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How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

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Abstract

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.

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Authors and Affiliations

  1. Department of Mathematics, ISEL - Engineering Superior Institute of Lisbon, Rua Conselheiro Emídio Navarro 1, 1949-014, Lisbon, Portugal

    Jorge Duarte & Cristina Januário

  2. Mathematics Department, Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Jorge Duarte & Nuno Martins

  3. Department of Mathematics, ESTS - Technology Superior School of Setubal, Campus do IPS, Rua Vale de Chaves, Estefanilha, 2914-761, Setubal, Portugal

    Carla Rodrigues

  4. ICREA-Complex Systems Lab, Universitat Pompeu Fabra (UPF), Parc de Recerca Biomèdica de Barcelona (PRBB), Dr. Aiguader, 88, 08003, Barcelona, Spain

    Josep Sardanyés

  5. Institut de Biologia Evolutiva (UPF-CSIC-PRBB), Pg. Maritim de la Barceloneta 37, 08003, Barcelona, Spain

    Josep Sardanyés

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  1. Jorge Duarte
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  3. Cristina Januário
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Correspondence to Jorge Duarte or Josep Sardanyés.

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Duarte, J., Rodrigues, C., Januário, C. et al. How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?. Acta Biotheor 63, 341–361 (2015). https://doi.org/10.1007/s10441-015-9254-z

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  • DOI: https://doi.org/10.1007/s10441-015-9254-z

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