# 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 and Continuous Dynamical Systems - B, 2013, 18 (1) : 259-271. doi: 10.3934/dcdsb.2013.18.259
##### References:
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##### References:
 [1] H. Behncke, Periodical cicadas, J. Math. Biol., 40 (2000), 413-431. doi: 10.1007/s002850000024. [2] M. G. Bulmer, Periodic insects, Am. Nat., 111 (1977), 1099-1117. doi: 10.1086/283240. [3] J. M. Cushing, Nonlinear semelparous Leslie models, Math. Biosci. Eng., 3 (2006), 17-36. [4] J. M. Cushing, Three stage semelparous Leslie models, J. Math. Biol., 59 (2009), 75-104. doi: 10.1007/s00285-008-0208-9. [5] N. V. Davydova, O. Diekmann and S. A. van Gils, Year class competition or competitive exclusion for strict biennials, J. Math. Biol., 46 (2003), 95-131. doi: 10.1007/s00285-002-0167-5. [6] N. V. Davydova, "Old and Young. Can They Coexist," Thesis, University of Utrecht, 2004, http://igitur-archive.library.uu.nl/dissertations/2004-0115-092805/UUindex.html. [7] N. V. Davydova, O. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra Apl., 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020. [8] O. Diekmann and S. A. van Gils, Invariance and symmetry in a year-class model, in "Bifurcations, Symmetry and Patterns", (Porto, 2000), Birkhäuser, Basel, (2003), 141-150. [9] O. Diekmann and S. A. van Gils, On the cyclic replicator equation and the dynamics of semelparous populations, SIAM J. Applied Dynamical Systems, 8 (2009), 1160-1189. [10] P. van den Drissche and M. L. Zeeman, Three-dimensional competitive Lotka-Volterra systems with no periodic orbits, SIAM J. Appl. Math., 58 (1998), 227-234. doi: 10.1137/S0036139995294767. [11] A. Edalat and E. C. Zeeman, The stable classes of the codimension-one bifurcations of the planar replicator system, Nonlinearity, 5 (1992), 921-939. [12] J. Hofbauer and K. Sigmund, "Evolutionary Games and Population Dynamics," Cambridge University Press, Cambridge, UK, 1998. [13] R. Kon, Nonexistence of synchronous orbits and class coexistence in matrix population models, SIAM J. Appl. Math., 66 (2005), 616-626. doi: 10.1137/05062353X. [14] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations, J. Math. Biol., 55 (2007), 781-802. doi: 10.1007/s00285-007-0111-9. [15] E. Mjolhus, A. Wikan and T. Solberg, On synchronization in semelparous populations, J. Math. Biol., 50 (2005), 1-21. doi: 10.1007/s00285-004-0275-5. [16] J. D. Murry, "Mathematical Biology," Springer-Verlag, New York, 2003. [17] Y. Wang, Necessary and sufficient conditions for the existence of periodic orbits in a Lotka-Volterra system, J. Math. Anal. Appl., 284 (2003), 236-249. doi: 10.1016/S0022-247X(03)00340-8. [18] Y. Wang, H. Wu and S. Ruan, Periodic orbits near heteroclinic cycles in a cyclic replicator system, J. Math. Biol., 64 (2012), 855-872. doi: 10.1007/s00285-011-0435-3.
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