Title :

Feasibility of sparse large Lotka-Volterra ecosystems

Author(s) :

Akjouj, Imane [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Université de Lille
Najim, Jamal [Auteur]
Laboratoire d'Informatique Gaspard-Monge [LIGM]

Journal title :

Journal of Mathematical Biology

Pages :

1-28

Publisher :

Springer

Publication date :

2022

ISSN :

0303-6812

English keyword(s) :

92D40
Secondary 60B20
60G70
Gaussian concentration. MSC Classification 2010: Primary 15B52
Large random matrices
Lotka-Volterra systems
Feasibility and stability
Foodwebs
Theoretical ecology

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]
Sciences de l'environnement/Biodiversité et Ecologie

English abstract : [en]

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with d other species and that their interaction ...
Show more >Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka-Volterra system of coupled differential equations. We assume that each species interacts with d other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n. We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution (component-wise) to the equilibrium linear equation. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Proc. AMS 2021.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL