Darwinian dynamics of a juvenile-adult model

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

Darwinian dynamics of a juvenile-adult model

J. M. Cushing ^1, and Simon Maccracken Stump ^2,

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

Abstract
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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) $.

Keywords: equilibrium, Structured population dynamics, evolutionary game theory., bifurcation, dynamic dichotomy, semelparity, Darwinian dynamics, synchronous cycles, juvenile-adult population model.

Mathematics Subject Classification: Primary: 92D25, 92D15; Secondary: 37N2.

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:

[1]	T. S. Bellows, Jr., Analytical model for laboratory populations of Callosobruchus Chinensis and C. Maculatus, (Coleoptera: Bruchidae), Journal of Animal Ecology 51 (1982), 263-287. doi: 10.2307/4324.
[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), 687-713.
[3]	J. M. Cushing and Zhou Yicang, The net reproductive value and stability in structured population models, Natural Resource Modeling, 8 (1994), 1-37.
[4]	J. M. Cushing, Cycle chains and the LPA model, Journal of Difference Equations and Applications, 9 (2003), 655-670. doi: 10.1080/1023619021000042216.
[5]	J. M. Cushing, Nonlinear semelparous Leslie models, Mathematical Biosciences and Engineering, 3 (2006), 17-36. doi: 10.3934/mbe.2006.3.17.
[6]	J. M. Cushing, Three stage semelparous Leslie models, Journal of Mathematical Biology, 59 (2009), 75-104. doi: 10.1007/s00285-008-0208-9.
[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), 277-297. doi: 10.1080/17513758.2010.491583.
[8]	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.
[9]	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.
[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), 95-131. doi: 10.1007/s00285-002-0167-5.
[11]	N. V. Davydova, "Old and Young: Can They Coexist?" Ph.D Dissertation, University of Utrecht, The Netherlands, 2004.
[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), 185-243. doi: 10.1016/j.laa.2004.12.020.
[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), 327-335. doi: 10.1080/10236190412331335409.
[14]	B. Ebenman, Niche differences between age and classes and intraspecific competition in age-structured populations, Journal of Theoretical Biology, 124 (1987), 25-33. doi: 10.1016/S0022-5193(87)80249-7.
[15]	B. Ebenman, Competition between age classes and population dynamics, Journal of Theoretical Biology, 131 (1988), 389-400. doi: 10.1016/S0022-5193(88)80036-5.
[16]	J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density-dependent population models, Journal of Mathematical Biology, 4 (1977), 101-147. doi: 10.1007/BF00275980.
[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), 471-486. doi: 10.2307/3886.
[18]	H. Keilhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences 156, Springer, New York, 2004.
[19]	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.
[20]	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.
[21]	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. doi: 10.1007/978-0-8176-4556-4.
[22]	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.
[23]	P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika, 45 (1958), 316-330.
[24]	W. Liu, Global analysis of an Ebenman's model of population with two competing age classes, Acta Mathematicae Applicatae Sinica, 11 (1995), 160-171. doi: 10.1007/BF02013151.
[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), 567-571. doi: 10.1016/S0022-5193(05)80090-6.
[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), 747-770. doi: 10.2307/3535.
[27]	R. M. May, Biological populations obeying difference equations: Stable points, stable cycles, and chaos, Journal of Theoretical Biology, 49 (1975), 645-647. doi: 10.1016/0022-5193(75)90078-8.
[28]	R. M. Nisbet and L. C. Onyiah, Population dynamic consequences of competition within and between age classes, Journal of Mathematical Biology, 32 (1994), 329-344. doi: 10.1007/BF00160164.
[29]	D. A. Roff, "The Evolution of Life Histories: Theory and Analysis," Chapman and Hall, New York, 1992.
[30]	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.
[31]	T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics," Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633.
[32]	A. Wikan and E. Mjølhus, Overcompensatory recruitment and generation delay in discrete age-structured population models, Journal of Mathematical Biology, 35 (1996), 195-239. doi: 10.1007/s002850050050.
[33]	A. Wikan, Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities, Mathematical Biosciences, 146 (1997), 37-62. doi: 10.1016/S0025-5564(97)00074-6.

