Dynamically evolved community size and stability of random Lotka-Volterra ecosystems ^(a)

Abstract

We use dynamical generating functionals to study the stability and size of communities evolving in Lotka-Volterra systems with random interaction coefficients. The size of the eco-system is not set from the beginning. Instead, we start from a set of possible species, which may undergo extinction. How many species survive depends on the properties of the interaction matrix; the size of the resulting food web at stationarity is a property of the system itself in our model, and not a control parameter as in most studies based on random matrix theory. We find that prey-predator relations enhance stability, and that variability of species interactions promotes instability. Complexity of inter-species couplings leads to reduced sizes of ecological communities. Dynamically evolved community size and stability are hence positively correlated.

Metacommunity‐scale biodiversity regulation and the self‐organised emergence of macroecological patterns

“…Community assembly models, therefore, converge on dynamic steady states defined by the onset of structural instability. By applying alternative mathematical approaches to studying this phenomenon (Yodzis 1988;Tokita 2004;Rossberg 2013;Dougoud et al 2018;Barbier et al 2018;Galla 2018), structural instability can be interpreted as resulting from the amplification of perturbations through complex indirect interactions in large communities.…”

Ecological communities from random generalized Lotka-Volterra dynamics with nonlinear feedback

“…In the following we will set r i = 1 and K i = 1 for all species, following [39,43]. The coefficients α ij describe the interactions between the different species.…”

“…These are typically models in which the (relative) growth of the abundance of one species depends linearly on the disordered fitness function. Examples can be found in [30,31,39]. One notable exception are so-called Sato-Crutchfield dynamics in the context of game learning [38,40]; in these cases the fixed point relations contain logarithmic terms in the degrees of freedom.…”

Stabilization of extensive fine-scale diversity by spatio-temporal chaos

“…). An important point, however, is that for a large number of interacting types the niche-like interactions must be much stronger than the interactions between different types: as the sum over the (random) effects of all the other types will be of order √ K, in order to substantially affect the ecological dynamics, Q must be of order √ K. Most theoretical work, motivated particularly by competition for a mixture of resources [20,21,48,74], has focused on such large Q [27,28,75,76] and positive correlations between interactions, γ > 0 [23,24,29,30,31,32]. In this regime, for a large number of types, there is a unique stable, uninvadable, community, corresponding to a stable fixed point of the dynamics, with a substantial fraction (O(K)) types surviving and the other types unable to invade.…”

“…Most recent work using has focused on stable communities caused by niche-like self interactions, [26,28,30,31,75]. (Note that [31,75] use a model with a third parameter but this can essentially be eliminated by use of the Lagrange multiplier, Υ(t).) For Q > Q c (γ) = √ K(1 + γ)/ √ 2 there is a unique large stable community, with O(K) types surviving.…”