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Global dynamics and bifurcations in a four-dimensional replicator system

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  • In this paper, the four-dimensional cyclic replicator system $\dot{u}_i = {u}_i [-(Bu)_i + \sum_{j=1}^{4} u_j (Bu)_j ],1\le i \le 4$, with $b_1 = b_3$ is considered, in which the first row of the matrix $B$ is $(0~ b_1~ b_2~ b_3)$ and the other rows of $B$ are cyclic permutations of the first row. Our aim is to study the global dynamics and bifurcations in the system, and to show how and when all but one species go to extinction. By reducing the four-dimensional system to a three-dimensional one, we show that there is no periodic orbit in the system. For the case $b_1 b_2 < 0$, we give complete analysis on the global dynamics. For the case $b_1 b_2 \ge 0$, we extend some results obtained by Diekmann and van Gils (2009). By combining our work with that in Diekmann and van Gils (2009), we present the dynamics and bifurcations of the system on the whole $(b_1, b_2)$-plane. The analysis leads to explanations for the phenomena that in some semelparous species, all but one brood go extinct.
    Mathematics Subject Classification: 34C23, 92D25, 37N25.


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