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November  2012, 17(8): 2671-2689. doi: 10.3934/dcdsb.2012.17.2671

Steady states in hierarchical structured populations with distributed states at birth

József Z. Farkas 1, and  Peter Hinow 2,

1. 

Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, United Kingdom
2. 

Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413

Received  March 2011 Revised  August 2011 Published  July 2012

We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individual's size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.
Keywords: steady states, fixed points of nonlinear maps, Hierarchical structured populations, spectral methods, stability., semigroups of linear operators.
Mathematics Subject Classification: 92D25, 47N60, 47D06, 35B3.
Citation: József Z. Farkas, Peter Hinow. Steady states in hierarchical structured populations with distributed states at birth. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2671-2689. doi: 10.3934/dcdsb.2012.17.2671
References:
[1]

A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness andapproximation, Appl. Math. Optim., 51 (2005), 35-59.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Differential Equations, 217 (2005), 431-455.  Google Scholar

[3]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[4]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  Google Scholar

[5]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986.  Google Scholar

[6]

À. Calsina and J. Saldań, Asymptotic behavior of a model ofhierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987.  Google Scholar

[7]

À. Calsina and J. Saldañ, Basic theory for a class of models ofhierarchically structured population dynamics with distributed states in the recruitment, Math. Models Methods Appl. Sci., 16 (2006), 1695-1722.  Google Scholar

[8]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729.  Google Scholar

[9]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, PA, 1998.  Google Scholar

[10]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.  Google Scholar

[11]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York 2000.  Google Scholar

[12]

J. Z. Farkas, D. M. Green and P. Hinow, Semigroup analysis ofstructured parasite populations, Math. Model. Nat. Phenom., 5 (2010), 94-114.  Google Scholar

[13]

J. Z. Farkas and T. Hagen, Stability and regularity results for asize-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  Google Scholar

[14]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8 (2009), 1825-1839.  Google Scholar

[15]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 639-657.  Google Scholar

[16]

J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514.  Google Scholar

[17]

A. Grabosch and H. J. A. M. Heijmans, Cauchy problems with state-dependent time evolution, Japan J. Appl. Math., 7 (1990), 433-457.  Google Scholar

[18]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  Google Scholar

[19]

S. M. Henson and J. M. Cushing, Hierarchical models of intra-specific competition: Scramble versus contest, J. Math. Biol., 34 (1996), 755-772. Google Scholar

[20]

S. R.-J. Jang and J. M. Cushing, A discrete hierarchical model ofintra-specific competition, J. Math. Anal. Appl., 280 (2003), 102-122.  Google Scholar

[21]

S. R.-J. Jang and J. M. Cushing, Dynamics of hierarchical models in discrete-time, J. Difference Equ. Appl., 11 (2005), 95-115.  Google Scholar

[22]

N. Kato, A principle of linearized stability for nonlinear evolution equations, Trans. Amer. Math. Soc., 347 (1995), 2851-2868.  Google Scholar

[23]

J. A. J. Metz and O. Diekmann, Age dependence, in "The Dynamics of Physiologically Structured Populations" (Amsterdam, 1983), Lecture Notes in Biomath., 68, Springer, Berlin, 1986.  Google Scholar

[24]

J. Prüss, On the qualitative behavior of populations with age-specific interactions, Comput. Math. Appl., 9 (1983), 327-339.  Google Scholar

[25]

S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math., 48 (1988), 549-591.  Google Scholar

[26]

Ch. Walker, Global bifurcation of positive equilibria in nonlinear population models, J. Diff. Eq., 248 (2010), 1756-1776.  Google Scholar

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.  Google Scholar

[28]

K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

show all references

References:
[1]

A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness andapproximation, Appl. Math. Optim., 51 (2005), 35-59.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Differential Equations, 217 (2005), 431-455.  Google Scholar

[3]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[4]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18 (1976), 620-709.  Google Scholar

[5]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, "One-Parameter Semigroups of Positive Operators," Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986.  Google Scholar

[6]

À. Calsina and J. Saldań, Asymptotic behavior of a model ofhierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987.  Google Scholar

[7]

À. Calsina and J. Saldañ, Basic theory for a class of models ofhierarchically structured population dynamics with distributed states in the recruitment, Math. Models Methods Appl. Sci., 16 (2006), 1695-1722.  Google Scholar

[8]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729.  Google Scholar

[9]

J. M. Cushing, "An Introduction to Structured Population Dynamics," CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, PA, 1998.  Google Scholar

[10]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985.  Google Scholar

[11]

K.-J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York 2000.  Google Scholar

[12]

J. Z. Farkas, D. M. Green and P. Hinow, Semigroup analysis ofstructured parasite populations, Math. Model. Nat. Phenom., 5 (2010), 94-114.  Google Scholar

[13]

J. Z. Farkas and T. Hagen, Stability and regularity results for asize-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  Google Scholar

[14]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8 (2009), 1825-1839.  Google Scholar

[15]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 639-657.  Google Scholar

[16]

J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514.  Google Scholar

[17]

A. Grabosch and H. J. A. M. Heijmans, Cauchy problems with state-dependent time evolution, Japan J. Appl. Math., 7 (1990), 433-457.  Google Scholar

[18]

M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  Google Scholar

[19]

S. M. Henson and J. M. Cushing, Hierarchical models of intra-specific competition: Scramble versus contest, J. Math. Biol., 34 (1996), 755-772. Google Scholar

[20]

S. R.-J. Jang and J. M. Cushing, A discrete hierarchical model ofintra-specific competition, J. Math. Anal. Appl., 280 (2003), 102-122.  Google Scholar

[21]

S. R.-J. Jang and J. M. Cushing, Dynamics of hierarchical models in discrete-time, J. Difference Equ. Appl., 11 (2005), 95-115.  Google Scholar

[22]

N. Kato, A principle of linearized stability for nonlinear evolution equations, Trans. Amer. Math. Soc., 347 (1995), 2851-2868.  Google Scholar

[23]

J. A. J. Metz and O. Diekmann, Age dependence, in "The Dynamics of Physiologically Structured Populations" (Amsterdam, 1983), Lecture Notes in Biomath., 68, Springer, Berlin, 1986.  Google Scholar

[24]

J. Prüss, On the qualitative behavior of populations with age-specific interactions, Comput. Math. Appl., 9 (1983), 327-339.  Google Scholar

[25]

S. L. Tucker and S. O. Zimmerman, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math., 48 (1988), 549-591.  Google Scholar

[26]

Ch. Walker, Global bifurcation of positive equilibria in nonlinear population models, J. Diff. Eq., 248 (2010), 1756-1776.  Google Scholar

[27]

G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.  Google Scholar

[28]

K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

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