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

Abstract

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.

Alternative cliques of coexisting species in complex ecosystems

“…In section 3.2 we also explore the properties of cliques for systems with increasing S and close to the critical line separating stable coexistence from the cliques regime. This line begins at the point (µ, σ) = (−1, 0) and ends at (µ, σ) = (0, √ 2/S) [12,20,29] (figure 1(A)). The equation for the line is…”

“…In the original parameter space, the condition is a line crossing from points [12,20,29]). The first point does not depend on system size: neutral coexistence (σ = 0) is always disrupted when competition overcomes self-regulation, no matter the number of species [15,36].…”

Stock patterns in a class of delayed discrete-time population models

“…Further, in the logistic LV model, prey population follows a logistic growth function. These two models and their variations are effective for developing large and complex LV models [5], describing population evolution and trait dynamics [28], explain ecological resilience and population harvesting [46], etc. On the other hand, fundamental contributions from May [26,27], and Shapiro [39] using recurrence relations proved the existence of chaos in population ecology.…”