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.
Similar content being viewed by others
References
Alarcón T, Getto Ph, Nakata Y (2014) Stability analysis of a renewal equation for cell population dynamics with quiescence. SIAM J Appl Math 74(4):1266–1297
Allgower EL, Georg K (2003) Introduction to numerical continuation methods. SIAM Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia
Ascher UM, Mattheij RMM, Russell RD (1995) Numerical solution of boundary value problems for ordinary differential equations. SIAM Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia
Boldin B (2006) Introducing a population into a steady community: the critical case, the center manifold, and the direction of bifurcation. SIAM J Appl Math 66(4):1424–1453
Breda D, Maset S, Vermiglio R (2009) Trace-DDE: a tool for robust analysis and characteristic equations for delay differential equations. In: Loiseau JJ, Michiels W, Niculescu SI, Sipahi R (eds) Topics in time delay systems: analysis, algorithms, and control. Lecture notes in control and information sciences, vol 388. Springer, New York, pp 145–155
Breda D, Getto Ph, Sánchez Sanz J, Vermiglio R (2015) Computing the eigenvalues of realistic Daphnia models by pseudospectral methods. SIAM J Sci Comput 37(6):2607–2629
Calsina À, Saldaña J (1995) A model of physiologically structured population dynamics with a nonlinear individual growth rate. J Math Biol 33:335–364
Claessen D, de Roos AM (2003) Bistability in a size-structured population model of cannibalistic fish—a continuation study. Theor Popul Biol 64:49–65
Claessen D, de Roos AM, Persson L (2004) Population dynamic theory of size-dependent cannibalism. Proc Biol Sci 271(1537):333–340
de Roos AM, Persson L (2002) Size-dependent life-history traits promote catastrophic collapses of top predators. Proc Natl Acad Sci 99(20):12,907–12,912
de Roos AM, Persson L (2013) Population and community ecology of ontogenetic development. No. 51 in monographs in population biology. Princeton University Press, Princeton
de Roos AM, Metz JAJ, Evers E, Leipoldt A (1990) A size dependent predator-prey interaction: who pursues whom? J Math Biol 28:609–643
de Roos AM, Diekmann O, Getto Ph, Kirkilionis MA (2010) Numerical equilibrium analysis for structured consumer resource models. Bull Math Biol 72:259–297
Dhooge A, Govaerts W, Kuznetsov YA, Mestrom W, Riet AM, Sautois B (2006) MATCONT and CL_ MATCONT: continuation toolboxes in MATLAB. User guide. http://www.matcontugentbe/manualpdf
Diekmann O, Gyllenberg M (2012) Equations with infinite delay: blending the abstract and the concrete. J Differ Equ 252:819–851
Diekmann O, Korvasova K (2016) Linearization of solution operators for state-dependent delay equations: a simple example. Discret Contin Dyn Syst 36:137–149
Diekmann O, Gyllenberg M, Huang H, Kirkilionis M, Metz JAJ, Thieme HR (2001) On the formulation and analysis of general deterministic structured population models II. Nonlinear theory. J Math Biol 43:157–189
Diekmann O, Gyllenberg M, Metz JAJ (2003) Steady-state analysis of structured population models. Theor Popul Biol 63:309–338
Diekmann O, Getto Ph, Gyllenberg M (2007) Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J Math Anal 39(4):1023–1069
Diekmann O, Gyllenberg M, Metz JAJ, Nakaoka S, de Roos AM (2010) Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J Math Biol 61:277–318
Dormand JR, Prince PJ (1980) A family of embedded Runge–Kutta formulae. J Comput Appl Math 6:19–26
Engelborghs K, Luzyanina T, Samaey G (2001) PDDE-BIFTOOL v. 2.00: a MATLAB package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, KU Leuven, Leuven, Belgium
Getto Ph, Diekmann O, de Roos AM (2005) On the (dis) advantages of cannibalism. J Math Biol 51:695–712
Hairer E, Norsett SP, Wanner G (1993) Solving ordinary differential equations I. Nonstiff problems, 2nd edn. Springer Series in Computational Mathematics, Springer, Berlin
Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. No. 99 in applied mathematical sciences. Springer, New York
Kelley C (1995) Iterative methods for linear and nonlinear equations. No. 16 in frontiers in applied mathematics. SIAM, Philadelphia
Kirkilionis MA, Diekmann O, Lisser B, Nool M, Sommejier B, de Roos AM (2001) Numerical continuation of equilibria of physiologically structured population models I. Theory Math Mod Meth Appl Sci 11(6):1101–1127
Kuznetsov YA (2004) Elements of applied bifurcation theory, 3rd edn. No. 112 in applied mathematical sciences. Springer, New York
McCauley E, Nisbet RM, Murdoch WW, de Roos AM, Gurney WSC (1999) Large-amplitude cycles of daphnia and its algal prey in enriched environments. Lett Nat 402:653–656
Meng X, Lundström NLP, Bodin M, Brännström A (2013) Dynamics and management of stage-structured fish stocks. Bull Math Biol 75:1–23
Perko L (2001) Differential equations and dynamical systems, 3rd edn. No. 7 in texts in applied mathematics. Springer, New York
van den Bosch F, de Roos AM, Gabriel W (1988) Cannibalism as a life boat mechanism. J Math Biol 26:619–633
Zhang L, Lin Z, Pedersen M (2012) Effects of growth curve plasticity on size-structured population dynamics. Bull Math Biol 74:327–345
Acknowledgments
For helpful discussions, we thank Odo Diekmann and Andre de Roos.
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-016-0194-9
Keywords
- Numerical bifurcation analysis
- Equilibria
- Curve continuation
- Structured populations
- Consumer–resource
- Cannibalism
- Trophic