NONLINEAR BOUNDARY CONDITIONS DERIVED BY SINGULAR PERTUBATION IN AGE STRUCTURED POPULATION DYNAMICS MODEL

Arnaud Ducrot; Pierre Magal; Ousmane Seydi

doi:10.11948/2011026

2011 Volume 1 Issue 3

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Arnaud Ducrot, Pierre Magal, Ousmane Seydi. NONLINEAR BOUNDARY CONDITIONS DERIVED BY SINGULAR PERTUBATION IN AGE STRUCTURED POPULATION DYNAMICS MODEL[J]. Journal of Applied Analysis & Computation, 2011, 1(3): 373-395. doi: 10.11948/2011026

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Arnaud Ducrot, Pierre Magal, Ousmane Seydi. NONLINEAR BOUNDARY CONDITIONS DERIVED BY SINGULAR PERTUBATION IN AGE STRUCTURED POPULATION DYNAMICS MODEL[J]. Journal of Applied Analysis & Computation, 2011, 1(3): 373-395. doi: 10.11948/2011026

NONLINEAR BOUNDARY CONDITIONS DERIVED BY SINGULAR PERTUBATION IN AGE STRUCTURED POPULATION DYNAMICS MODEL

UMR CNRS 5251, IBM & INRIA Bordeaux sud-ouest Anubis, Université de Bordeaux, 3 ter, Place de la Victoire 33000 Bordeaux, France

Abstract

Abstract

In this article, we derive Ricker's[22, 23] type nonlinear boundary condition for an age structured population dynamic model by using a singular perturbation. The question addressed in this paper is the convergence of the singularly perturbed system. We first obtain a finite time convergence for a fixed initial distribution. Then we focus on the convergence uniformly of the singularly perturbed system with respect to the initial distribution in bounded sets.
- Population dynamics /
- age structured models /
- density dependent models /
- singular perturbations
MSC: 35B25;35F30;47D62;92D25

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References

[1]	S. Anita, Analysis and Control of Age-Dependent Population Dynamics, Kluwer, Dordrecht, 2000. Google Scholar
[2]	O. Arino, E. Sanchez, R. Bravo de la Parra and P. Auger, A singular perturbation in an age-structured population model, SIAM J. Appl. Math., 60(1999), 408-436. Google Scholar
[3]	D.P. Armtsrong, R. S. Davidson, J.K. Perrott, J. Roygard and L. Buchanan, Density-dependent population growth in a reintroduced population of North Island saddlebacks, Journal of Animal Ecology, 74(2005), 160-170. Google Scholar
[4]	A.G. Blundell and D.R. Peart, Density-dependent population dynamics of a dominant rain forest canopy tree,Ecology, 85(2004), 704-715. Google Scholar
[5]	R. Bravo de la Parra, E. Sanchez, O. Arino and P. Auger, Time Scales in Density Dependent Discrete Models, Journal of Biological Systems, 5(1997), 111-129. Google Scholar
[6]	R. Bravo de la Parra, E. Sanchez, O. Arino and P. Auger, A discrete model with density dependent fast migration,Mathematical Biosciences, 157(1999), 91-109. Google Scholar
[7]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2010. Google Scholar
[8]	T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Sciences Publications, 2006. Google Scholar
[9]	J. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. Google Scholar
[10]	K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. Google Scholar
[11]	W.S. Gurney and R. M. Nisbet, Age-and density-dependent population dynamics in static and variable environments, Theor. Popul. Biol., 17(1980), 321-344. Google Scholar
[12]	J.E. Franke and A.Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, Journal of Mathematical Biology, 30(1991), 161-168. Google Scholar
[13]	M. Iannelli, Mathematical theory of age-structured population dynimics, Giadini Editori in Pisa, 1994. Google Scholar
[14]	L. Liu and J. E. Cohen, Equilibrium and local stability in a logistic matrix model for age-structured populations, Journal of Mathematical Biology, 25(1987), 73-88. Google Scholar
[15]	K. Lorenzen and K. Enberg, Density-dependent growth as a key mechanism in the regulation of fish populations:evidence from among-population comparisons, Proc. R. Soc. Lond. B, 269(2002), 49-54. Google Scholar
[16]	P. Magal, and S. Ruan, On Integrated Semigroups and Age Structured Models in Lp Spaces, Differential and Integral Equations, 20(2007), 197-139. Google Scholar
[17]	P. Magal and S. Ruan, On Semilinear Cauchy Problems with Non-dense Domain, Advances in Differential Equations, 14(2009), 1041-1084. Google Scholar
[18]	P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Nondense Domain and Applications to Hopf Bifurcation in Age Structured Models, Memoirs of the American Mathematical Society, 202:951(2009). Google Scholar
[19]	R. H. Martin, Nonlinear Operators and Differential Equations in Banach spaces, John Wiley & Sons, 1976. Google Scholar
[20]	B. Perthame, Transport equations in biology, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007. Google Scholar
[21]	R.J. Putman, J. Langbein, A.J. M. Hewison and S.K. Sharma, Relative roles of density-dependent and density-independent factors in population dynamics of British deer, Mammal Review, 26(2008), 81-101. Google Scholar
[22]	W.E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11(1954), 559-623. Google Scholar
[23]	W.E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can., 191(1975). Google Scholar
[24]	C. Rorres, Stability of an age specific population with density dependent fertility, Theor. Popul. Biol., 10(1976), 26-46. Google Scholar
[25]	C. Rorres, Local stability of a population with density-dependent fertility, Theor. Popul. Biol., 16(1979), 283-300. Google Scholar
[26]	C. Rorres, A nonlinear model of population growth in which fertility is dependent on birth rate, SIAM J. Appl. Math., 37:2(1979), 423-432. Google Scholar
[27]	G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, New York, 2002. Google Scholar
[28]	H.R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152(1990), 416-447. Google Scholar
[29]	H. R. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ., 8(2008), 283-305. Google Scholar
[30]	H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. Google Scholar
[31]	S. Tang and L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, Journal of Mathematical Biology, 44(2002), 185-199. Google Scholar
[32]	G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, 1985. Google Scholar