Coexistence and extinction for stochastic Kolmogorov systems

Hening, Alexandru; Nguyen, Dang H.

doi:10.1214/17-aap1347

Cited by 148 publications

(178 citation statements)

References 37 publications

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“…and β 2 > b 1 a 2 (b 2 R − α 2 ) b 1 R − α 1 − b 2 a 2 . If both invasion rates are positive we get by Hening & Nguyen (2018a) that the species coexist. > 0,…”

Section: Discussionmentioning

confidence: 99%

“…This makes it a particularly relevant paper to our work. For proofs of this theorem for stochastic differential equations see Hening & Nguyen (2018a)[Theorem 4.1 and Example 2.4] as well as Benaim (2018) [Theorem 4.4 and Definition 4.3]. In the setting of PDMP see Benaïm & Lobry (2016) and Benaim (2018) [Theorem 4.4 and Definition 4.3].…”

Section: Stochastic Coexistence Theorymentioning

confidence: 99%

“…2 > 0 so that, according to the result by Hening & Nguyen (2018a), none of the species go extinct on their own, as well as…”

Section: Stochastic Differential Equationsmentioning

confidence: 99%

“…x 1 persists while x 2 goes extinct. The following scenarios are possible (Turelli & Gillespie (1980), Kesten & Ogura (1981), Hening & Nguyen (2018a), Evans et al (2015), Hening & Nguyen (2018b))…”

Section: Stochastic Differential Equationsmentioning

confidence: 99%

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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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“…and β 2 > b 1 a 2 (b 2 R − α 2 ) b 1 R − α 1 − b 2 a 2 . If both invasion rates are positive we get by Hening & Nguyen (2018a) that the species coexist. > 0,…”

Section: Discussionmentioning

confidence: 99%

Section: Stochastic Coexistence Theorymentioning

confidence: 99%

“…2 > 0 so that, according to the result by Hening & Nguyen (2018a), none of the species go extinct on their own, as well as…”

Section: Stochastic Differential Equationsmentioning

confidence: 99%

Section: Stochastic Differential Equationsmentioning

confidence: 99%

See 2 more Smart Citations

The competitive exclusion principle in stochastic environments

Hening

Nguyen

2020

J. Math. Biol.

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“…Conditions for the coexistence and extinction of the differenmt species can be found in [HN18]. Suppose that δ = 0.05, f 1 (x) = 1, f 2 (x) = 1, g 1 (x) = 6, g 2 (x) = 6 for all x ∈ [0, ∞) 2 .…”

Section: Numerical Examplesmentioning

confidence: 99%

Harvesting of interacting stochastic populations

et al. 2019

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We analyze the optimal harvesting problem for an ecosystem of species that experience environmental stochasticity. Our work generalizes the current literature significantly by taking into account non-linear interactions between species, state-dependent prices, and species injections. The key generalization is making it possible to not only harvest, but also 'seed' individuals into the ecosystem. This is motivated by how fisheries and certain endangered species are controlled. The harvesting problem becomes finding the optimal harvesting-seeding strategy that maximizes the expected total income from the harvest minus the lost income from the species injections. Our analysis shows that new phenomena emerge due to the possibility of species injections.It is well-known that multidimensional harvesting problems are very hard to tackle. We are able to make progress, by characterizing the value function as a viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equations. Moreover, we provide a verification theorem, which tells us that if a function has certain properties, then it will be the value function. This allows us to show heuristically, as was shown in Lungu and Øksendal (Bernoulli '01), that it is almost surely never optimal to harvest or seed from more than one population at a time.It is usually impossible to find closed-form solutions for the optimal harvesting-seeding strategy. In order to by-pass this obstacle we approximate the continuous-time systems by Markov chains. We show that the optimal harvesting-seeding strategies of the Markov chain approximations converge to the correct optimal harvesting strategy. This is used to provide numerical approximations to the optimal harvesting-seeding strategies and is a first step towards a full understanding of the intricacies of how one should harvest and seed interacting species. In particular, we look at three examples: one species modeled by a Verhulst-Pearl diffusion, two competing species and a two-species predator-prey system.

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Toward a “modern coexistence theory” for the discrete and spatial

Ellner

Snyder

Adler

et al. 2022

Ecological Monographs

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The usual theoretical condition for coexistence is that each species in a community can increase when it is rare (mutual invasibility). Traditional coexistence theory implicitly assumes that the invading species is common enough that we can ignore demographic stochasticity but rare enough that it does not compete with itself, even after it has reached a stationary spatial distribution. However, short‐distance dispersal of discrete individuals leads to locally dense population clusters, and existing theory breaks down. We have an intuition that when we account for invader–invader competition, shorter‐range dispersal should reduce the invader's ability to escape competition, but exactly how does this translate into lower population growth? And how will invader discreteness affect outcomes? We need a way of partitioning the contributions to coexistence, but current modern coexistence theory (MCT) does not apply under these conditions. Here we present a computationally based partitioning method to quantify the contributions to coexistence from different mechanisms, as in MCT. We also build up an intuition for how invader clumping and discreteness will affect these contributions by analyzing a case study, a lattice‐based spatial lottery model. We first consider fluctuation‐dependent coexistence, partitioning the contributions of variable environment, variable competition, demographic stochasticity, and their correlations and interactions. Our second example examines fluctuation‐independent coexistence maintained by a fecundity–survival trade‐off, and partitions the contributions to coexistence from interspecific differences in fecundity, in mortality, and in dispersal. We find that demographic stochasticity harms an invader, but only slightly. Localized invader dispersal, on the other hand, can have a strong effect. When invaders are more clumped, they compete with each other more intensely when rare, so they too become limited by environment‐competition covariance. More invader clumping also means that variation in competition changes from helping the invader to harming it. More broadly, invader clumping is likely to weaken any coexistence mechanism that relies on the invader escaping competition from the resident, because invader clumping means that the resident is no longer the only source of competition.

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Coexistence and extinction for stochastic Kolmogorov systems

Cited by 148 publications

References 37 publications

The competitive exclusion principle in stochastic environments

The competitive exclusion principle in stochastic environments

Harvesting of interacting stochastic populations

Toward a “modern coexistence theory” for the discrete and spatial

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