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# Modeling population dynamics and control strategies for a unique species evolving in heterogenous landscape

Manon de la Tousche, Pierre-Alexandre Bliman, Yves Dumont

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Manon de la Tousche, Pierre-Alexandre Bliman, Yves Dumont. Modeling population dynamics and control strategies for a unique species evolving in heterogenous landscape. 15 th Conference on Dynamical Systems Applied to Biology and Natural Sciences (DSABNS 2024), Feb 2024, Lisbonne, Portugal. pp.194. hal-04495187

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Submitted on 22 Mar 2024

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MODELING POPULATION DYNAMICS AND CONTROL STRATEGIES
FOR A UNIQUE SPECIES EVOLVING IN HETEROGENOUS LANDSCAPE

Manon de la Tousche *$^{1,3}$, Pierre-Alexandre Bliman $^{1}$ and Yves Dumont $^{2,3,4}$

¹ Sorbonne Université, Université Paris-Cité, Inria, CNRS, Laboratoire Jacques-Louis Lions,
équipe Mamba, Paris, France

² CIRAD, UMR AMAP, 97410 Saint-Pierre, Reunion Island, France

³ AMAP, Univ Montpellier, CIRAD, CNRS, INRAE, IRD, Montpellier, France

⁴ Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

manon.de-la-tousche@inria.fr (*presenter)

pierre-alexandre.bliman@inria.fr, yves.dumont@cirad.fr

We study a single-species metapopulation model with patches connected by linear diffusion. Inspired by the framework inaugurated by Takeuchi in [7] and Takeuchi and Lu in [4], and using tools of cooperative system theory [6], we show that under appropriate conditions, the sign of the stability modulus of the Jacobian of the system at the origin determines the asymptotic behaviour of the solutions. If it is non-positive, then the population becomes extinct in every patch. Conversely, if it is positive, then there exists a unique nonnegative equilibrium, which is positive and globally asymptotically stable.

In the latter case, given a subset of 'controlled' patches where human intervention is allowed, we study whether introducing additional mortality terms in these patches can result in population elimination in every patch. We characterize this possibility by an algebraic property on the graph of the residual, uncontrolled, system. When the population persists whatever the control, we assess the minimal attainable positive equilibrium value. When extinction is possible, we study the optimization problem consisting in achieving this task while minimizing a certain cost function, chosen as a nondecreasing and convex function of the mortalities added in the controlled patches. Using the (strict) convexity properties of the spectral radius of a non-negative matrix with respect to its diagonal elements [1, 3, 5], we show that such minimization problem admits a global minimizer, which is unique when not every patch is controlled.

This presentation stands within the framework of an ongoing project, AttracTIS, in Réunion island,
which aims at studying a combination of vector control tools against *Bactrocera dorsalis*, including the
Sterile Insect Technique [2]. This pest, also called oriental fruit fly [8], invaded Réunion island in 2017 and
since then has been impacting significantly the production of fruits, in particular mangos.

References

[1] Cohen, J.E. (1981). Convexity of the dominant eigenvalue of an essentially nonnegative matrix. *Proceedings of the American Mathematical Society* 81(4): 657-658. https://doi.org/10.1090/S0002-9939-1981-0601750-2
[2] Dyck V.A., Hendrichs J., Robinson A.S. (2005). *Sterile insect technique: principles and practice in area-wide integrated pest management*. Springer, Dordrecht, The Netherlands

[3] Friedland, S. (1981). Convex spectral functions. *Linear and multilinear algebra* 9(4): 299–316. https://doi.org/10.1080/03081088108817381

[4] Lu, Z., Takeuchi, Y. (1993). Global asymptotic behavior in single-species discrete diffusion systems. *Journal of Mathematical Biology* 321: 67–77. https://doi.org/10.1007/BF00160375

[5] Nussbaum, R.D. (1986). Convexity and log convexity for the spectral radius. *Linear Algebra and its Applications* 73: 59–122. https://doi.org/10.1016/0024-3795(86)90233-8

[6] Smith, Hal L. (1995). *Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems*. American Mathematical Society, Providence.

[7] Takeuchi, Y. (1989). Cooperative systems theory and global stability of diffusion models. *Acta Applicandae Mathematica* 14: 49–57. https://doi.org/10.1007/BF00046673

[8] Vargas R.I., Piñero, J.C., Leblanc, L. (2015). An overview of pest species of bactrocera fruit flies (Diptera: Tephritidae) and the integration of biopesticides with other biological approaches for their management with a focus on the pacific region. *Insects* 6(2):297–318 https://doi.org/10.3390/insects6020297