Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Steady states in hierarchical structured populations with distributed states at birth

Pages: 2671 - 2689, Volume 17, Issue 8, November 2012      doi:10.3934/dcdsb.2012.17.2671

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József Z. Farkas - Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, United Kingdom (email)
Peter Hinow - Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413, United States (email)

Abstract: 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:  Hierarchical structured populations, steady states, fixed points of nonlinear maps, semigroups of linear operators, spectral methods, stability.
Mathematics Subject Classification:  92D25, 47N60, 47D06, 35B35.

Received: March 2011;      Revised: August 2011;      Available Online: July 2012.