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Adaptive dynamics analysis of a predator–prey model with selective disturbance

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  • Meng, Xin-zhu
  • Zhao, Sheng-nan
  • Zhang, Wen-yan

Abstract

Evolution problem is always a hot topic in the mathematical biology field. In this paper, we investigate the evolutionary effects of selective disturbance on an evolving trait (e.g. body size and maturation age) of the predator individuals in one-predator two-prey community. By using methods of adaptive dynamics and population dynamics we construct an invasion fitness function and obtain the conditions for evolutionary branching and evolutionary stability under selective disturbance in both monomorphic and dimorphic populations. We further conduct a size-selective disturbance function founded on chi-square distribution to study evolutionary stable coexistence, and considering the evolutionary branching and evolutionary stability by using theoretic analysis and numerical simulations. The evolutionary results from a biological point of view show that (1) two strategies could gradually evolve to form a single ancestral strategy, moreover, higher levels of polymorphism cannot build up during evolution, that is, following first evolutionary branching two species will eventually evolve into two generalist species and reach an evolutionary stable coexistence; (2) smaller disturbance could touch off higher levels of dimorphism during evolution, while large disturbance can go against evolutionary branching and advance evolutionary stability.

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  • Meng, Xin-zhu & Zhao, Sheng-nan & Zhang, Wen-yan, 2015. "Adaptive dynamics analysis of a predator–prey model with selective disturbance," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 946-958.
    • Handle: RePEc:eee:apmaco:v:266:y:2015:i:c:p:946-958
    DOI: 10.1016/j.amc.2015.06.020
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    References listed on IDEAS

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    1. Mougi, Akihiko, 2012. "Predator–prey coevolution driven by size selective predation can cause anti-synchronized and cryptic population dynamics," Theoretical Population Biology, Elsevier, vol. 81(2), pages 113-118.
    2. David W. Coltman & Paul O'Donoghue & Jon T. Jorgenson & John T. Hogg & Curtis Strobeck & Marco Festa-Bianchet, 2003. "Undesirable evolutionary consequences of trophy hunting," Nature, Nature, vol. 426(6967), pages 655-658, December.
    3. P. Marrow & U. Dieckmann & R. Law, 1996. "Evolutionary Dynamics of Predator-Prey Systems: An Ecological Perspective," Working Papers wp96002, International Institute for Applied Systems Analysis.
    4. U. Dieckmann & R. Law, 1996. "The Dynamical Theory of Coevolution: A Derivation from Stochastic Ecological Processes," Working Papers wp96001, International Institute for Applied Systems Analysis.
    5. Zu, Jian & Wang, Jinliang, 2013. "Adaptive evolution of attack ability promotes the evolutionary branching of predator species," Theoretical Population Biology, Elsevier, vol. 89(C), pages 12-23.
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    2. Liu, Xia & Zhang, Tonghua & Meng, Xinzhu & Zhang, Tongqian, 2018. "Turing–Hopf bifurcations in a predator–prey model with herd behavior, quadratic mortality and prey-taxis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 446-460.
    3. Li, Hongjie & Zhu, Yinglian & jing, Liu & ying, Wang, 2018. "Consensus of second-order delayed nonlinear multi-agent systems via node-based distributed adaptive completely intermittent protocols," Applied Mathematics and Computation, Elsevier, vol. 326(C), pages 1-15.
    4. Shan, Yaonan & She, Kun & Zhong, Shouming & Zhong, Qishui & Shi, Kaibo & Zhao, Can, 2018. "Exponential stability and extended dissipativity criteria for generalized discrete-time neural networks with additive time-varying delays," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 145-168.

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