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Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles

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Abstract

We investigate the appearance of chaos in a microbial 3-species model motivated by a potentially chaotic real world system (as characterized by positive Lyapunov exponents (Becks et al., Nature 435, 2005). This is the first quantitative model that simulates characteristic population dynamics in the system. A striking feature of the experiment was three consecutive regimes of limit cycles, chaotic dynamics and a fixed point. Our model reproduces this pattern. Numerical simulations of the system reveal the presence of a chaotic attractor in the intermediate parameter window between two regimes of periodic coexistence (stable limit cycles). In particular, this intermediate structure can be explained by competition between the two distinct periodic dynamics. It provides the basis for stable coexistence of all three species: environmental perturbations may result in huge fluctuations in species abundances, however, the system at large tolerates those perturbations in the sense that the population abundances quickly fall back onto the chaotic attractor manifold and the system remains. This mechanism explains how chaos helps the system to persist and stabilize against migration. In discrete populations, fluctuations can push the system towards extinction of one or more species. The chaotic attractor protects the system and extinction times scale exponentially with system size in the same way as with limit cycles or in a stable situation.

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Acknowledgments

We thank Lutz Becks for useful discussion and helpful comments. Funding was provided by SFB TR12 of the Deutsche Forschungsgemeinschaft (AA, FG) and by SPP 1162 (AR 288/14) and SPP 1374 (AR 288/16) of the Deutsche Forschungsgemeinschaft (HA).

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Authors and Affiliations

  1. Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77, D-50937, Köln, Germany

    Fanny Groll & Alexander Altland

  2. General Ecology, Institute for Zoology, Biocenter, University of Cologne, Zülpicher Str. 47b, D-50674, Köln, Germany

    Hartmut Arndt

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  1. Fanny Groll
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Correspondence to Fanny Groll.

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Groll, F., Arndt, H. & Altland, A. Chaotic attractor in two-prey one-predator system originates from interplay of limit cycles. Theor Ecol 10, 147–154 (2017). https://doi.org/10.1007/s12080-016-0317-9

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  • DOI: https://doi.org/10.1007/s12080-016-0317-9

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