Forecasting Transitions in Systems with High-Dimensional Stochastic Complex Dynamics: A Linear Stability Analysis of the Tangled Nature Model

Abstract

We propose a new procedure to monitor and forecast the onset of transitions in high dimensional complex systems. We describe our procedure by an application to the Tangled Nature model of evolutionary ecology. The quasi-stable configurations of the full stochastic dynamics are taken as input for a stability analysis by means of the deterministic mean field equations. Numerical analysis of the high dimensional stability matrix allows us to identify unstable directions associated with eigenvalues with positive real part. The overlap of the instantaneous configuration vector of the full stochastic system with the eigenvectors of the unstable directions of the deterministic mean field approximation is found to be a good early-warning of the transitions occurring intermittently.Introduction -Many complex high dimensional systems are characterised by intermittent dynamics, where relatively long quiescent periods are interrupted by sudden and quick bursts of activity during which the system suffers hectic rearrangements. [6] can exhibit sudden changes both overall or in one of its subsystems, like when a bloom of harmful algae suddenly forms in the sea [7]. Due to their widespread occurrence, these transitions have gathered a huge interest in the last decade, with research mainly focused on developing statistical methods to forecast them from the observed time series [8][9][10] and on the development of a general mathematical framework to describe them [11]. In the present paper we contribute to both efforts by developing a mathematical analysis by use of a paradigmatic model exhibiting intermittent stochastic evolution and by identifying systemic observables that can deliver early-warning of impending transitions. We focus on the Tangled Nature (TaNa) model [12][13][14] of evolutionary ecology. The initial aim of the model was to establish a sound and simple mathematically framework for "punctuated equilibrium", i.e. the observed intermittent mode of macro-evolution.

Linear stability theory as an early warning sign for transitions in high dimensional complex systems

“…In Fig. (3) of [15] it was discussed how these two quantities behave in the TaNa model and we demonstrated the forecasting power of the indicator Q(t) and we gave an explanation on why we missed some of the transitions. Here we illustrate in Fig.…”

“…In the current paper we elaborate the method we have presented in [15], we discuss the mathematical details both of the mean field approximation and of a Linear Stability Analysis (LSA) and explore in greater detail its forecasting power. As a first test case we consider the Tangled Nature (TaNa) model of evolutionary ecology [16], which has had considerable success in reproducing both macro-evolutionary aspects such as the intermittent mode of extinctions [17] and ecological aspects such as species abundance distributions [18] and species area laws [19].…”

“…As a first test case we consider the Tangled Nature (TaNa) model of evolutionary ecology [16], which has had considerable success in reproducing both macro-evolutionary aspects such as the intermittent mode of extinctions [17] and ecological aspects such as species abundance distributions [18] and species area laws [19]. A much more succinct and schematic discussion was given in [15].…”

“…As said in this paper we start by elaborating on the procedure presented in [15] checking the performance of an alarm threshold built on the stability indicator. This procedure is to be considered a starting point, its weakness consists in the need of full information on the system (one needs to know the structure of the whole configuration space).…”

“…In Section 4 we analyse the results coming from the forecasting procedure first introduced in [15] applied to both models. In Section 5A to study the robustness of the method, we introduce a level of error in the mean field interaction matrix.…”