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Numerical equilibrium analysis for structured consumer resource models

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Roos, A. M. de, Diekmann, O., Getto, P. and Kirkilionis, Markus, 1962-. (2010) Numerical equilibrium analysis for structured consumer resource models. Bulletin of Mathematical Biology, Vol.72 (No.2). pp. 259-297. ISSN 0092-8240

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Official URL: http://dx.doi.org/10.1007/s11538-009-9445-3

Abstract

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Age-structured populations, Delay differential equations, Consumers -- Mathematical models, Daphnia -- Mathematical models
Journal or Publication Title:	Bulletin of Mathematical Biology
Publisher:	Springer New York LLC
ISSN:	0092-8240
Date:	February 2010
Volume:	Vol.72
Number:	No.2
Number of Pages:	29
Page Range:	pp. 259-297
Identification Number:	10.1007/s11538-009-9445-3
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
References:	Allgower, E.L., Georg, K., 1990. Numerical Continuation Methods, an Introduction. SCM, vol. 13. Springer, Berlin. de Roos, A.M., 1997. A gentle introduction to models of physiologically structured populations. In: Tuljapurkar, S., Caswell, H. (Eds.), Structured-Population Models in Marine, Terrestrial, and Freshwater Systems, pp. 119–204. Chapman and Hall, New York. de Roos, A.M., Metz, J.A.J., Evers, E., Leipoldt, A., 1990. A size-dependent predator prey interaction: Who pursues whom? J. Math. Biol. 28, 609–643. de Roos, A.M., Diekmann, O., Gyllenberg, M., Metz, J.A.J., Nakaoka, S., 2009. Daphnia revisited. Submitted to J. Math. Biol. Diekmann, O., van Gils, S., Verduyn Lunel, S.M., Walther, H.-O., 1995. Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York. Diekmann, O., Getto, P., Gyllenberg, M., 2007. Stability and bifurcation analysis of Volterra functional equations in the light of suns and star. SIAM J. Math. Anal. 39(4), 1023–1069. Hale, J., 1977. Functional Differential Equations. Springer, New York. Hairer, E., Nørsett, S.P., Wanner, G., 1987. Solving Ordinary Differential Equations I. Nonstiff Problems. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin. Kirkilionis, M.A., Diekmann, O., Lisser, B., Nool, M., Sommeijer, B., de Roos, A.M., 2001. Numerical continuation of equilibria of physiologically structured population models. I. Theory. Math. Mod. Meth. Appl. Sci. 11(6), 1101–1127. Kuznetsov, Y.A., 1994. Elements of Applied Bifurcation Theory. Springer, New York. Metz, J.A.J., Diekmann, O. (Eds.), 1986. The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, vol. 68. Springer, Berlin. Rosenzweig, M.L., 1971. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science (Wash., DC) 171, 385–387.
URI:	http://wrap.warwick.ac.uk/id/eprint/3221