# 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 & 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. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar [2] P. A. Abrams, 'Adaptive Dynamics' vs. 'adaptive dynamics',, Journal of Evolutionary Biology, 18 (2005), 1162. Google Scholar [3] L. J. S. Allen, A density-dependent Leslie matrix model,, Mathematical Biosciences, 95 (1989), 179. doi: 10.1016/0025-5564(89)90031-X. Google Scholar [4] H. Behncke, Periodical cicadas,, Journal of Mathematical Biology, 40 (2000), 413. doi: 10.1007/s002850000024. Google Scholar [5] M. G. Bulmer, Periodical insects,, The American Naturalist, 111 (1977), 1099. doi: 10.1086/283240. Google Scholar [6] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar [7] J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models,, Natural Resource Modelling, 8 (1994), 297. Google Scholar [8] J. M. Cushing, An Introduction to Structured Population Dynamics,, Conference Series in Applied Mathematics Vol. 71, (1998). doi: 10.1137/1.9781611970005. Google Scholar [9] J. M. Cushing, Cycle chains and the LPA model,, Journal of Difference Equations and Applications, 9 (2003), 655. doi: 10.1080/1023619021000042216. Google Scholar [10] J. M. Cushing, Nonlinear semelparous Leslie models,, Mathematical Biosciences and Engineering, 3 (2006), 17. doi: 10.3934/mbe.2006.3.17. Google Scholar [11] J. M. Cushing, Three stage semelparous Leslie models,, Journal of Mathematical Biology, 59 (2009), 75. doi: 10.1007/s00285-008-0208-9. Google Scholar [12] J. M. Cushing, Matrix Models and Population Dynamics,, a chapter in Mathematical Biology (eds. Mark Lewis, (2009), 47. Google Scholar [13] J. M. Cushing, A bifurcation theorem for Darwinian matrix models,, Nonlinear Studies, 17 (2010), 1. Google Scholar [14] J. M. Cushing, A dynamic dichotomy for a system of hierarchical difference equations,, Journal of Difference Equations and Applications, 18 (2012), 1. doi: 10.1080/10236198.2011.628319. Google Scholar [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. doi: 10.1080/17513758.2014.899638. Google Scholar [16] J. M. Cushing and S. M. Henson, Stable bifurcations in semelparous Leslie models,, Journal of Biological Dynamics, 6 (2012), 80. doi: 10.1080/17513758.2012.716085. Google Scholar [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. Google Scholar [18] J. M. Cushing and S. M. Stump, Darwinian dynamics of a juvenile-adult model,, Mathematical Biosciences and Engineering, 10 (2013), 1017. doi: 10.3934/mbe.2013.10.1017. Google Scholar [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. doi: 10.1007/s00285-002-0167-5. Google Scholar [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. doi: 10.1016/j.laa.2004.12.020. Google Scholar [21] O. Diekmann and S. A. van Gils, Invariance and symmetry in a year-class model,, in Trends in Mathematics: Bifurcations, (2003), 141. Google Scholar [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. doi: 10.1080/10236190412331335409. Google Scholar [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. doi: 10.1137/080722734. Google Scholar [24] J. A. Endler, Multiple-trait coevolution and environmental gradients in guppies,, Trends in Ecology and Evolution, 10 (1995), 22. doi: 10.1016/S0169-5347(00)88956-9. Google Scholar [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. doi: 10.1093/icb/43.3.431. Google Scholar [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. doi: 10.1086/421412. Google Scholar [27] P. R. Grant and B. R. Grant, Predicting microevolutionary responses to directional selection on heritable variation,, Evolution, 49 (1995), 241. doi: 10.2307/2410334. Google Scholar [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. doi: 10.1126/science.290.5491.516. Google Scholar [29] R. Kon, Nonexistence of synchronous orbits and class coexistence in matrix population models,, SIAM Journal of Applied Mathematics, 66 (2005), 616. doi: 10.1137/05062353X. Google Scholar [30] R. Kon, Competitive exclusion between year-classes in a semelparous biennial population,, in Mathematical Modeling of Biological Systems, (2008), 79. Google Scholar [31] R. Kon, Permanence induced by life-cycle resonances: The periodical cicada problem,, Journal of Biological Dynamics, 6 (2012), 855. doi: 10.1080/17513758.2011.594098. Google