# American Institute of Mathematical Sciences

May  2019, 39(5): 2933-2960. doi: 10.3934/dcds.2019122

## Generalized linear models for population dynamics in two juxtaposed habitats

 1 Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France 2 USTHB, LAMNEDP, Faculté de Mathématiques, BP.32, El Alia, Bab Ezzouar, 16111 Alger, Algérie 3 Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France

* Corresponding author: Alexandre Thorel

Received  September 2018 Revised  October 2018 Published  January 2019

Fund Project: The last author is supported by CIFRE contract 2014/1307 with Qualiom Eco company

In this work we introduce a generalized linear model regulating the spread of population displayed in a $d$-dimensional spatial region $\Omega$ of $\mathbb{R}^{d}$ constituted by two juxtaposed habitats having a common interface $\Gamma$. This model is described by an operator $\mathcal{L}$ of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in $L^{p}$-spaces using the $H^{\infty }$-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by $\mathcal{L}$ which is important for the study of the complete nonlinear generalized diffusion equation associated to it.

Citation: Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Alexandre Thorel. Generalized linear models for population dynamics in two juxtaposed habitats. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2933-2960. doi: 10.3934/dcds.2019122
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