# Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations

1 MAMBA - Modelling and Analysis for Medical and Biological Applications
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
4 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : We analyse the asymptotic behaviour of integro-differential equations modelling $N$ populations in interaction, all structured by different traits. Interactions are modelled by non-local terms involving linear combinations of the total number of individuals in each population. These models have already been shown to be suitable for the modelling of drug resistance in cancer, and they generalise the usual Lotka-Volterra ordinary differential equations. Our aim is to give conditions under which there is persistence of all species. Through the analysis of a Lyapunov function, our first main result gives a simple and general condition on the matrix of interactions, together with a convergence rate. The second main result establishes another type of condition in the specific case of mutualistic interactions. When either of these conditions is met, we describe which traits are asymptotically selected.
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Journal articles

Cited literature [36 references]

https://hal.archives-ouvertes.fr/hal-01470722
Contributor : Camille Pouchol <>
Submitted on : Friday, April 14, 2017 - 10:34:02 AM
Last modification on : Tuesday, December 10, 2019 - 3:08:21 PM
Long-term archiving on: Saturday, July 15, 2017 - 12:27:00 PM

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### Identifiers

• HAL Id : hal-01470722, version 4
• ARXIV : 1702.06187

### Citation

Camille Pouchol, Emmanuel Trélat. Global stability with selection in integro-differential Lotka-Volterra systems modelling trait-structured populations. Journal of Biological Dynamics, Taylor & Francis Open, 2018, 12 (1), pp.872--893. ⟨hal-01470722v4⟩

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