Optimal harvesting strategy based on rearrangements of functions

Emamizadeh, Behrouz; Farjudian, Amin; Liu, Yichen

doi:10.1016/j.amc.2017.10.006

Cited by 7 publications

(12 citation statements)

References 17 publications

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“…The next step will be the computable analysis of a complete PDE-constrained rearrangement optimization problem. Note that linear rearrangement optimization is one of the two main components of some numerical methods for solving PDE-constrained rearrangement optimization problems [7,10], the other being PDE solving. Apart from some isolated work (e. g., [2,16]) computable analysis of PDE solving is largely an unexplored area.…”

Section: Discussionmentioning

confidence: 99%

“…In response, G. R. Burton laid out a theory for optimization of convex functionals over rearrangement classes [3,5]. Although PDE-constrained rearrangement optimization problems have their origins in fluid mechanics, the abstract formulation of these problems has shown greater potential, and Burton's theory has been used in the study of PDE-constrained rearrangement optimization problems in several areas, e. g., finance [12], free boundary problems [8], non-local problems [11], population biology [10], and eigenvalue problems [9], to name a few. Of special interest is the case when the generator of the rearrangement class is a two-valued function.…”

Section: Introductionmentioning

confidence: 99%

“…1.1 Contributions of the paper On the computational side of PDE-constrained rearrangement optimization problems, only sporadic attempts have been made, with the main focus on numerical algorithms based on floating-point arithmetic [7,14,11,10]. The current paper is meant to serve as a starting point towards answering questions of the type just listed.…”

Section: Introductionmentioning

confidence: 99%

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Computable Analysis of Linear Rearrangement Optimization

Farjudian

2019

Lecture Notes in Computer Science

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Optimization problems over rearrangement classes arise in various areas such as mathematics, fluid mechanics, biology, and finance. When the generator of the rearrangement class is two-valued, they reduce to shape optimization and free boundary problems which can exhibit intriguing symmetry breaking phenomena. A robust framework is required for computable analysis of these problems. In this paper, as a first step towards such a robust framework, we provide oracle Turing machines that compute the distribution function, decreasing rearrangement, and linear rearrangement optimizers, with respect to functions that are continuous and have no significant flat zones. This assumption on the reference function is necessary, as otherwise, the aforementioned operations may not be computable. We prove that the results can be computed to within any degree of accuracy, conforming to the framework of Type-II Theory of Effectivity.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computable Analysis of Linear Rearrangement Optimization

Farjudian

2019

Lecture Notes in Computer Science

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“…These facts plus the variational formulation of u E are stated in the following lemma. For proofs see Proposition 2.1 in [12], and Lemma 8 in [8].…”

Section: Preliminariesmentioning

confidence: 99%

“…For details, we refer the readers to [2,8,12]. A natural question arises: What would be the best hunting strategy so that the species will not extinct and, at the same time, the residents near the inhabitant can benefit from this supply of food?…”

Section: Introductionmentioning

confidence: 99%

Analyticity of free boundary in an optimal harvesting problem

Emamizadeh

Liu

2020

Lett Math Phys

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This note revisits an optimisation problem pertaining to the optimal harvesting of a marine species. The existence of solutions and the corresponding optimal conditions they satisfy have already been proved. It is known that the optimal solutions can be identified with n-dimensional shapes. We will obtain an interesting result concerning the free boundary of the optimal shapes. Indeed, we will prove that if a parameter in the admissible set is kept sufficiently small then the free boundaries will be real analytic hypersurfaces.

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A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat

Searle

Vuuren

2022

Partial Differ. Equ. Appl.

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The objective in this paper is to determine analytically a maximally sustainable pro-rata density-dependent harvest rate of a hypothetical biological species on the spatial boundary of a protected habitat (i.e. no harvesting of the species is allowed in the interior of the spatial domain). This is achieved by analysing an initial-boundary value model involving a reaction-diffusion equation in which the reaction term is a concave function of the population density, depends periodically on time, and varies spatially. The model is equipped with Robin boundary conditions modelling a continuum of potential pro-rata density-dependent harvest rates on the spatial boundary. A set of necessary and sufficient criteria for the existence of a unique positive periodic attractor of solutions to the model is established by employing the theory of comparison principles and monotone iteration schemes. Thereafter, a long-time asymptotic analysis of the population density is undertaken by invoking classical results from the theory of eigenproblems. In this way, the aforementioned necessary and sufficient conditions are rendered more practical in terms of an upper bound on the maximal allowable pro-rata density-dependent harvest rate. Moreover, important properties of the time-periodic attractor of solutions are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium.

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Optimal harvesting strategy based on rearrangements of functions

Cited by 7 publications

References 17 publications

Computable Analysis of Linear Rearrangement Optimization

Computable Analysis of Linear Rearrangement Optimization

Analyticity of free boundary in an optimal harvesting problem

A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat

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