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Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in \(L^1\)

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Abstract

In the present paper, we address the asymptotic behavior of a fish population system structured in age and weight, while also incorporating spatial effects. Initially, we develop an abstract perturbation result concerning the essential spectral radius, employing the regular systems approach. Following that, we present the model in the form of a perturbed boundary problem, which involves unbounded operators on the boundary. Using time-invariant regular techniques, we construct the corresponding semigroup solution. Then, we designate an operator characteristic equation of the primary system via the radius of a bounded linear operator defined on the boundary space. Moreover, we provide a characterization of the uniform exponential stability and the asynchronous exponential growth property (AEG) by localizing the essential radius and proving the irreducibility of the perturbed semigroup. Finally, we precise the projection that emerged from the (AEG) property; this depends on the developed characteristic equation.

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Acknowledgements

We would like to express our heartfelt gratitude to the anonymous referee for their insightful comments, especially for suggesting a shortened proof of Theorem 13.

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Authors and Affiliations

  1. Faculty of Sciences Semlalia, Cadi Ayyad University, LMDP, UMMISCO (IRD-UPMC), PB 2390,  Marrakesh, Morocco

    Samir Boujijane, Mohamed Halloumi & Lahcen Maniar

  2. National School of Applied Sciences, Cadi Ayyad University, LMDP, UMMISCO (IRD-UPMC), PB 575,  Marrakesh, Morocco

    Said Boulite

  3. Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, Sa, Italy

    Abdelaziz Rhandi

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  1. Samir Boujijane
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Correspondence to Said Boulite.

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S. Boujijane is supported by the project I-MAROC ‘Artificial Intelligence/Applied Mathematics, Health/Environment: Simulation for Decision Support’, the R &D MULTITHÉMATIQUE -APR &D2020 program financed by the Moroccan “Département de l’Enseignement Supérieur et de la Recherche Scientifique” (DESRS) and “Fondation OCP” (FOCP).

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Boujijane, S., Boulite, S., Halloumi, M. et al. Well-posedness and asynchronous exponential growth of an age-weighted structured fish population model with diffusion in \(L^1\). J. Evol. Equ. 24, 14 (2024). https://doi.org/10.1007/s00028-023-00942-7

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