Scholar [32] R. Kon, Y. Saito and Y. Takeuchi, Permanence of single-species stage-structured models,, Journal of Mathematical Biology, 48 (2004), 515. doi: 10.1007/s00285-003-0239-1. Google Scholar [33] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations,, Journal of Mathematical Biology, 55 (2007), 781. doi: 10.1007/s00285-007-0111-9. Google Scholar [34] R. Lande, Natural selection and random genetic drift in phenotypic evolution,, Evolution, 30 (1976), 314. Google Scholar [35] R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607. Google Scholar [36] B. J. McGill and J. S. Brown, Evolutionary Game Theory and Adpative Dynamics of Continuous Traits,, Annual Review of Ecology, 38 (2007), 403. Google Scholar [37] E. Mjølhus, A. Wikan and T. Solberg, On synchronization in semelparous populations,, Journal of Mathematical Biology, 50 (2005), 1. doi: 10.1007/s00285-004-0275-5. Google Scholar [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. Google Scholar [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. doi: 10.1007/978-94-010-0585-2_30. Google Scholar [40] D. A. Roff, The Evolution of Life Histories: Theory and Analysis,, Chapman and Hall, (1992). Google Scholar [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. doi: 10.1111/j.1469-1795.1999.tb00055.x. Google Scholar [42] C. A. Stockwell, A. P. Hendry and M. T. Kinnison, Contemporary evolution meets conservation biology,, Trends in Ecology and Evolution, 18 (2003), 94. doi: 10.1016/S0169-5347(02)00044-7. Google Scholar [43] S. C. Stearns, The Evolution of Life Histories,, Oxford University Press, (1992). Google Scholar [44] G. Strang, Linear Algebra and Its Applications,, 3rd edition, (1988). Google Scholar [45] J. N. Thompson, Rapid evolution as an ecological process,, Trends in Ecology and Evolution, 13 (1998), 329. doi: 10.1016/S0169-5347(98)01378-0. Google Scholar [46] W. O. Tschumy, Competition between juveniles and adults in age-structured populations,, Theoretical Population Biology, 21 (1982), 255. doi: 10.1016/0040-5809(82)90017-X. Google Scholar [47] T. L Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar

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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. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar [2] P. A. Abrams, 'Adaptive Dynamics' vs. 'adaptive dynamics',, Journal of Evolutionary Biology, 18 (2005), 1162. Google Scholar [3] L. J. S. Allen, A density-dependent Leslie matrix model,, Mathematical Biosciences, 95 (1989), 179. doi: 10.1016/0025-5564(89)90031-X. Google Scholar [4] H. Behncke, Periodical cicadas,, Journal of Mathematical Biology, 40 (2000), 413. doi: 10.1007/s002850000024. Google Scholar [5] M. G. Bulmer, Periodical insects,, The American Naturalist, 111 (1977), 1099. doi: 10.1086/283240. Google Scholar [6] F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar [7] J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models,, Natural Resource Modelling, 8 (1994), 297. Google Scholar [8] J. M. Cushing, An Introduction to Structured Population Dynamics,, Conference Series in Applied Mathematics Vol. 71, (1998). doi: 10.1137/1.9781611970005. Google Scholar [9] J. M. Cushing, Cycle chains and the LPA model,, Journal of Difference Equations and Applications, 9 (2003), 655. doi: 10.1080/1023619021000042216. Google Scholar [10] J. M. Cushing, Nonlinear semelparous Leslie models,, Mathematical Biosciences and Engineering, 3 (2006), 17. doi: 10.3934/mbe.2006.3.17. Google Scholar [11] J. M. Cushing, Three stage semelparous Leslie models,, Journal of Mathematical Biology, 59 (2009), 75. doi: 10.1007/s00285-008-0208-9. Google Scholar [12] J. M. Cushing, Matrix Models and Population Dynamics,, a chapter in Mathematical Biology (eds. Mark Lewis, (2009), 47. Google Scholar [13] J. M. Cushing, A bifurcation theorem for Darwinian matrix models,, Nonlinear Studies, 17 (2010), 1. Google Scholar [14] J. M. Cushing, A dynamic dichotomy for a system of hierarchical difference equations,, Journal of Difference Equations and Applications, 18 (2012), 1. doi: 10.1080/10236198.2011.628319. Google Scholar [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. doi: 10.1080/17513758.2014.899638. Google Scholar [16] J. M. Cushing and S. M. Henson, Stable bifurcations in semelparous Leslie