×

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.

MSC:

92D25 Population dynamics (general)
65P30 Numerical bifurcation problems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Alarcón, T; Getto, Ph; Nakata, Y, Stability analysis of a renewal equation for cell population dynamics with quiescence, SIAM J Appl Math, 74, 1266-1297, (2014) · Zbl 1320.37040
[2] Allgower EL, Georg K (2003) Introduction to numerical continuation methods. SIAM Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia · Zbl 1036.65047
[3] 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 · Zbl 0843.65054
[4] Boldin, B, Introducing a population into a steady community: the critical case, the center manifold, and the direction of bifurcation, SIAM J Appl Math, 66, 1424-1453, (2006) · Zbl 1096.37054
[5] Breda, D; Maset, S; Vermiglio, R; Loiseau, JJ (ed.); Michiels, W (ed.); Niculescu, SI (ed.); Sipahi, R (ed.), Trace-DDE: a tool for robust analysis and characteristic equations for delay differential equations, No. 388, 145-155, (2009), New York
[6] Breda, D; Getto, Ph; Sánchez Sanz, J; Vermiglio, R, Computing the eigenvalues of realistic daphnia models by pseudospectral methods, SIAM J Sci Comput, 37, 2607-2629, (2015) · Zbl 1335.92072
[7] Calsina, À; Saldaña, J, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J Math Biol, 33, 335-364, (1995) · Zbl 0828.92025
[8] Claessen, D; Roos, AM, Bistability in a size-structured population model of cannibalistic fish—a continuation study, Theor Popul Biol, 64, 49-65, (2003) · Zbl 1103.92040
[9] Claessen, D; Roos, AM; Persson, L, Population dynamic theory of size-dependent cannibalism, Proc Biol Sci, 271, 333-340, (2004)
[10] Roos, AM; Persson, L, Size-dependent life-history traits promote catastrophic collapses of top predators, Proc Natl Acad Sci, 99, 12,907-12,912, (2002)
[11] de Roos AM, Persson L (2013) Population and community ecology of ontogenetic development. No. 51 in monographs in population biology. Princeton University Press, Princeton
[12] Roos, AM; Metz, JAJ; Evers, E; Leipoldt, A, A size dependent predator-prey interaction: who pursues whom?, J Math Biol, 28, 609-643, (1990) · Zbl 0718.92026
[13] Roos, AM; Diekmann, O; Getto, Ph; Kirkilionis, MA, Numerical equilibrium analysis for structured consumer resource models, Bull Math Biol, 72, 259-297, (2010) · Zbl 1185.92088
[14] 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 · Zbl 1237.34133
[15] Diekmann, O; Gyllenberg, M, Equations with infinite delay: blending the abstract and the concrete, J Differ Equ, 252, 819-851, (2012) · Zbl 1237.34133
[16] Diekmann, O; Korvasova, K, Linearization of solution operators for state-dependent delay equations: a simple example, Discret Contin Dyn Syst, 36, 137-149, (2016) · Zbl 1371.37046
[17] Diekmann, O; Gyllenberg, M; Huang, H; Kirkilionis, M; Metz, JAJ; Thieme, HR, On the formulation and analysis of general deterministic structured population models II. nonlinear theory, J Math Biol, 43, 157-189, (2001) · Zbl 1028.92019
[18] Diekmann, O; Gyllenberg, M; Metz, JAJ, Steady-state analysis of structured population models, Theor Popul Biol, 63, 309-338, (2003) · Zbl 1098.92062
[19] Diekmann, O; Getto, Ph; Gyllenberg, M, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J Math Anal, 39, 1023-1069, (2007) · Zbl 1149.39021
[20] Diekmann, O; Gyllenberg, M; Metz, JAJ; Nakaoka, S; Roos, AM, Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example, J Math Biol, 61, 277-318, (2010) · Zbl 1208.92082
[21] Dormand, JR; Prince, PJ, A family of embedded Runge-Kutta formulae, J Comput Appl Math, 6, 19-26, (1980) · Zbl 0448.65045
[22] 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 · Zbl 1096.37054
[23] Getto, Ph; Diekmann, O; Roos, AM, On the (dis) advantages of cannibalism, J Math Biol, 51, 695-712, (2005) · Zbl 1077.92047
[24] Hairer E, Norsett SP, Wanner G (1993) Solving ordinary differential equations I. Nonstiff problems, 2nd edn. Springer Series in Computational Mathematics, Springer, Berlin · Zbl 0789.65048
[25] Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. No. 99 in applied mathematical sciences. Springer, New York · Zbl 0787.34002
[26] Kelley C (1995) Iterative methods for linear and nonlinear equations. No. 16 in frontiers in applied mathematics. SIAM, Philadelphia · Zbl 0832.65046
[27] Kirkilionis, MA; Diekmann, O; Lisser, B; Nool, M; Sommejier, B; Roos, AM, Numerical continuation of equilibria of physiologically structured population models I, Theory Math Mod Meth Appl Sci, 11, 1101-1127, (2001) · Zbl 1013.92036
[28] Kuznetsov YA (2004) Elements of applied bifurcation theory, 3rd edn. No. 112 in applied mathematical sciences. Springer, New York · Zbl 1082.37002
[29] McCauley, E; Nisbet, RM; Murdoch, WW; Roos, AM; Gurney, WSC, Large-amplitude cycles of daphnia and its algal prey in enriched environments, Lett Nat, 402, 653-656, (1999)
[30] Meng, X; Lundström, NLP; Bodin, M; Brännström, A, Dynamics and management of stage-structured fish stocks, Bull Math Biol, 75, 1-23, (2013) · Zbl 1402.92356
[31] Perko L (2001) Differential equations and dynamical systems, 3rd edn. No. 7 in texts in applied mathematics. Springer, New York · Zbl 0973.34001
[32] Bosch, F; Roos, AM; Gabriel, W, Cannibalism as a life boat mechanism, J Math Biol, 26, 619-633, (1988) · Zbl 0714.92019
[33] Zhang, L; Lin, Z; Pedersen, M, Effects of growth curve plasticity on size-structured population dynamics, Bull Math Biol, 74, 327-345, (2012) · Zbl 1238.92036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.