Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback

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November 2009, 8(6): 1825-1839. doi: 10.3934/cpaa.2009.8.1825

Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback

József Z. Farkas ^1, and Thomas Hagen ^2,

1.	Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA, United Kingdom
2.	Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152

Received October 2008 Revised February 2009 Published August 2009

Abstract
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In this work we consider a size-structured cannibalism model with the model ingredients (fertility, growth, and mortality rate) depending on size (ranging over an infinite domain) and on a general function of the standing population (environmental feedback). Our focus is on the asymptotic behavior of the system, in particular on the effect of cannibalism on the long-term dynamics. To this end, we formally linearize the system about steady state and establish conditions in terms of the model ingredients which yield uniform exponential stability of the governing linear semigroup. We also show how the point spectrum of the linearized semigroup generator can be characterized in the special case of a separable attack rate and establish a general instability result. Further spectral analysis allows us to give conditions for asynchronous exponential growth of the linear semigroup.

Keywords: linear semigroup methods, cannibalism, Size-structured populations, asynchronous exponential growth..

Mathematics Subject Classification: 92D25, 47D06, 35B3.

Citation: József Z. Farkas, Thomas Hagen. Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1825-1839. doi: 10.3934/cpaa.2009.8.1825

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