# American Institute of Mathematical Sciences

January  2013, 18(1): 259-271. doi: 10.3934/dcdsb.2013.18.259

## Global dynamics and bifurcations in a four-dimensional replicator system

 1 School of Mathematics and Computational Science, Sun Yat-sen University, Guangzhou 510275, China 2 Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250

Received  October 2011 Revised  April 2012 Published  September 2012

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.
Citation: Yuanshi Wang, Hong Wu, Shigui Ruan. Global dynamics and bifurcations in a four-dimensional replicator system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 259-271. doi: 10.3934/dcdsb.2013.18.259
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