Abstract
We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism.
Highlights
Academic Editor: Pavel GrabarnikReceived: 14 January 2021Accepted: 22 February 2021Published: 26 February 2021In recent decades, there has been an increasing interest in the mathematical modelling of several biological phenomena: pattern formation, epidemic disease transmission, blood circulation, atherosclerosis, species competency, migration of species, and so on, see for instance [1,2,3,4,5,6]
This paper presents a the application of Dubovitskii and Milyutin formalism to study a control problem for a reaction-diffusion system arising in the competency of three species in a bounded ecosystem
The control problem is reformulated as an optimal control problem by incorporating a cost functional which maximizes the total density of beneficial species to the ecosystem and minimizes the pests and the exposition to the human intervention by incorporation of some substances to control pests
Summary
There has been an increasing interest in the mathematical modelling of several biological phenomena: pattern formation, epidemic disease transmission, blood circulation, atherosclerosis, species competency, migration of species, and so on, see for instance [1,2,3,4,5,6]. In [14,15,22] the authors study the solution of inverse problems related with the reconstruction of coefficients in reactiondiffusion systems arising in epidemiology, from observations of the state solution in a fixed time, by application of some results of standard optimal control theory. To the best of our knowledge, there is not yet in the literature an application of Dubovitskii and Milyutin to study optimal control problems in reaction-diffusion systems arising in the competition of organisms or species in an ecosystem.
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