Physiologically structured populations with diffusion and dynamic
boundary conditions

József Z. Farkas; Peter Hinow; József Z. Farkas; Peter Hinow

doi:10.3934/mbe.2011.8.503

Mathematical Biosciences and Engineering

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

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Physiologically structured populations with diffusion and dynamic boundary conditions

1.
Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA
2.
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413

Received: 01 March 2010 Accepted: 29 June 2018 Published: 01 April 2011
MSC : 92D25, 47N60, 47D06, 35B35.

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.
- diffusion,
- stability.,
- semigroups of linear operators,
- Structured populations,
- spectral methods,
- Wentzell-Robin boundary condition
Citation: József Z. Farkas, Peter Hinow. Physiologically structured populations with diffusion and dynamicboundary conditions[J]. Mathematical Biosciences and Engineering, 2011, 8(2): 503-513. doi: 10.3934/mbe.2011.8.503

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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.
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