Linear stability and positivity results for a generalized size-structured Daphnia model with inflow§

§Dedicated to Professor Miklós Farkas on the occasion of his 75th birthday.

Abstract

We employ semigroup and spectral methods to analyze the linear stability of positive stationary solutions of a generalized size-structured Daphnia model. Using the regularity properties of the governing semigroup, we are able to formulate a general stability condition, which permits an intuitively clear interpretation in a special case of model ingredients. Moreover, we derive a comprehensive instability criterion that reduces to an elegant instability condition for the classical Daphnia population model in terms of the inherent net reproduction rate of Daphnia individuals.

§Dedicated to Professor Miklós Farkas on the occasion of his 75th birthday.

Keywords:

• Predator–prey interaction
• Structured population dynamics
• Semigroup methods
• Principle of linearized stability

Keywords:

• 2000 MSC: 92D25
• 47D06
• 35B35

Acknowledgements

JZF was supported by The Carnegie Trust for the Universities of Scotland while visiting the University of Memphis. TH acknowledges support through a Faculty Research Grant of The University of Memphis. We thank the reviewers for their thoughtful comments and suggestions.

Notes

§Dedicated to Professor Miklós Farkas on the occasion of his 75th birthday.

¹Here we refer to the following version of the Desch–Schappacher Perturbation Theorem (see Corollary III.3.4 in 9) which we state verbatim for the reader's convenience: THEOREM Let be the generator of a strongly continuous semigroup on the Banach space and let . Moreover, assume that there exists t ₀ > 0 and p ∈ [1, ∞) such that

for all functions . Then generates a strongly continuous semigroup on .