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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mfrac> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> represents the number of species and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> , the abundance of species <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> . Among these families of coupled differential equations, Lotka–Volterra (LV) equations, corresponding to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mrow> <mml:mtext mathvariant="bold">x</mml:mtext> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:math> play a privileged role, as the LV model represents an acceptable trade-off between complexity and tractability. Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>r</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> is the intrinsic growth of species <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> stands for the interaction matrix: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> </mml:math> represents the effect of species <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>j</mml:mi> </mml:math> over species <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>i</mml:mi> </mml:math> . For large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , estimating matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> is often an overwhelming task and an alternative is to draw <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> 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.

Topics & Concepts