We analyse the linear stability of fixed points in large dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling, among others, 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. In particular, we show that fixed points of dynamical systems defined on random graphs are always unstable if they are large enough, except if all interactions are of the predator-prey type. This qualitatively new 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 we find that there exist a dynamical phase transition and critical mean degree above which the response becomes non-oscillatory also for antagonistic systems .
We analyse the linear stability of fixed points in large dynamical systems defined on sparse, random graphs with predator-prey, competitive, and mutualistic interactions. These systems are aimed at modelling, among others, 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. In particular, we show that fixed points of dynamical systems defined on random graphs are always unstable if they are large enough, except if all interactions are of the predator-prey type. This qualitatively new 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 .
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, for antagonistic systems we also
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also for antagonistic systems