show all references

References:

[1]	T. S. Bellows, Jr., Analytical model for laboratory populations of Callosobruchus Chinensis and C. Maculatus, (Coleoptera: Bruchidae), Journal of Animal Ecology 51 (1982), 263-287. doi: 10.2307/4324.
[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), 687-713.
[3]	J. M. Cushing and Zhou Yicang, The net reproductive value and stability in structured population models, Natural Resource Modeling, 8 (1994), 1-37.
[4]	J. M. Cushing, Cycle chains and the LPA model, Journal of Difference Equations and Applications, 9 (2003), 655-670. doi: 10.1080/1023619021000042216.
[5]	J. M. Cushing, Nonlinear semelparous Leslie models, Mathematical Biosciences and Engineering, 3 (2006), 17-36. doi: 10.3934/mbe.2006.3.17.
[6]	J. M. Cushing, Three stage semelparous Leslie models, Journal of Mathematical Biology, 59 (2009), 75-104. doi: 10.1007/s00285-008-0208-9.
[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), 277-297. doi: 10.1080/17513758.2010.491583.
[8]	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.
[9]	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.
[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), 95-131. doi: 10.1007/s00285-002-0167-5.
[11]	N. V. Davydova, "Old and Young: Can They Coexist?" Ph.D Dissertation, University of Utrecht, The Netherlands, 2004.
[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), 185-243. doi: 10.1016/j.laa.2004.12.020.
[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), 327-335. doi: 10.1080/10236190412331335409.
[14]	B. Ebenman, Niche differences between age and classes and intraspecific competition in age-structured populations, Journal of Theoretical Biology, 124 (1987), 25-33. doi: 10.1016/S0022-5193(87)80249-7.
[15]	B. Ebenman, Competition between age classes and population dynamics, Journal of Theoretical Biology, 131 (1988), 389-400. doi: 10.1016/S0022-5193(88)80036-5.
[16]	J. Guckenheimer, G. Oster and A. Ipaktchi, The dynamics of density-dependent population models, Journal of Mathematical Biology, 4 (1977), 101-147. doi: 10.1007/BF00275980.
[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), 471-486. doi: 10.2307/3886.
[18]	H. Keilhöfer, "Bifurcation Theory: An Introduction with Applications to PDEs," Applied Mathematical Sciences 156, Springer, New York, 2004.
[19]	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.
[20]	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.
[21]	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. doi: 10.1007/978-0-8176-4556-4.
[22]	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.
[23]	P. H. Leslie and J. C. Gower, The properties of a stochastic model for two competing species, Biometrika, 45 (1958), 316-330.
[24]	W. Liu, Global analysis of an Ebenman's model of population with two competing age classes, Acta Mathematicae Applicatae Sinica, 11 (1995), 160-171. doi: 10.1007/BF02013151.
[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), 567-571. doi: 10.1016/S0022-5193(05)80090-6.
[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), 747-770. doi: 10.2307/3535.
[27]	R. M. May, Biological populations obeying difference equations: Stable points, stable cycles, and chaos, Journal of Theoretical Biology, 49 (1975), 645-647. doi: 10.1016/0022-5193(75)90078-8.
[28]	R. M. Nisbet and L. C. Onyiah, Population dynamic consequences of competition within and between age classes, Journal of Mathematical Biology, 32 (1994), 329-344. doi: 10.1007/BF00160164.
[29]	D. A. Roff, "The Evolution of Life Histories: Theory and Analysis," Chapman and Hall, New York, 1992.
[30]	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.
[31]	T. L. Vincent and J. S. Brown, "Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics," Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633.
[32]	A. Wikan and E. Mjølhus, Overcompensatory recruitment and generation delay in discrete age-structured population models, Journal of Mathematical Biology, 35 (1996), 195-239. doi: 10.1007/s002850050050.
[33]	A. Wikan, Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities, Mathematical Biosciences, 146 (1997), 37-62. doi: 10.1016/S0025-5564(97)00074-6.

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