# American Institute of Mathematical Sciences

March  2016, 21(2): 655-676. doi: 10.3934/dcdsb.2016.21.655

## Evolutionary dynamics of a multi-trait semelparous model

 1 Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, Arizona 85721, United States 2 Interdisciplinary Program in Applied Mathematics and Department of Mathematics, University of Arizona, 617 N Santa Rita, Tucson, Arizona 85721, United States

Received  December 2014 Revised  April 2015 Published  November 2015

We consider a multi-trait evolutionary (game theoretic) version of a two class (juvenile-adult) semelparous Leslie model. We prove the existence of both a continuum of positive equilibria and a continuum of synchronous 2-cycles as the result of a bifurcation that occurs from the extinction equilibrium when the net reproductive number $R_{0}$ increases through $1$. We give criteria for the direction of bifurcation and for the stability or instability of each bifurcating branch. Semelparous Leslie models have imprimitive projection matrices. As a result (unlike matrix models with primitive projection matrices) the direction of bifurcation does not solely determine the stability of a bifurcating continuum. Only forward bifurcating branches can be stable and which of the two is stable depends on the intensity of between-class competitive interactions. These results generalize earlier results for single trait models. We give an example that illustrates how the dynamic alternative can change when the number of evolving traits changes from one to two.
Citation: Amy Veprauskas, J. M. Cushing. Evolutionary dynamics of a multi-trait semelparous model. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 655-676. doi: 10.3934/dcdsb.2016.21.655
##### References:
 [1] P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods. Ecology Letters, 4 (2001), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x. [2] P. A. Abrams, 'Adaptive Dynamics' vs. 'adaptive dynamics', Journal of Evolutionary Biology, 18 (2005), 1162-1165. [3] L. J. S. Allen, A density-dependent Leslie matrix model, Mathematical Biosciences, 95 (1989), 179-187. doi: 10.1016/0025-5564(89)90031-X. [4] H. Behncke, Periodical cicadas, Journal of Mathematical Biology, 40 (2000), 413-431. doi: 10.1007/s002850000024. [5] M. G. Bulmer, Periodical insects, The American Naturalist, 111 (1977), 1099-1117. doi: 10.1086/283240. [6] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001. [7] J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Natural Resource Modelling, 8 (1994), 297-333. [8] J. M. Cushing, An Introduction to Structured Population Dynamics, Conference Series in Applied Mathematics Vol. 71, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005. [9] J. M. Cushing, Cycle chains and the LPA model, Journal of Difference Equations and Applications, 9 (2003), 655-670. doi: 10.1080/1023619021000042216. [10] J. M. Cushing, Nonlinear semelparous Leslie models, Mathematical Biosciences and Engineering, 3 (2006), 17-36. doi: 10.3934/mbe.2006.3.17. [11] J. M. Cushing, Three stage semelparous Leslie models, Journal of Mathematical Biology, 59 (2009), 75-104. doi: 10.1007/s00285-008-0208-9. [12] J. M. Cushing, Matrix Models and Population Dynamics, a chapter in Mathematical Biology (eds. Mark Lewis, A.J. Chaplain, James P. Keener, and Philip K. Maini), IAS/Park City Mathematics Series Vol 14, American Mathematical Society, Providence, RI, 2009, 47-150. [13] J. M. Cushing, A bifurcation theorem for Darwinian matrix models, Nonlinear Studies, 17 (2010), 1-13. [14] J. M. Cushing, A dynamic dichotomy for a system of hierarchical difference equations, Journal of Difference Equations and Applications, 18 (2012), 1-26. doi: 10.1080/10236198.2011.628319. [15] J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, Journal of Biological Dynamics, 8 (2014), 57-73. doi: 10.1080/17513758.2014.899638. [16] J. M. Cushing and S. M. Henson, Stable bifurcations in semelparous Leslie models, Journal of Biological Dynamics, 6 (2012), 80-102. doi: 10.1080/17513758.2012.716085. [17] J. M. Cushing and J. Li, On Ebenman's model for the dynamics of a population with competing juveniles and adults, Bulletin of Mathematical Biology, 51 (1989), 687-713. [18] J. M. Cushing and S. M. Stump, Darwinian dynamics of a juvenile-adult model, Mathematical Biosciences and Engineering, 10 (2013), 1017-1044. doi: 10.3934/mbe.2013.10.1017. [19] N. V. Davydova, O. Diekmann and S. A. van Gils, Year class coexistence or competitive exclusion for strict biennials?, Journal of Mathematical Biology, 46 (2003), 95-131. doi: 10.1007/s00285-002-0167-5. [20] N. V. Davydova, O. