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Sliding Shilnikov connection in Filippov-type predator–prey model

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Abstract

Recently, a piecewise smooth differential system was derived as a model of a 1 predator–2 prey interaction where the predator feeds adaptively on its preferred prey and an alternative prey. In such a model, strong evidence of chaotic behavior was numerically found. Here, we revisit this model and prove the existence of a Shilnikov sliding connection when the parameters are taken in a codimension one submanifold of the parameter space. As a consequence of this connection, we conclude, analytically, that the model behaves chaotically for an open region of the parameter space.

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Acknowledgements

The authors are very grateful to Professor Sérgio Furtado dos Reis for meaningful discussion and constructive criticism on the manuscript. The authors also thank the referees for their comments and suggestions which helped us to improve the presentation of this paper. Finally, the authors thank Espaço da Escrita—Pró-Reitoria de Pesquisa—UNICAMP for the language services provided. Tiago Carvalho is supported by São Paulo Research Foundation (FAPESP Grants 2017/00883-0 and 2019/10450-0) and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Grant 304809/2017-9). Douglas Novaes is supported by São Paulo Research Foundation (FAPESP Grants 2018/16430-8, 2018/13481-0, and 2019/10269-3) and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq Grants 306649/2018-7 and 438975/2018-9). Luiz F. Gonçalves is supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES), Finance Code 001.

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  1. Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Av. Bandeirantes, 3900, Ribeirão Preto, SP, CEP 14040-901, Brazil

    Tiago Carvalho

  2. Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda 651, Cidade Universitária Zeferino Vaz, Campinas, SP, CEP 13083-859, Brazil

    Douglas Duarte Novaes

  3. Instituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista (UNESP), Rua Cristóvão Colombo, 2265, São José do Rio Preto, SP, CEP 15054-000, Brazil

    Luiz Fernando Gonçalves

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  1. Tiago Carvalho
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Carvalho, T., Duarte Novaes, D. & Gonçalves, L.F. Sliding Shilnikov connection in Filippov-type predator–prey model. Nonlinear Dyn 100, 2973–2987 (2020). https://doi.org/10.1007/s11071-020-05672-w

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