Control theory and the management of ecosystems: A threshold policy with hysteresis is robust (original) (raw)
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Optimal Control of a Fishery Utilizing Compensation and Critical Depensation Models
Applied Mathematics and Information Sciences, 2020
This study proposes optimal control problems with two different biological dynamics: a compensation model and a critical depensation model. The static equilibrium reference points of the models are defined and discussed. Also, bifurcation analyses on the models show the existence of transcritical and saddle-node bifurcations for the compensation and critical depensation models respectively. Pontyagin's maximum principle is employed to determine the necessary conditions of the model. In addition, sufficiency conditions that guarantee the existence and uniqueness of the optimality system are defined. The characterization of the optimal control gives rise to both the boundary and interior solutions, with the former indicating that the resource should be harvested if and only if the value of the net revenue per unit harvest (due to the application of up to the maximum fishing effort) is at least the value of the shadow price of fish stock. Numerical simulations with empirical data on the sardinella are carried out to validate the theoretical results.
The stabilizability of a controlled system describing the dynamics of a fishery
Comptes Rendus Biologies, 2005
This work presents two stock-effort dynamical models describing the evolution of a fish population growing and moving between two fishing zones, on which it is harvested by a fishing fleet, distributed on the two zones. The first model corresponds to the case of constant displacement rates of the fishing effort, and the second one to fish stock-dependent displacement rates. In equations of the fishing efforts, a control function is introduced as the proportion of the revenue to be invested, for each fleet. The stabilizability analysis of the aggregated model, in the neighborhood of the equilibrium point, enables the determination of a Lyapunov function, which ensures the existence of a stabilizing discontinuous feedback for this model. This enables us to control the system and to lead, in an uniform way, any solution of this system towards this desired equilibrium point. To cite this article: R. Mchich et al., C. R. Biologies 328 (2005). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Applied Mathematical Sciences, 2015
The aim of the paper is to design a fishing strategy in order to regulate an exploited fish population. To this end, we consider a stage structured continuous model of a harvested fish population. This model includes 1 n stages represented by their abundance () i Xt , stage 0 being the pre-recruits stage. Each stage is characterized by its fecundity, mortality and predation rates. We gave a formula for the fishing effort as an output feedback control that allows to stabilize the system. We first gave a sufficient condition for the system to be globally stable around the reference equilibrium. Then the global stability behavior of the positive equilibrium is studied via the output feedback control. The same problem for the continuous stock recruitment model has been addressed in our previous work [9]. Simulations results of the continuous fishery systems confirm the effectiveness of the proposed design.
MALAYSIAN JOURNAL OF COMPUTING, 2021
In the marine ecosystem, the time delay or lag may occur in the predator response function, which measures the rate of capture of prey by a predator. This is because, when the growth of the prey population is null at the time delay period, the predator’s growth is affected by its population and prey population densities only after the time delay period. Therefore, the generalized Gause type predator-prey fishery models with a selective proportional harvesting rate of fish and time lag in the Holling type II predator response function are proposed to simulate and solve the population dynamical problem. From the mathematical analysis of the models, a certain dimension of time delays in the predator response or reaction function can change originally stable non-trivial critical points to unstable ones. This is due to the existence of the Hopf bifurcation that measures the critical values of the time lag, which will affect the stabilities of the non-trivial critical points of the models...
Optimal Control Strategies in Ecosystem-Based Fishery Models
2019
This dissertation considers the use of food chain models coupled with optimal control theory as a new approach for the problem of implementing ecosystem-based fishery management (EBFM) strategies. We consider the Black Sea anchovy on the southern part of the Black Sea as a case study of the implementation of EBFM. Because of the availability of temporal data, we build our first food chain model using ordinary differential equations to describe the anchovy dynamics, and then build our second food chain model using partial differential equations to include spatial features of the anchovy dynamics. In the study, we use the harvest rate of the anchovy fishery as our control that corresponds to number of fishing fleets. In the first model, the Black Sea anchovy stock was coupled with a prey and a predator species, using a system of nonlinear differential equations. The objective for the problem is to find the ecosystem-based optimal harvesting strategy that maximizes the discounted net v...
Sliding and oscillations in fisheries with on-off harvesting and different switching times
2014
In this paper, we propose a fishery model with a discontinuous on-off harvesting policy, based on a very simple and well known rule: stop fishing when the resource is too scarce, i.e. whenever fish biomass is lower than a given threshold. The dynamics of the one-dimensional continuous time model, represented by a discontinuous piecewise-smooth ordinary differential equation, converges to the Schaefer equilibrium or to the threshold through a sliding process. We also consider the model with discrete time impulsive on-off switching that shows oscillations around the threshold value. Finally, a discrete-time version of the model is considered, where on-off harvesting switchings are decided with the same discrete time scale of non overlapping reproduction seasons of the harvested fish species. In this case the border collision bifurcations leading to the creations and destruction of periodic oscillations of the fish biomass are studied.
CONTROL OF NONLINEAR DYNAMIC MODELS OF PREDATOR-PREY TYPE
Oecologia Australis, 2012
In the ecological context, a large class of population dynamics models can be written as dynamical systems of one or two variables, i.e., each variable represents a population density of a species. When one or more species is removed from the system (harvested), it is necessary to introduce a control (harvest policy) in order to avoid the extinction of species, due to harvesting. A threshold policy (TP) and a threshold policy with hysteresis (TPH) reviewed and discussed in this paper can be used to avoid the collapse of population densities, governed by predator-prey models. A threshold policy changes the dynamics of a predator-prey dynamical system in such a way that a stable positive equilibrium point is achieved. In other words, coexistence of both species occurs. A threshold policy with hysteresis changes the dynamics so that a limit cycle (bounded oscillation) is achieved, i.e., coexistence of species with a bounded oscillation in population densities occurs. This paper studies the continuous and discrete logistic model for one species and the Lotka-Volterra and Rosenzweig-MacArthur models for two species. The TP and TPH are seen to be versatile and useful in renewable resources management, being simple to design and implement, with some advantages in a situation of overexploitation, as well as in the presence of different types of uncertainties. The design of the policies is carried out by appropriate choice of virtual equilibria in a simple and intuitive manner, and the mathematics used is simple. Keywords: Nonlinear predator-prey models; threshold policy; threshold policy with hysteresis.
A predator-prey model for the optimal control of fish harvesting through the imposition of a tax
An International Journal of Optimization and Control: Theories & Applications (IJOCTA)
This paper is devoted to the study of ecosystem based fisheries management. The model represents the interaction between prey and predator population with Holling II functional response consisting of different carrying capacities and constant intrinsic growth rates. We have considered the continuous harvesting of predator only. It is observed that if the intrinsic growth rate of predator population crosses a certain critical value, the system enters into Hopf bifurcation. Our observations indicate that tax, the management object in fisheries system play huge impacts on this system. The optimal harvesting policy is disposed by imposing a tax per unit of predator biomass. The optimal harvest strategy is determined using Pontryagin's maximum principle, which is subject to state equations and control limitations. The implications of tax are also examined. We have derived different bifurcations and global stability of the system. Finally, numerical simulations are used to back up the...
Instabilities and robust control in fisheries
2003
Abstract: Demand and supply analysis in fisheries often indicates the presence of instabilities and multiple equilibria, both in open access conditions and in the socially optimal solution. The associated management problems are further intensified by uncertainty on the evolution of the resource stock or on demand conditions. In this paper the fishery management problem is handled using robust optimal control, where the objective is to choose a harvesting rule that will work, in the sense of preventing instabilities and ...