Abstract
In this work, we propose a multifishing area prey-predator discrete-time model which describes the interaction between the prey and middle and top predators in various areas, which are connected by their movements to their neighbors, to provide realistic description prey effects of two predators. A grid of colored cells is presented to illustrate the entire domain; each cell may represent a subdomain or area. Next, we propose two harvesting control strategies that focus on maximizing the biomass of prey, in the targeted area, and minimizing the biomass of middle and top predators coming from the neighborhood of this targeted area to ensure sustainability and maintain a differential chain system. Theoretically, we have proved the existence of optimal controls, and we have given a characterization of controls in terms of states and adjoint functions based on a discrete version of Pontryagin’s maximum principle. To illustrate the theoretical results obtained, we propose numerical simulations for several scenarios applying the forward-backward sweep method (FBSM) to solve our optimality system in an iterative process.
Highlights
In this work, we propose a multifishing area prey-predator discrete-time model which describes the interaction between the prey and middle and top predators in various areas, which are connected by their movements to their neighbors, to provide realistic description prey effects of two predators
We propose two harvesting control strategies that focus on maximizing the biomass of prey, in the targeted area, and minimizing the biomass of middle and top predators coming from the neighborhood of this targeted area to ensure sustainability and maintain a differential chain system. eoretically, we have proved the existence of optimal controls, and we have given a characterization of controls in terms of states and adjoint functions based on a discrete version of Pontryagin’s maximum principle
In [7], the authors have presented a bioeconomic model for several fish populations. ey have assumed that these fish populations compete with each other for space or food. e techniques and issues associated with the bioeconomic modeling for the exploitation of marine resources have been discussed in detail in [10,11,12,13,14]
Summary
2.1.1. e Model and Simulation without Controls. we consider a multifishing area prey-predator discrete-time model which describes the spatial-temporal evolution of fish populations within a global domain of interest Ω, split to M × M areas or cells, uniform in size. erefore, our domain Ω can be represented as follows: M. Erefore, our domain Ω can be represented as follows: M. is model classifies the population into three compartments in the area Cpq. Let xCi pq , yCi pq , and zCi pq denote the population densities of the prey, middle predator, and top predator, respectively. E aim is to make fishing effort controls by the fishing fleets during fishing in the studied area Cpq while adding one harvesting function to harvest the middle and top predators yCi pq and zCi pq which threaten the prey xCi pq over time to ensure environmental sustainability and maintain a differential chain system. Our goal is to minimize the number of middle predators yCpq and top predators zCpq while maximizing the harvesting function and increasing the number of prey xCpq in Cpq. In other words, we are seeking optimal controls uCrs∗ and vCrs∗ such that.
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