Global eradication for spatially structured populations by regional control

Sebastian Aniţa; Vincenzo Capasso; Ana-Maria Moşneagu

doi:10.3934/dcdsb.2018263

Article Contents

2019, Volume 24, Issue 6: 2511-2533. Doi: 10.3934/dcdsb.2018263

This issue Previous Article Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations Next Article Stability and bifurcation in an age-structured model with stocking rate and time delays

Global eradication for spatially structured populations by regional control

1.
Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, "Octav Mayer" Institute of Mathematics of the Romanian Academy, Iaşi 700506, Romania
2.
ADAMSS (Centre for Advanced Applied Mathematical and Statistical Sciences), Universitá degli Studi di Milano, 20133 Milano, Italy
3.
Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iaşi, Iaşi 700506, Romania

* Corresponding author: Vincenzo Capasso
* Corresponding author: Vincenzo Capasso

Received: November 2017

Revised: May 2018

Early access: October 2018

Published: June 2019

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Abstract

This paper concerns problems for the eradication of apopulation by acting on a subregion ω. The dynamics isdescribed by a general reaction-diffusion system, including one ormore populations, subject to a vital dynamics with either locallogistic or nonlocal logistic terms. For the one populationcase, a necessary condition and a sufficient condition foreradicability (zero-stabilizability) are obtained, in terms of the sign of the principaleigenvalue of a suitable elliptic operator acting on the domain$Ω \setminus \overline{ω }$. A feedbackharvesting-like control with a large constant harvesting raterealizes eradication of the population. The problem oferadication is then reformulated in a more convenient way, bytaking into account the total cost of the damages produced by apest population and the costs related to the choice of therelevant subregion, and approximated by a regional optimalcontrol problem with a finite horizon. A conceptual iterativealgorithm is formulated for the simulation of the proposedoptimal control problem. Numerical tests are given to illustratethe effectiveness of the results. Relevant regional controlproblems for two populations reaction-diffusion models, such asprey-predator system, and an SIR epidemic system with spatialstructure and local/nonlocal force of infection, have beenanalyzed too.

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Mathematics Subject Classification: Primary: 49K20, 49Q10, 92D25, 93C20, 93B40.

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Figure 1. The representation of final iteration of $\omega$ for $\alpha \in \{5,6,7,8,9,10\}$ and $\beta = 0$

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Figure 2. The representation of final iteration of $\omega$ for $\alpha \in \{0.2, 0.3, 0.4, 0.45, 0.5\}$ and $\beta = 0.3$

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Figure 3. The representation of initial and final iterations of $\omega$ for $\alpha \in \{0.5, 1, 2.5\}$ and $\beta = 0.001$

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