Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations

Abstract

We analyze the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modeling the stability of fixed points in large systems defined on complex networks, such as ecosystems consisting of a large number of species that interact through a food web. We develop an exact theory for the spectral distribution and the leading eigenvalue of the corresponding sparse Jacobian matrices. This theory reveals that the nature of local interactions has a strong influence on a system's stability. We show that, in general, linear dynamical systems defined on random graphs with a prescribed degree distribution of unbounded support are unstable if they are large enough, implying a tradeoff between stability and diversity. Remarkably, in contrast to the generic case, antagonistic systems that contain only interactions of the predator-prey type can be stable in the infinite size limit. This feature for antagonistic systems is accompanied by a peculiar oscillatory behavior of the dynamical response of the system after a perturbation, when the mean degree of the graph is small enough. Moreover, for antagonistic systems we also find that there exist a dynamical phase transition and critical mean degree above which the response becomes nonoscillatory.

Local sign stability and its implications for spectra of sparse random graphs and stability of ecosystems

“…Since in this paper we consider large complex systems, following [25], we introduce here two variants of linear stability in the limit of large N. Consider a sequence of matrices M N growing in size N ∈ N. In this case, we can distinguish two classes of matrix sequences, viz., those for which the real part of the leading eigenvalue converges to a finite value, i.e.…”

“…Although this fueled significant interest [11], it is only recently that the influence of sparse network structure on dynamical stability has been studied. Indeed, following pioneering work on the spectra of symmetric Erdős-Rényi graphs [12][13][14][15], recent papers studied the spectra of random, directed graphs [16][17][18][19][20][21][22][23][24] and the spectra of random graphs with predator-prey, mutualistic, or competitive interactions [25].…”

“…whether sign(A ij A ji ) = 0 (unidirectional interactions), sign(A ij A ji ) = −1 (sign-antisymmetric interactions), or sign(A ij A ji ) = 1 (sign-symmetric interactions). Notably, the spectra of (infinitely large) sparse random graphs are confined to a region in the complex plane with bounded real part if sign(A ij A ji ) ∈ {0, −1} for all pairs i, j of nodes [21,25], whereas the spectra of (infinitely large) sparse random graphs encompass the full real axis if there exists a finite proportion of links with sign(A ij A ji ) = 1 [25].…”

Uncovering the non-equilibrium stationary properties in sparse Boolean networks

“…However, it is not yet clear what happens in the case where a combination of many of these patterns is present in a network. The presence of bi-directional links with opposite signs of the interactions (such as in predator-prey systems) have been shown to produce oscillatory behaviour in the context of continuous linear dynamics [74], but little is known in the context of discrete state dynamics, e.g., for the linear threshold model. The dynamics investigated in Sect.…”

Local and collective transitions in sparsely-interacting ecological communities

“…In one, the dynamics is linearized around a fixed point, and the parameters describing the dynamics of coexisting species are sampled at random. This approach predicts stability bounds [11,14], and has also been applied to sparse interactions [20].…”