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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 |
References:
[1] |
H. Behncke, Periodical cicadas, J. Math. Biol., 40 (2000), 413-431.
doi: 10.1007/s002850000024. |
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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. |
show all references
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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