Mathematical Biosciences and Engineering (MBE)

Physiologically structured populations with diffusion and dynamic boundary conditions

Pages: 503 - 513, Volume 8, Issue 2, April 2011      doi:10.3934/mbe.2011.8.503

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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 consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. We equip the model with generalized Wentzell-Robin (or dynamic) boundary conditions. This approach allows the modelling of populations in which individuals may have distinguished physiological states. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. These results are obtained by establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is, our model admits a finite-dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.

Keywords:  Structured populations, diffusion, Wentzell-Robin boundary condition, semigroups of linear operators, spectral methods, stability.
Mathematics Subject Classification:  92D25, 47N60, 47D06, 35B35.

Received: March 2010;      Accepted: December 2010;      Available Online: April 2011.