On a nonlinear age-structured model of  semelparous species

Ryszard Rudnicki; Radosław Wieczorek

doi:10.3934/dcdsb.2014.19.2641

Article Contents

2014, Volume 19, Issue 8: 2641-2656. Doi: 10.3934/dcdsb.2014.19.2641

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On a nonlinear age-structured model of semelparous species

Ryszard Rudnicki^1, and
Radosław Wieczorek^2,

1.
Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice
2.
Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice

Received: November 2013

Revised: April 2014

Published: October 2014

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Abstract

We study a nonlinear age-structured model of a population such that individuals may give birth only at a given age. Properties of measure-valued periodic solutions of this system are investigated. We show that in some cases the age profile of the population tends to a Dirac measure, which means that the population asymptotically consists of individuals at the same age. This phenomenon is observed in nature in some insects populations.

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Mathematics Subject Classification: Primary: 92D25; Secondary: 35F20, 45G10.

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References

[1]	M. G. Bulmer, Periodical insects, The American Naturalist, 111 (1977), 1099-1117.doi: 10.1086/283240.
[2]	J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277.doi: 10.1016/j.jde.2011.11.003.
[3]	P. Cull and A. Vogt, The periodic limit for the Leslie model, Math. Biosci., 21 (1974), 39-54.doi: 10.1016/0025-5564(74)90103-5.
[4]	J. M. Cushing, Three stage semelparous Leslie models, J. Math. Biol., 59 (2009), 75-104.
[5]	O. Diekmann, N. Davydova and S. van Gils, On a boom and bust year class cycle, J. Difference Equ. Appl., 11 (2005), 327-335.doi: 10.1080/10236190412331335409.
[6]	R. Dilão and A. Lakmeche, On the weak solutions of the McKendrick equation: Existence of demography cycles, Math. Model. Nat. Phenom., 1 (2006), 1-32.doi: 10.1051/mmnp:2006001.
[7]	J. Z. Farkas, Stability conditions for the non-linear McKendrick equations, Appl. Math. Comput., 156 (2004), 771-777.doi: 10.1016/j.amc.2003.06.019.
[8]	J. E. Franke and A.-A. Yakubu, Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models, Math. Biosci., 204 (2006), 1-20.doi: 10.1016/j.mbs.2006.08.016.
[9]	G. Gripenberg, S. O. Londen and O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, New York, 1990.doi: 10.1017/CBO9780511662805.
[10]	M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent populations dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.
[11]	P. Gwiazda and A. Marciniak-Czochra, Structured population equations in metric spaces, J. Hyperbolic Differ. Equ., 7 (2010), 733-773.doi: 10.1142/S021989161000227X.
[12]	P. E. Jabin, Large time concentration for solutions to kinetic equations with energy dissipation, Comm. Partial Differential Equations, 25 (2000), 541-557.doi: 10.1080/03605300008821523.
[13]	P. Michel, A singular asymptotic behavior of a transport equation, C. R. Math. Acad. Sci. Paris, 346 (2008), 155-159.doi: 10.1016/j.crma.2007.12.010.
[14]	B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser, Basel 2007.
[15]	S. I. Rubinow, A maturity time representation for cell populations, Biophy. J., 8 (1968), 1055-1073.doi: 10.1016/S0006-3495(68)86539-7.
[16]	R. Rudnicki and R. Wieczorek, Asymptotic analysis of a semelparous species model, Fund. Inform., 103 (2010), 219-233.
[17]	S. K. Tumuluri, Steady state analysis of a nonlinear renewal equation, Math. Comput. Modelling, 53 (2011), 1420-1435.doi: 10.1016/j.mcm.2010.02.050.