Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

Michel Benaïm; Antoine Bourquin; Dang H. Nguyen

doi:10.3934/dcdsb.2022023

Article Contents

2022, Volume 27, Issue 11: 6841-6863. Doi: 10.3934/dcdsb.2022023

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Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

1.
Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland
2.
Department of Mathematics, University of Alabama, Tuscaloosa, Al 35487-0350, USA

^*Corresponding author: Michel Benaïm
^*Corresponding author: Michel Benaïm

Received: April 2021

Revised: November 2021

Early access: February 2022

Published: November 2022

Michel Benaim and Antoine Bourquin are supported in part by the SNF grant 200020-196999. Dang H. Nguyen is supported in part by NSF through the grant DMS-1853467.

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Abstract

We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A. Hening and D. Nguyen in [9, 10] where they provided conditions for stochastic persistence and extinction. In this paper, we extend their results to the degenerate situation in which the top or the bottom species is subject to random perturbations. Under the persistence condition, there exists a unique invariant probability measure supported by the interior of $ {{\mathbb R}}_+^n $ having a smooth density.

Moreover, we study a more general model, in which we give new conditions which make it possible to characterize the convergence of the semi-group towards the unique invariant probability measure either at an exponential rate or at a polynomial one. This will be used in the stochastic Lotka-Volterra food chain to see that if intra-specific competition occurs for all species, the rate of convergence is exponential while in the other cases it is polynomial.

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Mathematics Subject Classification: Primary: 60J60, 60H10; Secondary: 37H15, 92D25.

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References

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