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra and its Applications, 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020. [21] O. Diekmann and S. A. van Gils, Invariance and symmetry in a year-class model, in Trends in Mathematics: Bifurcations, Symmetry and Patterns (eds/ J. Buescu, S. D. Castro, A. P. da Silva Dias and I. S. Labouriau), Birkhäuser Verlag, Basel, Switzerland, 141-150, 2003. [22] O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, Journal of Difference Equations and Applications, 11 (2005), 327-335. doi: 10.1080/10236190412331335409. [23] O. Diekmann and S. A. van Gils, On the cyclic replicator equations and the dynamics of semelparous populations, SIAM Journal of Applied Dynamical Systems, 8 (2009), 1160-1189. doi: 10.1137/080722734. [24] J. A. Endler, Multiple-trait coevolution and environmental gradients in guppies, Trends in Ecology and Evolution, 10 (1995), 22-29. doi: 10.1016/S0169-5347(00)88956-9. [25] C. K. Ghalambor, J. A. Walker and D. N. Reznick, Multi-trait selection, adaptation, and constraints on the evolution of burst swimming performance, Integrative and Comparative Biology, 43 (2003), 431-438. doi: 10.1093/icb/43.3.431. [26] C. K. Ghalambor, D. N. Reznick and J. A. Walker, Constraints on adaptive evolution: The functional trade-off between reproduction and fast-start swimming performance in the Trinidadian guppy (Poecilia reticulata), American Naturalist, 164 (2004), 38-50. doi: 10.1086/421412. [27] P. R. Grant and B. R. Grant, Predicting microevolutionary responses to directional selection on heritable variation, Evolution, 49 (1995), 241-251. doi: 10.2307/2410334. [28] A. P. Hendry, J. K. Wenburg, P. Bentzen, E. C. Volk and T. P. Quinn, Rapid evolution of reproductive isolation in the wild: evidence from introduced salmon, Science, 290 (2000), 516-518. doi: 10.1126/science.290.5491.516. [29] R. Kon, Nonexistence of synchronous orbits and class coexistence in matrix population models, SIAM Journal of Applied Mathematics, 66 (2005), 616-626. doi: 10.1137/05062353X. [30] R. Kon, Competitive exclusion between year-classes in a semelparous biennial population, in Mathematical Modeling of Biological Systems, Volume II. (eds. A. Deutsch, R. Bravo de la Parra, R. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz). Birkhäuser, Boston, 2008, 79-90. [31] R. Kon, Permanence induced by life-cycle resonances: The periodical cicada problem, Journal of Biological Dynamics, 6 (2012), 855-890. doi: 10.1080/17513758.2011.594098. [32] R. Kon, Y. Saito and Y. Takeuchi, Permanence of single-species stage-structured models, Journal of Mathematical Biology, 48 (2004), 515-528. doi: 10.1007/s00285-003-0239-1. [33] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations, Journal of Mathematical Biology, 55 (2007), 781-802. doi: 10.1007/s00285-007-0111-9. [34] R. Lande, Natural selection and random genetic drift in phenotypic evolution, Evolution, 30 (1976), 314-334. [35] R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615. [36] B. J. McGill and J. S. Brown, Evolutionary Game Theory and Adpative Dynamics of Continuous Traits, Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 403-435. [37] E. Mjølhus, A. Wikan and T. Solberg, On synchronization in semelparous populations, Journal of Mathematical Biology, 50 (2005), 1-21. doi: 10.1007/s00285-004-0275-5. [38] S. O'Steen, A. J. Cullum and A. F. Bennett, Rapid evolution of escape ability in Trinidadian guppies(Poecilia reticulata), Evolution, 56 (2002), 776-784. [39] T. P. Quinn, M. T. Kinnison and M. J. Unwin, Evolution of chinook salmon (Oncorhynchus tshawytscha) populations in New Zealand: Pattern, rate, and process, Genetica, 8 (2001), 493-513. doi: 10.1007/978-94-010-0585-2_30. [40] D. A. Roff, The Evolution of Life Histories: Theory and Analysis, Chapman and Hall, New York, 1992. [41] C. A. Stockwell and S. C. Weeks, Translocations and rapid evolutionary responses in recently established populations of western mosquitofish, (Gambusia affinis) Animal Conservation, 2 (1999), 103-110. doi: 10.1111/j.1469-1795.1999.tb00055.x. [42] C. A. Stockwell, A. P. Hendry and M. T. Kinnison, Contemporary evolution meets conservation biology, Trends in Ecology and Evolution, 18 (2003), 94-101. doi: 10.1016/S0169-5347(02)00044-7. [43] S. C. Stearns, The Evolution of Life Histories, Oxford University Press, Oxford, UK, 1992. [44] G. Strang, Linear Algebra and Its Applications, 3rd edition, Harcourt Brace Jovanovich, Forth Worth, 1988. [45] J. N. Thompson, Rapid evolution as an ecological process, Trends in Ecology and Evolution, 13 (1998), 329-332. doi: 10.1016/S0169-5347(98)01378-0. [46] W. O. Tschumy, Competition between juveniles and adults in age-structured populations, Theoretical Population Biology, 21 (1982), 255-268. doi: 10.1016/0040-5809(82)90017-X. [47] T. L Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542633.

