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Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models

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Abstract

With the aim of applying numerical methods, we develop a formalism for physiologically structured population models in a new generality that includes consumer–resource, cannibalism and trophic models. The dynamics at the population level are formulated as a system of Volterra functional equations coupled to ODE. For this general class, we develop numerical methods to continue equilibria with respect to a parameter, detect transcritical and saddle-node bifurcations and compute curves in parameter planes along which these bifurcations occur. The methods combine curve continuation, ODE solvers and test functions. Finally, we apply the methods to the above models using existing data for Daphnia magna consuming Algae and for Perca fluviatilis feeding on Daphnia magna. In particular, we validate the methods by deriving expressions for equilibria and bifurcations with respect to which we compute errors, and by comparing the obtained curves with curves that were computed earlier with other methods. We also present new curves to show how the methods can easily be applied to derive new biological insight. Schemes of algorithms are included.

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Acknowledgments

For helpful discussions, we thank Odo Diekmann and Andre de Roos.

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Correspondence to Julia Sánchez Sanz.

Additional information

The research of Julia Sánchez Sanz was funded by Ministerio de Economía y Competitividad, Gobierno de España (MINECO) under the FPI Grant BES-2011-047867, the research of Philipp Getto by the Deutsche Forschungsgemeinschaft (DFG) under the project “Delay Equations and Structured Population Dynamics.” Julia Sánchez Sanz and Philipp Getto received additional support from the MINECO under the project MTM-2010-18318, Julia Sánchez Sanz by the MINECO under internship Grant EEBB-I-2013-05933, project MTM2013-46553-C3-1-P and the Severo Ochoa excellence accreditation SEV-2013-0323, Philipp Getto from the ERC Starting Grant 658 No. 259559 and the Fields Institute for Research in Mathematical Sciences under the Short Thematic Program on Delay Differential Equations.

Appendix

Appendix

This appendix contains the pseudo-code schemes of the algorithms that correspond to the numerical methods presented in Section 5. We here use the pseudo-code language established in Allgower and Georg (2003). Before the continuation of an equilibrium or a bifurcation, Algorithm 1 reduces the dimension of \(u_0\) to obtain \(\hat{u}_0\). Under one-parameter variation, Algorithm 6 computes equilibrium curves, where \(\hat{H}(\hat{u}_i)\) is obtained with Algorithm 3, and the predicted point \(\hat{v}_{i+1}\) with Algorithm 4. \(R_0(I,E,p)\) and \(\varTheta (I,E,p)\) are computed with Algorithm 2. For detecting saddle-node bifurcations, we use as test function the last component of \(t_i\) obtained with Algorithm 4, and for transcritical bifurcations the output of Algorithm 5. Under two-parameter variation, Algorithm 9 computes bifurcation curves, where \(\hat{L}(\hat{u}_i)\) is obtained with Algorithm 7 for transcriticals and with Algorithm 8 for saddle-nodes.

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Sánchez Sanz, J., Getto, P. Numerical Bifurcation Analysis of Physiologically Structured Populations: Consumer–Resource, Cannibalistic and Trophic Models. Bull Math Biol 78, 1546–1584 (2016). https://doi.org/10.1007/s11538-016-0194-9

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