Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations
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arXiv
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| Format: | Preprint |
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2020
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| author | Mambuca, Andrea Marcello Cammarota, Chiara Neri, Izaak |
| author_facet | Mambuca, Andrea Marcello Cammarota, Chiara Neri, Izaak |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2009_11211 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations Mambuca, Andrea Marcello Cammarota, Chiara Neri, Izaak Statistical Mechanics Disordered Systems and Neural Networks Populations and Evolution 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. |
| title | Dynamical systems on large networks with predator-prey interactions are stable and exhibit oscillations |
| topic | Statistical Mechanics Disordered Systems and Neural Networks Populations and Evolution |
| url | https://arxiv.org/abs/2009.11211 |