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Dynamics of a Population with Two Equal Dominated Species

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Abstract

We consider a population with two equal dominated species, dynamics of which is defined by an one-dimensional piecewise-continuous, two parametric function. It is shown that for any non-zero parameters this function has two fixed points and several periodic points. We prove that all periodic (in particular fixed) points are repelling, and find an invariant set which asymptotically involves the trajectories of any initial point except fixed and periodic ones. We showed that the orbits are unstable and chaotic because Lyapunov exponent is non-negative. The limit sets analyzed by bifurcation diagrams. We give biological interpretations of our results.

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Notes

  1. https://en.wikipedia.org/wiki/Lyapunov_-exponent.

  2. https://en.wikipedia.org/wiki/Bifurcation_-theory.

  3. https://en.wikipedia.org/wiki/Bifurcation_-diagram.

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Acknowledgements

We thank the referee for careful reading of the manuscript and for useful suggestions.

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  1. Institute of Mathematics, 81, Mirzo Ulug’bek Str., 100170, Tashkent, Uzbekistan

    U. A. Rozikov & J. B. Usmonov

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  1. U. A. Rozikov
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  2. J. B. Usmonov
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Correspondence to U. A. Rozikov.

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Rozikov, U.A., Usmonov, J.B. Dynamics of a Population with Two Equal Dominated Species. Qual. Theory Dyn. Syst. 19, 62 (2020). https://doi.org/10.1007/s12346-020-00399-w

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