}} ~ \Omega,\

\frac{\partial u}{\partial \eta}&+qu = 0 &&\textcolor{black}{\rm{on}} ~ \partial\Omega,

\end{aligned}

\right.

\end{equation*}

where, $h\in U=\{h\in L^2(\Omega): 0\leq h(x)\leq H\}.$ 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

$J(h)=\int_{\Omega}h(x)u_h(x)~{\rm{d}}x-\int_{\Omega}(B_1+B_2 h(x))h(x)~{\rm{d}}x$

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.