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##### References:
 [1] P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods. Ecology Letters, 4 (2001), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x. [2] P. A. Abrams, 'Adaptive Dynamics' vs. 'adaptive dynamics', Journal of Evolutionary Biology, 18 (2005), 1162-1165. [3] L. J. S. Allen, A density-dependent Leslie matrix model, Mathematical Biosciences, 95 (1989), 179-187. doi: 10.1016/0025-5564(89)90031-X. [4] H. Behncke, Periodical cicadas, Journal of Mathematical Biology, 40 (2000), 413-431. doi: 10.1007/s002850000024. [5] M. G. Bulmer, Periodical insects, The American Naturalist, 111 (1977), 1099-1117. doi: 10.1086/283240. [6] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001. [7] J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Natural Resource Modelling, 8 (1994), 297-333. [8] J. M. Cushing, An Introduction to Structured Population Dynamics, Conference Series in Applied Mathematics Vol. 71, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005. [9] J. M. Cushing, Cycle chains and the LPA model, Journal of Difference Equations and Applications, 9 (2003), 655-670. doi: 10.1080/1023619021000042216. [10] J. M. Cushing, Nonlinear semelparous Leslie models, Mathematical Biosciences and Engineering, 3 (2006), 17-36. doi: 10.3934/mbe.2006.3.17. [11] J. M. Cushing, Three stage semelparous Leslie models, Journal of Mathematical Biology, 59 (2009), 75-104. doi: 10.1007/s00285-008-0208-9. [12] J. M. Cushing, Matrix Models and Population Dynamics, a chapter in Mathematical Biology (eds. Mark Lewis, A.J. Chaplain, James P. Keener, and Philip K. Maini), IAS/Park City Mathematics Series Vol 14, American Mathematical Society, Providence, RI, 2009, 47-150. [13] J. M. Cushing, A bifurcation theorem for Darwinian matrix models, Nonlinear Studies, 17 (2010), 1-13. [14] J. M. Cushing, A dynamic dichotomy for a system of hierarchical difference equations, Journal of Difference Equations and Applications, 18 (2012), 1-26. doi: 10.1080/10236198.2011.628319. [15] J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, Journal of Biological Dynamics, 8 (2014), 57-73. doi: 10.1080/17513758.2014.899638. [16] J. M. Cushing and S. M. Henson, Stable bifurcations in semelparous Leslie models, Journal of Biological Dynamics, 6 (2012), 80-102. doi: 10.1080/17513758.2012.716085. [17] J. M. Cushing and J. Li, On Ebenman's model for the dynamics of a population with competing juveniles and adults, Bulletin of Mathematical Biology, 51 (1989), 687-713. [18] J. M. Cushing and S. M. Stump, Darwinian dynamics of a juvenile-adult model, Mathematical Biosciences and Engineering, 10 (2013), 1017-1044. doi: 10.3934/mbe.2013.10.1017. [19] N. V. Davydova, O. Diekmann and S. A. van Gils, Year class coexistence or competitive exclusion for strict biennials?, Journal of Mathematical Biology, 46 (2003), 95-131. doi: 10.1007/s00285-002-0167-5. [20] N. V. Davydova, O. Diekmann and S. A. van Gils, On circulant populations. I. The algebra of semelparity, Linear Algebra and its Applications, 398 (2005), 185-243. doi: 10.1016/j.laa.2004.12.020. [21] O. Diekmann and S. A. van Gils, Invariance and symmetry in a year-class model, in Trends in Mathematics: Bifurcations, Symmetry and Patterns (eds/ J. Buescu, S. D. Castro, A. P. da Silva Dias and I. S. Labouriau), Birkhäuser Verlag, Basel, Switzerland, 141-150, 2003. [22] O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, Journal of Difference Equations and Applications, 11 (2005), 327-335. doi: 10.1080/10236190412331335409. [23] O. Diekmann and S. A. van Gils, On the cyclic replicator equations and the dynamics of semelparous populations, SIAM Journal of Applied Dynamical Systems, 8 (2009), 1160-1189. doi: 10.1137/080722734. [24] J. A. Endler, Multiple-trait coevolution and environmental gradients in guppies, Trends in Ecology and Evolution, 10 (1995), 22-29. doi: 10.1016/S0169-5347(00)88956-9. [25] C. K. Ghalambor, J. A. Walker