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

Abstract

We analyse the stability of linear dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling 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 have a strong influence on 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 only contain interactions of the predator-prey type can be stable in the infinite size limit. This qualitatively feature for antagonistic systems is accompanied by a peculiar oscillatory behaviour 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 non-oscillatory.

38 pages, 18 figures, last version fixes a typo in Eqs.(G11) and (G12)

References in corpus (34)

• Stability Criteria for Complex Ecosystems
• Dynamics of market correlations: Taxonomy and portfolio analysis
• Giant strongly connected component of directed networks
• Pathways towards instability in financial networks
• Marginally Stable Equilibria in Critical Ecosystems
• Transition to chaos in random neuronal networks
• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• Numerical implementation of dynamical mean field theory for disordered systems: application to the Lotka-Volterra model of ecosystems
• Non-Hermitian Localization in Biological Networks
• Loops of any size and Hamilton cycles in random scale-free networks
• The effect of population abundances on the stability of large random ecosystems
• Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• On the localization transition in symmetric random matrices
• May's Instability in Large Economies
• Spectral Theory of Sparse Non-Hermitian Random Matrices
• Heterogeneous connections induce oscillations in large scale networks
• Spectra of sparse non-Hermitian random matrices: an analytical solution
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Localization of eigenvectors in random graphs
• Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices
• Lifshitz tails for spectra of Erdős--Rényi random graphs
• Linear stability analysis for large dynamical systems on directed random graphs
• Universal transient behavior in large dynamical systems on networks
• Localization and universality of eigenvectors in directed random graphs
• Distribution of shortest cycle lengths in random networks
• Equivalence of replica and cavity methods for computing spectra of sparse random matrices
• Lifshitz tails on the Bethe lattice: a combinatorial approach
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Instability to a heterogeneous oscillatory state in randomly connected recurrent networks with delayed interactions
• On one generalization of the elliptic law for random matrices
• Random Antagonistic Matrices

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• A path integral approach to sparse non-Hermitian random matrices
• Route to Chaos in Multi-Species Ecosystems
• Uncovering the non-equilibrium stationary properties in sparse Boolean networks
• Feasibility and Single Parameter Scaling of Extinctions in Large Ecological Communities