models,, Journal of Biological Dynamics, 6 (2012), 80. doi: 10.1080/17513758.2012.716085. Google Scholar [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. Google Scholar [18] J. M. Cushing and S. M. Stump, Darwinian dynamics of a juvenile-adult model,, Mathematical Biosciences and Engineering, 10 (2013), 1017. doi: 10.3934/mbe.2013.10.1017. Google Scholar [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. doi: 10.1007/s00285-002-0167-5. Google Scholar [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. doi: 10.1016/j.laa.2004.12.020. Google Scholar [21] O. Diekmann and S. A. van Gils, Invariance and symmetry in a year-class model,, in Trends in Mathematics: Bifurcations, (2003), 141. Google Scholar [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. doi: 10.1080/10236190412331335409. Google Scholar [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. doi: 10.1137/080722734. Google Scholar [24] J. A. Endler, Multiple-trait coevolution and environmental gradients in guppies,, Trends in Ecology and Evolution, 10 (1995), 22. doi: 10.1016/S0169-5347(00)88956-9. Google Scholar [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. doi: 10.1093/icb/43.3.431. Google Scholar [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. doi: 10.1086/421412. Google Scholar [27] P. R. Grant and B. R. Grant, Predicting microevolutionary responses to directional selection on heritable variation,, Evolution, 49 (1995), 241. doi: 10.2307/2410334. Google Scholar [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. doi: 10.1126/science.290.5491.516. Google Scholar [29] R. Kon, Nonexistence of synchronous orbits and class coexistence in matrix population models,, SIAM Journal of Applied Mathematics, 66 (2005), 616. doi: 10.1137/05062353X. Google Scholar [30] R. Kon, Competitive exclusion between year-classes in a semelparous biennial population,, in Mathematical Modeling of Biological Systems, (2008), 79. Google Scholar [31] R. Kon, Permanence induced by life-cycle resonances: The periodical cicada problem,, Journal of Biological Dynamics, 6 (2012), 855. doi: 10.1080/17513758.2011.594098. Google Scholar [32] R. Kon, Y. Saito and Y. Takeuchi, Permanence of single-species stage-structured models,, Journal of Mathematical Biology, 48 (2004), 515. doi: 10.1007/s00285-003-0239-1. Google Scholar [33] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations,, Journal of Mathematical Biology, 55 (2007), 781. doi: 10.1007/s00285-007-0111-9. Google Scholar [34] R. Lande, Natural selection and random genetic drift in phenotypic evolution,, Evolution, 30 (1976), 314. Google Scholar [35] R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607. Google Scholar [36] B. J. McGill and J. S. Brown, Evolutionary Game Theory and Adpative Dynamics of Continuous Traits,, Annual Review of Ecology, 38 (2007), 403. Google Scholar [37] E. Mjølhus, A. Wikan and T. Solberg, On synchronization in semelparous populations,, Journal of Mathematical Biology, 50 (2005), 1. doi: 10.1007/s00285-004-0275-5. Google Scholar [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. Google Scholar [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. doi: 10.1007/978-94-010-0585-2_30. Google Scholar [40] D. A. Roff, The Evolution of Life Histories: Theory and Analysis,, Chapman and Hall, (1992). Google Scholar [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. doi: 10.1111/j.1469-1795.1999.tb00055.x. Google Scholar [42] C. A. Stockwell, A. P. Hendry and M. T. Kinnison, Contemporary evolution meets conservation biology,, Trends in Ecology and Evolution, 18 (2003), 94. doi: 10.1016/S0169-5347(02)00044-7. Google Scholar [43] S. C. Stearns, The Evolution of Life Histories,, Oxford University Press, (1992). Google Scholar [44] G. Strang, Linear Algebra and Its Applications,, 3rd edition, (1988). Google Scholar [45] J. N. Thompson, Rapid evolution as an ecological process,, Trends in Ecology and Evolution, 13 (1998), 329. doi: 10.1016/S0169-5347(98)01378-0. Google Scholar [46] W. O. Tschumy, Competition between juveniles and adults in age-structured populations,, Theoretical Population Biology, 21 (1982), 255. doi: 10.1016/0040-5809(82)90017-X. Google Scholar [47] T. L Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar
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