# American Institute of Mathematical Sciences

2013, 10(4): 1017-1044. doi: 10.3934/mbe.2013.10.1017

## Darwinian dynamics of a juvenile-adult model

 1 Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721 2 Department of Ecology and Evolutionary Biology, 1041 E. Lowell St, University of Arizona, Tucson, AZ 85721, United States

Received  October 2012 Revised  January 2013 Published  June 2013

The bifurcation that occurs from the extinction equilibrium in a basic discrete time, nonlinear juvenile-adult model for semelparous populations, as the inherent net reproductive number $R_{0}$ increases through $1$, exhibits a dynamic dichotomy with two alternatives: an equilibrium with overlapping generations and a synchronous 2-cycle with non-overlapping generations. Which of the two alternatives is stable depends on the intensity of competition between juveniles and adults and on the direction of bifurcation. We study this dynamic dichotomy in an evolutionary setting by assuming adult fertility and juvenile survival are functions of a phenotypic trait $u$ subject to Darwinian evolution. Extinction equilibria for the Darwinian model exist only at traits $u^{\ast }$ that are critical points of $R_{0}\left( u\right)$. We establish the simultaneous bifurcation of positive equilibria and synchronous 2-cycles as the value of $R_{0}\left( u^{\ast }\right)$ increases through $1$ and describe how the stability of these dynamics depend on the direction of bifurcation, the intensity of between-class competition, and the extremal properties of $R_{0}\left( u\right)$ at $u^{\ast }$. These results can be equivalently stated in terms of the inherent population growth rate $r\left( u\right)$.
Citation: J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1017-1044. doi: 10.3934/mbe.2013.10.1017
##### References:
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##### References:
 [1] T. S. Bellows, Jr., Analytical model for laboratory populations of Callosobruchus Chinensis and C. Maculatus,, (Coleoptera: Bruchidae), 51 (1982), 263.  doi: 10.2307/4324.  Google Scholar [2] J. M. Cushing and Jia 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 [3] J. M. Cushing and Zhou Yicang, The net reproductive value and stability in structured population models,, Natural Resource Modeling, 8 (1994), 1.   Google Scholar [4] J. M. Cushing, Cycle chains and the LPA model,, Journal of Difference Equations and Applications, 9 (2003), 655.  doi: 10.1080/1023619021000042216.  Google Scholar [5] J. M. Cushing, Nonlinear semelparous Leslie models,, Mathematical Biosciences and Engineering, 3 (2006), 17.  doi: 10.3934/mbe.2006.3.17.  Google Scholar [6] J. M. Cushing, Three stage semelparous Leslie models,, Journal of Mathematical Biology, 59 (2009), 75.  doi: 10.1007/s00285-008-0208-9.  Google Scholar [7] J. M. Cushing, On the relationship between $r$ and $R_{0}$ and its role in the bifurcation of equilibria of Darwinian matrix models,, Journal of Biological Dynamics, 5 (2011), 277.  doi: 10.1080/17513758.2010.491583.  Google Scholar [8] 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 [9] 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 [10] 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 [11] N. V. Davydova, "Old and Young: Can They Coexist?", Ph.D Dissertation, (2004).   Google Scholar [12] 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 [13] 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 [14] B. Ebenman, Niche differences between age and classes and intraspecific competition in age-structured populations,, Journal of Theoretical Biology, 124 (1987), 25.  doi: 10.1016/S0022-5193(87)80249-7.  Google Scholar [15] B. Ebenman, Competition between age classes and population dynamics,, Journal of Theoretical Biology, 131 (1988), 389.  doi: 10.1016/S0022-5193(88)80036-5.  Google Scholar [16] J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density-dependent population models,, Journal of Mathematical Biology, 4 (1977), 101.  doi: 10.1007/BF00275980.  Google Scholar [17] M. P. Hassell, J. H. Lawton and R. M. May, Patterns of dynamical behaviour in single-species populations,, Journal of Animal Ecology. 45 (1976), 45 (1976), 471.  doi: 10.2307/3886.  Google Scholar [18] H. Keilhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs,", Applied Mathematical Sciences 156, 156 (2004).   Google Scholar [19] 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 [20] R. Kon and Y. Iwasa, Single-class orbits in nonlinear Leslie matrix models for semelparous populations,, Journal of Mathematical Biology. 55 (2007), 55 (2007), 781.  doi: 10.1007/s00285-007-0111-9.  Google Scholar [21] R. Kon, Competitive exclusion between year-classes in a semelparous biennial population,, in, (2008), 79.  doi: 10.1007/978-0-8176-4556-4.  Google Scholar [22] 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 [23] P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species,, Biometrika, 45 (1958), 316.   Google Scholar [24] W. Liu, Global analysis of an Ebenman's model of population with two competing age classes,, Acta Mathematicae Applicatae Sinica, 11 (1995), 160.  doi: 10.1007/BF02013151.  Google Scholar [25] M. Loreau, Competition between age-classes and stability of stage structured populations: A re-examination of Ebenman's model,, Journal of Theoretical Biology, 144 (1990), 567.  doi: 10.1016/S0022-5193(05)80090-6.  Google Scholar [26] R. M. May, G. R. Conway, M. P. Hassell and T. R. E. Southwood, Time delays, density-dependence and single species oscillations,, Journal of Animal Ecology, 43 (1974), 747.  doi: 10.2307/3535.  Google Scholar [27] R. M. May, Biological populations obeying difference equations: Stable points, stable cycles, and chaos,, Journal of Theoretical Biology, 49 (1975), 645.  doi: 10.1016/0022-5193(75)90078-8.  Google Scholar [28] R. M. Nisbet and L. C. Onyiah, Population dynamic consequences of competition within and between age classes,, Journal of Mathematical Biology, 32 (1994), 329.  doi: 10.1007/BF00160164.  Google Scholar [29] D. A. Roff, "The Evolution of Life Histories: Theory and Analysis,", Chapman and Hall, (1992).   Google Scholar [30] 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 [31] 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 [32] A. Wikan and E. Mjølhus, Overcompensatory recruitment and generation delay in discrete age-structured population models,, Journal of Mathematical Biology, 35 (1996), 195.  doi: 10.1007/s002850050050.  Google Scholar [33] A. Wikan, Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities,, Mathematical Biosciences, 146 (1997), 37.  doi: 10.1016/S0025-5564(97)00074-6.  Google Scholar
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