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**Numerical bifurcation analysis of physiologically structured populations: consumer-resource, cannibalistic and trophic models.**
*(English)*
Zbl 1367.92108

In this programmatic paper, the authors are concerned with developing an abstract framework for formulating models describing the dynamics of physiologically structured populations and analyzing their bifurcations from a numerical viewpoint. This framework is understood as a coupling between systems of Volterra functional equations (arising from renewal equations) and ODEs, and incorporates both structured and unstructured populations, the former being characterized by certain interaction variables. Also, the framework describes both the individual i-level and the population p-level.

The renewal equations are formulated for the birth rates at the p-level and for the interaction variables. Linearities in the dynamics of the unstructured p-variables are exploited in order to reduce dimensions.

A numerical curve continuation method based on Euler tangent prediction and correction of the predicted point, and an ODE solver are presented, along with a method for computing the equilibrium branches. The computation of bifurcation points and bifurcation curves is one of the main objectives. Two types of transcritical bifurcation points are considered and defined in terms of the intersecting branches. A new approach to derive alternate test functions and to detect bifurcations is introduced, and detailed evaluation algorithms are presented in an appendix. Saddle node bifurcations are also considered.

The above-defined framework is then applied to consumer-resource models describing trees competing for light in a forest and Daphnia consuming Algae, respectively, to a predator-prey-resource food chain and to a model of cannibalism in fish populations. The abstract results are thereby validated through their application to relevant realistic models.

The renewal equations are formulated for the birth rates at the p-level and for the interaction variables. Linearities in the dynamics of the unstructured p-variables are exploited in order to reduce dimensions.

A numerical curve continuation method based on Euler tangent prediction and correction of the predicted point, and an ODE solver are presented, along with a method for computing the equilibrium branches. The computation of bifurcation points and bifurcation curves is one of the main objectives. Two types of transcritical bifurcation points are considered and defined in terms of the intersecting branches. A new approach to derive alternate test functions and to detect bifurcations is introduced, and detailed evaluation algorithms are presented in an appendix. Saddle node bifurcations are also considered.

The above-defined framework is then applied to consumer-resource models describing trees competing for light in a forest and Daphnia consuming Algae, respectively, to a predator-prey-resource food chain and to a model of cannibalism in fish populations. The abstract results are thereby validated through their application to relevant realistic models.

Reviewer: Paul Georgescu (Iaşi)

### Keywords:

structured populations; consumer-resource models; trophic models; cannibalism; bifurcations; Volterra functional equations; numerical analysis
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\textit{J. Sánchez Sanz} and \textit{P. Getto}, Bull. Math. Biol. 78, No. 7, 1546--1584 (2016; Zbl 1367.92108)

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### References:

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