and D. N. Reznick, Multi-trait selection, adaptation, and constraints on the evolution of burst swimming performance, Integrative and Comparative Biology, 43 (2003), 431-438. doi: 10.1093/icb/43.3.431. [26] C. K. Ghalambor, D. N. Reznick and J. A. Walker, Constraints on adaptive evolution: The functional trade-off between reproduction and fast-start swimming performance in the Trinidadian guppy (Poecilia reticulata), American Naturalist, 164 (2004), 38-50. doi: 10.1086/421412. [27] P. R. Grant and B. R. Grant, Predicting microevolutionary responses to directional selection on heritable variation, Evolution, 49 (1995), 241-251. doi: 10.2307/2410334. [28] A. P. Hendry, J. K. Wenburg, P. Bentzen, E. C. Volk and T. P. Quinn, Rapid evolution of reproductive isolation in the wild: evidence from introduced salmon, Science, 290 (2000), 516-518. doi: 10.1126/science.290.5491.516. [29] R. Kon, Nonexistence of synchronous orbits and class coexistence in matrix population models, SIAM Journal of Applied Mathematics, 66 (2005), 616-626. doi: 10.1137/05062353X. [30] R. Kon, Competitive exclusion between year-classes in a semelparous biennial population, in Mathematical Modeling of Biological Systems, Volume II. (eds. A. Deutsch, R. Bravo de la Parra, R. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky and H. Metz). Birkhäuser, Boston, 2008, 79-90. [31] R. Kon, Permanence induced by life-cycle resonances: The periodical cicada problem, Journal of Biological Dynamics, 6 (2012), 855-890. doi: 10.1080/17513758.2011.594098. [32] R. Kon, Y. Saito and Y. Takeuchi, Permanence of single-species stage-structured models, Journal of Mathematical Biology, 48 (2004), 515-528. doi: 10.1007/s00285-003-0239-1. [33] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations, Journal of Mathematical Biology, 55 (2007), 781-802. doi: 10.1007/s00285-007-0111-9. [34] R. Lande, Natural selection and random genetic drift in phenotypic evolution, Evolution, 30 (1976), 314-334. [35] R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615. [36] B. J. McGill and J. S. Brown, Evolutionary Game Theory and Adpative Dynamics of Continuous Traits, Annual Review of Ecology, Evolution, and Systematics, 38 (2007), 403-435. [37] E. Mjølhus, A. Wikan and T. Solberg, On synchronization in semelparous populations, Journal of Mathematical Biology, 50 (2005), 1-21. doi: 10.1007/s00285-004-0275-5. [38] S. O'Steen, A. J. Cullum and A. F. Bennett, Rapid evolution of escape ability in Trinidadian guppies(Poecilia reticulata), Evolution, 56 (2002), 776-784. [39] T. P. Quinn, M. T. Kinnison and M. J. Unwin, Evolution of chinook salmon (Oncorhynchus tshawytscha) populations in New Zealand: Pattern, rate, and process, Genetica, 8 (2001), 493-513. doi: 10.1007/978-94-010-0585-2_30. [40] D. A. Roff, The Evolution of Life Histories: Theory and Analysis, Chapman and Hall, New York, 1992. [41] C. A. Stockwell and S. C. Weeks, Translocations and rapid evolutionary responses in recently established populations of western mosquitofish, (Gambusia affinis) Animal Conservation, 2 (1999), 103-110. doi: 10.1111/j.1469-1795.1999.tb00055.x. [42] C. A. Stockwell, A. P. Hendry and M. T. Kinnison, Contemporary evolution meets conservation biology, Trends in Ecology and Evolution, 18 (2003), 94-101. doi: 10.1016/S0169-5347(02)00044-7. [43] S. C. Stearns, The Evolution of Life Histories, Oxford University Press, Oxford, UK, 1992. [44] G. Strang, Linear Algebra and Its Applications, 3rd edition, Harcourt Brace Jovanovich, Forth Worth, 1988. [45] J. N. Thompson, Rapid evolution as an ecological process, Trends in Ecology and Evolution, 13 (1998), 329-332. doi: 10.1016/S0169-5347(98)01378-0. [46] W. O. Tschumy, Competition between juveniles and adults in age-structured populations, Theoretical Population Biology, 21 (1982), 255-268. doi: 10.1016/0040-5809(82)90017-X. [47] T. L Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542633.
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