Equilibrium and surviving species in a large Lotka–Volterra system of differential equations

Clenet, Maxime; Massol, François; Najim, Jamal

doi:10.1007/s00285-023-01939-z

Cited by 5 publications

(5 citation statements)

References 38 publications

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“…Note that in this study we left aside the formal model problems of equilibrium stability, and of dynamic properties of model solutions. All these problems are an independent scientific area [62][63][64][65][66][67].…”

Section: Discussion: Coexistence Of Microbial Populations and Regulat...mentioning

confidence: 99%

Control Factors for the Equilibrium Composition of Microbial Communities in Open Systems: Theory and Experiments

Дегерменджи

Abakumov

2023

Mathematics

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The present paper is a summary of the authors’ theoretical and experimental research dealing with the patterns of stable equilibrium coexistence of microbial populations in flow systems interacting through specific density-dependent growth regulators (RFs). The discovered “paradoxical” lack of dependence of the background steady-state levels (concentrations) of RFs on their input values is confirmed experimentally and theoretically through the introduced sensitivity coefficients. This effect has been termed “autostabilization” of RFs. An important theorem (formula) of “quantization” suggesting the integer value of the sum of all sensitivity coefficients, which is equal to the difference between the number of RFs and the number of populations of one trophic level, has been proven. A modification of the “quantization” formula for an arbitrary trophic web is shown. A new criterion for intra- and inter-population microbial interactions for RFs is proposed—the response of growth acceleration to a perturbation in population size. This criterion makes it possible to quantify interspecific complex relationships, which has been previously impossible. The relationship between the new coefficients of inter-population interactions and the accuracy of model verification has been shown theoretically. Based on this criterion and the autostabilization effect, a method for experimental search for unknown RFs is proposed.

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Section: Discussion: Coexistence Of Microbial Populations and Regulat...mentioning

confidence: 99%

Control Factors for the Equilibrium Composition of Microbial Communities in Open Systems: Theory and Experiments

Дегерменджи

Abakumov

2023

Mathematics

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“…Additionally, we go beyond classic applications of this random-interaction approach, which focused on computing properties of a particular equilibrium (e.g., the equilibrium with all

species), to study the set of persistent species that are actually assembled through the dynamics of Eq. 2 , possibly following the extinction of some species from the pool ( 41 , 44 – 46 ).…”

Section: Methodsmentioning

confidence: 99%

“…Rather than asking whether a particular set of species can coexist, we examine metacommunities that emerge through the model dynamics from an initial random pool ( 41 – 43 ). This probabilistic approach is a powerful way to identify characteristic outcomes in complex models ( 44 – 46 ). Such outcomes are robust, in the sense that they do not depend on specific parameter choices.…”

mentioning

confidence: 99%

Coexistence of many species under a random competition–colonization trade-off

Miller,

Clenet,

Della Libera

et al. 2024

Proc. Natl. Acad. Sci. U.S.A.

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The competition–colonization (CC) trade-off is a well-studied coexistence mechanism for metacommunities. In this setting, it is believed that the coexistence of all species requires their traits to satisfy restrictive conditions limiting their similarity. To investigate whether diverse metacommunities can assemble in a CC trade-off model, we study their assembly from a probabilistic perspective. From a pool of species with parameters (corresponding to traits) sampled at random, we compute the probability that any number of species coexist and characterize the set of species that emerges through assembly. Remarkably, almost exactly half of the species in a large pool typically coexist, with no saturation as the size of the pool grows, and with little dependence on the underlying distribution of traits. Through a mix of analytical results and simulations, we show that this unlimited niche packing emerges as assembly actively moves communities toward overdispersed configurations in niche space. Our findings also apply to a realistic assembly scenario where species invade one at a time from a fixed regional pool. When diversity arises de novo in the metacommunity, richness still grows without bound, but more slowly. Together, our results suggest that the CC trade-off can support the robust emergence of diverse communities, even when coexistence of the full species pool is exceedingly unlikely.

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“…For the i.i.d. model, a simple order statistics argument can be found in [66,67]. These a priori different methods yield the same equations from which one can extract p(α).…”

Section: (I) Number Of Vanishing Speciesmentioning

confidence: 99%

“…Denote by Z ∼ N (0, 1) and by Φ the cumulative Gaussian distribution function: (Bunin [58], Galla [65], Clenet et al [66,67]). Let α ∈ √ 2, 2 log(N ) and assume that Γ follows Model (2.4)-(ii).…”

Section: Remarksmentioning

confidence: 99%

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

Akjouj,

Barbier,

Clenet

et al. 2024

Proc. R. Soc. A.

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Ecosystems represent archetypal complex dynamical systems, often modelled by coupled differential equations of the form d x i d t = x i ϕ i ( x 1 , … , x N ) , where N represents the number of species and x i , the abundance of species i . Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to ϕ i ( x 1 , … , x N ) = r i − x i + ( Γ x ) i , play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, r i is the intrinsic growth of species i and Γ stands for the interaction matrix: Γ i j represents the effect of species j over species i . For large N , estimating matrix Γ is often an overwhelming task and an alternative is to draw Γ at random, parameterizing its statistical distribution by a limited number of model features. Dealing with large random matrices, we naturally rely on random matrix theory (RMT). The aim of this review article is to present an overview of the work at the junction of theoretical ecology and large RMT. It is intended to an interdisciplinary audience spanning theoretical ecology, complex systems, statistical physics and mathematical biology.

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Equilibrium and surviving species in a large Lotka–Volterra system of differential equations

Cited by 5 publications

References 38 publications

Control Factors for the Equilibrium Composition of Microbial Communities in Open Systems: Theory and Experiments

Control Factors for the Equilibrium Composition of Microbial Communities in Open Systems: Theory and Experiments

Coexistence of many species under a random competition–colonization trade-off

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

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