Abstract
The aim of this work is to explore the optimal exploitation way for a biological resources model incorporating individual’s size difference and spatial effects. The existence of a unique nonnegative solution to the state system is shown by means of Banach’s fixed point theorem, and the continuous dependence of the population density with the harvesting effort is given. The optimal harvesting strategy is established via normal cone and adjoint system technique. Some conditions are found to assure that there is only one optimal policy.
Highlights
Introduction and Problem SettingSince the classical work by Skellam [1], dispersal or diffusion of biological individuals has been recognized as one of the most significant features, which affect the dynamics and evolution of populations
The objective of this paper is to investigate an optimal harvesting problem for a size- and space-structured population model and to analyze the structure of the optimal strategies
Suppose that u∗ ∈ U is a solution for the optimal control problem (1), and pu∗ is the corresponding solution of system (2)
Summary
Since the classical work by Skellam [1], dispersal or diffusion of biological individuals has been recognized as one of the most significant features, which affect the dynamics and evolution of populations. In [9], Botsford constructed a size-specific population model based on the continuity equation and suggested that the inclusion of individual growth rates could reveal optimal harvesting policies. Some linear optimal harvesting population models structured by size were introduced in [10, 11]. Kato in [16] sought the optimal harvesting rate in a profit maximization problem for a nonlinear size-structured model of two-species population. Other nonlinear size-specific models can be found in [17, 18] He and Liu [17] took fertility as the control variable and established the necessary optimality conditions of first order in the form of an Euler-Lagrange system. The objective of this paper is to investigate an optimal harvesting problem for a size- and space-structured population model and to analyze the structure of the optimal strategies.
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