### Mathematical Biosciences and Engineering

2013, Issue 4: 1017-1044. doi: 10.3934/mbe.2013.10.1017

# Darwinian dynamics of a juvenile-adult model

• Received: 01 October 2012 Accepted: 29 June 2018 Published: 01 June 2013
• MSC : Primary: 92D25, 92D15; Secondary: 37N25.

• The bifurcation that occurs from the extinction equilibrium in a basicdiscrete 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 withoverlapping generations and a synchronous 2-cycle with non-overlappinggenerations. Which of the two alternatives is stable depends on theintensity of competition between juveniles and adults and on the directionof bifurcation. We study this dynamic dichotomy in an evolutionary settingby assuming adult fertility and juvenile survival are functions of aphenotypic trait $u$ subject to Darwinian evolution. Extinction equilibriafor the Darwinian model exist only at traits $u^{\ast }$ that are criticalpoints of $R_{0}\left( u\right)$. We establish the simultaneous bifurcationof positive equilibria and synchronous 2-cycles as the value of $R_{0}\left(u^{\ast }\right)$ increases through $1$ and describe how the stability ofthese dynamics depend on the direction of bifurcation, the intensity ofbetween-class competition, and the extremal properties of $R_{0}\left(u\right)$ at $u^{\ast }$. These results can be equivalently stated in termsof the inherent population growth rate $r\left( u\right)$.

Citation: J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model[J]. Mathematical Biosciences and Engineering, 2013, 10(4): 1017-1044. doi: 10.3934/mbe.2013.10.1017

### Related Papers:

• The bifurcation that occurs from the extinction equilibrium in a basicdiscrete 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 withoverlapping generations and a synchronous 2-cycle with non-overlappinggenerations. Which of the two alternatives is stable depends on theintensity of competition between juveniles and adults and on the directionof bifurcation. We study this dynamic dichotomy in an evolutionary settingby assuming adult fertility and juvenile survival are functions of aphenotypic trait $u$ subject to Darwinian evolution. Extinction equilibriafor the Darwinian model exist only at traits $u^{\ast }$ that are criticalpoints of $R_{0}\left( u\right)$. We establish the simultaneous bifurcationof positive equilibria and synchronous 2-cycles as the value of $R_{0}\left(u^{\ast }\right)$ increases through $1$ and describe how the stability ofthese dynamics depend on the direction of bifurcation, the intensity ofbetween-class competition, and the extremal properties of $R_{0}\left(u\right)$ at $u^{\ast }$. These results can be equivalently stated in termsof the inherent population growth rate $r\left( u\right)$.

 [1] (Coleoptera: Bruchidae), Journal of Animal Ecology 51 (1982), 263-287. [2] Bulletin of Mathematical Biology, 51 (1989), 687-713. [3] Natural Resource Modeling, 8 (1994), 1-37. [4] Journal of Difference Equations and Applications, 9 (2003), 655-670. [5] Mathematical Biosciences and Engineering, 3 (2006), 17-36. [6] Journal of Mathematical Biology, 59 (2009), 75-104. [7] Journal of Biological Dynamics, 5 (2011), 277-297. [8] Journal of Difference Equations and Applications, 18 (2012), 1-26. [9] Journal of Biological Dynamics, 6 (2012), 80-102. [10] Journal of Mathematical Biology, 46 (2003), 95-131. [11] Ph.D Dissertation, University of Utrecht, The Netherlands, 2004. [12] Linear Algebra and its Applications, 398 (2005), 185-243. [13] Journal of Difference Equations and Applications, 11 (2005), 327-335. [14] Journal of Theoretical Biology, 124 (1987), 25-33. [15] Journal of Theoretical Biology, 131 (1988), 389-400. [16] Journal of Mathematical Biology, 4 (1977), 101-147. [17] Journal of Animal Ecology. 45 (1976), 471-486. [18] Applied Mathematical Sciences 156, Springer, New York, 2004. [19] SIAM Journal of Applied Mathematics, 66 (2005), 616-626. [20] Journal of Mathematical Biology. 55 (2007), 781-802. [21] 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. [22] Journal of Biological Dynamics, 6 (2012), 855-890. [23] Biometrika, 45 (1958), 316-330. [24] Acta Mathematicae Applicatae Sinica, 11 (1995), 160-171. [25] Journal of Theoretical Biology, 144 (1990), 567-571. [26] Journal of Animal Ecology, 43 (1974), 747-770. [27] Journal of Theoretical Biology, 49 (1975), 645-647. [28] Journal of Mathematical Biology, 32 (1994), 329-344. [29] Chapman and Hall, New York, 1992. [30] Theoretical Population Biology, 21 (1982), 255-268. [31] Cambridge University Press, New York, 2005. [32] Journal of Mathematical Biology, 35 (1996), 195-239. [33] Mathematical Biosciences, 146 (1997), 37-62.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.080 2.1