Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Benaı̈m, Michel; Lobry, Claude

doi:10.1214/16-aap1192

Cited by 69 publications

(115 citation statements)

References 34 publications

(55 reference statements)

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“…simultaneously in an all environments then the competitive exclusion principle holds. If this condition does not hold, we construct an example, based on the work by Benaïm & Lobry (2016), with two species x 1 , x 2 competing for one resource, and two environments E 1 and E 2 such that in the switched system the two species coexist. We note that in this setting we do have that the growth rates of the species depend linearly on the resource.…”

Section: Discussionmentioning

confidence: 99%

The competitive exclusion principle in stochastic environments

Hening

Nguyen

2020

J. Math. Biol.

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In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's 'paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times.In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the percapita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.2010 Mathematics Subject Classification. 92D25, 37H15, 60H10, 60J05, 60J99.

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Section: Discussionmentioning

confidence: 99%

The competitive exclusion principle in stochastic environments

Hening

Nguyen

2020

J. Math. Biol.

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“…As stated before, we pick two environments ε 1 and ε 2 and we model the environmental variation of a biosystem by randomly switching the chemostat the two species and the resource evolve in. This idea and its mathematical resolution has been introduced in [2]. In this previous article, the authors exhibit counterintuitive phenomenon on the behavior of a two-species Lotka-Volterra model of competition where the environment switches between two environments that are both favorable to the same species.…”

Section: The Probabilistic Model : a Pdmp Systemmentioning

confidence: 99%

“…It is clear that the process (Z t ) leaves invariant all the extinction set and the interior set M \ M 0 . In order to describe the behavior of the process (Z t ) when Z 0 ∈ M \ M 0 , [2] suggests to study the invasion rates of species w defined as:…”

Section: The Probabilistic Model : a Pdmp Systemmentioning

confidence: 99%

Comparison of the global dynamics for two chemostat-like models: Random temporal variation versus spatial heterogeneity

Lagasquie

Madec

2020

Nonlinear Analysis: Hybrid Systems

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This article is dedicated to the study and comparison of two chemostat-like competition models in a heterogeneous environment. The first model is a probabilistic model where we build a PDMP simulating the effect of the temporal heterogeneity of an environment over the species in competition. Its study uses classical tools in this field. The second model is a gradostat-like model simulating the effect of the spatial heterogeneity of an environment over the same species. Despite the fact that the nature of the two models is very different, we will see that their long time behavior is globally very similar. We define for both model quantities called invasion rates which model the growth rate of a species when it is near to extinction. We show that the signs of these invasion rates essentially determine the long time behavior for both systems. In particular, we exhibit a new example of bistability between a coexistence steady state and a semi-trivial steady state.

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“…for any ε > 0. As a result, To prove the permanence of the species when λ > 0, we use the techniques in [2]. We need some lemmas.…”

Section: )mentioning

confidence: 99%

Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate

Tuong

Nguyen

Dieu

et al. 2019

Nonlinear Analysis: Hybrid Systems

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In this paper, we consider a stochastic SIRS model with general incidence rate and perturbed by both white noise and color noise. We determine the threshold λ that is used to classify the extinction and permanence of the disease. In particular, λ < 0 implies that the disease-free (K, 0, 0) is globally asymptotic stable, i.e., the disease will eventually disappear. If λ > 0 the epidemic is strongly stochastically permanent. Our result is considered as a significant generalization and improvement over the results in [10,11,16,20,21].

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Lotka–Volterra with randomly fluctuating environments or “how switching between beneficial environments can make survival harder”

Cited by 69 publications

References 34 publications

The competitive exclusion principle in stochastic environments

The competitive exclusion principle in stochastic environments

Comparison of the global dynamics for two chemostat-like models: Random temporal variation versus spatial heterogeneity

Extinction and permanence in a stochastic SIRS model in regime-switching with general incidence rate

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