Optimal harvesting for a logistic model with grazing
Mohan Mallick, Ardra A, Sarath Sasi
TL;DR
The paper investigates an optimal harvesting problem for a logistic-growth population with grazing, modeled by a semi-linear elliptic equation with Robin boundary conditions. It proves the existence and uniqueness of a positive stationary state $u_h$ for large diffusion parameter $\\lambda$, and shows there exists an optimal grazing control $h^*\\in U$ that maximizes $J(h)=\\int_{\\Omega} h u_h \,dx-\\int_{\\Omega} (B_1+B_2 h) h \,dx$, with $U=\\{h:0\\le h\\le H\\}$. The optimality system, linking the state and adjoint $p$, is derived via sensitivity analysis and a pointwise gradient condition $h=\\min\\{H,\\max\\{0,(u_h-\\lambda p u_h-B_1)/(2B_2)\\}\}$. Uniqueness of the optimality system is established in dimensions $N=2,3$ for large $B_2$, ensuring a unique optimal control and state-adjoint pair under those regimes.
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, $h\in U=\{h\in L^2(Ω): 0\leq h(x)\leq H\}.$ We prove the existence and uniqueness of the positive solution for large $λ.$ Further, we establish the existence of an optimal control $h\in U$ that maximizes the functional $J(h)=\int_Ωh(x)u_h(x)~\rm{d}x-\int_Ω(B_1+B_2 h(x))h(x)~\rm{d}x$ 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.