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The discrete dynamics of symmetric competition in the plane

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Abstract

We consider the generalized Lotka-Volterra two-species system

$$\begin{gathered} x_{n + 1} = x_n \exp \left( {r_1 \left( {1 - x_n } \right) - s_1 y_n } \right) \hfill \\ y_{n + 1} = y_n \exp \left( {r_2 \left( {1 - y_n } \right) - s_2 x_n } \right) \hfill \\ \end{gathered} $$

originally proposed by R. M. May as a model for competitive interaction. In the symmetric case that r 1 = r 2 and s 1 = s 2, a region of ultimate confinement is found and the dynamics therein are described in some detail. The bifurcations of periodic points of low period are studied, and a cascade of period-doubling bifurcations is indicated. Within the confinement region, a parameter region is determined for the stable Hopf bifurcation of a pair of symmetrically placed period-two points, which imposes a second component of oscillation near the stable cycles. It is suggested that the symmetric competitive model contains much of the dynamical complexity to be expected in any discrete two-dimensional competitive model.

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Authors and Affiliations

  1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Canada

    Hong Jiang & Thomas D. Rogers

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  1. Hong Jiang
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  2. Thomas D. Rogers
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Research supported in part by an Izaak Walton Killiam Memorial Scholarship

Research supported by a grant from the Natural Research and Engineering Council of Canada

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Jiang, H., Rogers, T.D. The discrete dynamics of symmetric competition in the plane. J. Math. Biology 25, 573–596 (1987). https://doi.org/10.1007/BF00275495

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  • DOI: https://doi.org/10.1007/BF00275495

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