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We use nonlinear model predictive control to find the optimal harvesting effort of a renewable resource system with a nonlinear state equation that maximizes a nonlinear profit function. A solution approach is proposed and discussed and satisfactory numerical illustrations are provided.

Consumption of the world’s natural resources is increasing at a disturbing rate. The United Nations Environment Programme (UNEP) warned that current voracious consumption of resources cannot be sustained.

Unlike petroleum, oil, copper, and gold, fish are renewable resources. However, more people are eating fish than ever before, and fish stocks are declining alarmingly. Aquaculture is failing to fill the gaps between the supply and the demand, for lack of a better management, as reported by a recent Food and Agriculture Organization (FAO) review.

Better management of fisheries in the high seas, conservation of the biodiversity of ecosystems and species related to it, and reduction of illegal catch of popular and consumed worldwide fish are required to reverse the negative trends threatening fish and the ocean environment on which they depend. Collective actions at all levels and extensive cooperation optimizing the use of depleted resources are needed to help the world abandon the race for fish and adopt an ecosystemic approach that is crucial to ensure the health and future productivity of these key marine ecosystems.

Clark [

Since the earliest models of Gordon [

Ganguly and Chaudhuri [

Fan and Wang [

Peng [

Joshi et al. [

E. Braverman and L. Braverman [

In Halkos and Papageorgiou [

Duncan et al. [

Optimal control theory has been extensively used to determine the optimal harvesting policy for renewable resources such as fish stocks. Not only in the extensions of the basic model that we have described above, but also in more integrated models which involve two or more species, structured models, a population of consumers, predator-prey models, reserve-unreserve areas, and so forth.

Our intention in this paper is to use a different approach, model predictive control (MPC). Model predictive control for linear constrained systems provides excellent control solutions both theoretically and practically. Many systems, however, such as in renewable resources, are inherently nonlinear. This motivates the use of nonlinear model predictive control. Basically, in model predictive control an optimal control problem is solved for the current system state. MPC is based on an iterative process over finite horizon. At time

MPC is an advanced method of process control that has been successfully used in the process industries, especially in chemical processes; see, for instance, Goodwin et al. [

Clark [

The model is formulated in Section

Let

We use the logistic growth rate function

We also use the rate of harvest

An NMPC approach is used in the next section to determine the control variable at time

Different techniques have been proposed in the literature to speed up the calculation of the optimal control variable of the problem stated above. We use an approximate calculation of the integral in the objective function (

Substituting (

Since the harvesting effort

We provide in this section simulation examples to show different types of solutions that can be obtained using the results obtained in the previous section. For a given time horizon

Consider the following values of parameters:

Variations of

We keep all the parameters constant as in Case

Variations of

Again, we keep all the parameters constants as in Case

Variations of

Once again, we keep all the parameters constants as in Case

Variations of

Optimal control theory has been plentifully used to determine at time

The method described in this paper is quite robust and we propose to further experiment it on more complex models. For example, the catch-rate function (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the referees for carefully reading the paper and making relevant suggestions to improve its final version. The authors extend their appreciations to the Deanship of Scientific Research at King Saud University for funding the work through the research group Project no. RGP-024.