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Global eradication for spatially structured populations by regional control

  • * Corresponding author: Vincenzo Capasso

    * Corresponding author: Vincenzo Capasso 
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  • 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.

    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 $\beta = 0$

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

    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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