# American Institute of Mathematical Sciences

September  2020, 13(9): 2619-2640. doi: 10.3934/dcdss.2020165

## Numerical continuation and delay equations: A novel approach for complex models of structured populations

 1 CDLab – Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics – University of Udine, via delle scienze 206, 33100 Udine, Italy 2 Department of Mathematics and Statistics – University of Helsinki, P.O. Box 68 (Pietari Kalmin katu 5) FI-00014 Helsinki, Finland 3 CDLab – Computational Dynamics Laboratory

* Corresponding author: Dimitri Breda

Received  January 2019 Revised  April 2019 Published  September 2020 Early access  December 2019

Fund Project: All the authors are members of INdAM Research group GNCS. AA is supported by the PhD program in Computer Science, Mathematics and Physics, University of Udine; DB is supported by the INdAM GNCS project "Approssimazione numerica di problemi di evoluzione: aspetti deterministici e stocastici" (2018) and by the project PSD_2015_2017_DIMA_PRID_2017_ZANOLIN "SIDIA – SIstemi DInamici e Applicazioni" (UNIUD); FS is supported by Domast (Doctoral Programme in Mathematics and Statistics), University of Helsinki, and by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

Citation: Alessia Andò, Dimitri Breda, Francesca Scarabel. Numerical continuation and delay equations: A novel approach for complex models of structured populations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2619-2640. doi: 10.3934/dcdss.2020165
##### References:
 [1] AUTO, URL http://indy.cs.concordia.ca/auto/. [2] MatCont, URL https://sourceforge.net/projects/matcont/. [3] E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, Classics in Applied Mathematics, 45. SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898719154. [4] A. Andò and D. Breda, Collocation techniques for structured populations modeled by delay equations, SEPA SIMAI series, Springer. [5] D. Breda, O. Diekmann, W. de Graaf, A. Pugliese and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454. [6] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Sys., 15 (2016), 1-23.  doi: 10.1137/15M1040931. [7] D. Breda, P. Getto, J. Sánchez Sanz and R. Vermiglio, Computing the eigenvalues of realistic Daphnia models by pseudospectral methods, SIAM J. Sci. Comput., 37 (2015), A2607–A2629. doi: 10.1137/15M1016710. [8] C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp., 19 (1965), 577-593.  doi: 10.2307/2003941. [9] H. Dankowicz and F. Schilder, Recipes for Continuation, Computational Science & Engineering, 11. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972573. [10] A. M. de Roos, PSPManalysis, URL https://cran.r-project.org/package=PSPManalysis. [11] A. M. de Roos, A gentle introduction to models of physiologically structured populations, in Structured-Population Models in Marine, Terrestrial and Freshwater Systems, Chapman and Hall, New York, (1997), 119–204. [12] A. M. de Roos, O. Diekmann, P. Getto and M. A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, B. Math. Biol., 72 (2010), 259-297.  doi: 10.1007/s11538-009-9445-3. [13] A. M. de Roos, J. A. J. Metz, E. Evers and A. Leipoldt, A size-dependent predator prey interaction: Who pursues whom?, J. Math. Biol., 28 (1990), 609-643.  doi: 10.1007/BF00160229. [14] P. Deuflhard, B. Fiedler and P. Kunkel, Efficient numerical pathfollowing beyond critical points, SIAM J. Numer. Anal., 24 (1987), 912-927.  doi: 10.1137/0724059. [15] A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MatCont: A MATLAB package for numerical bifurcation analysis of ODEs, ACM T. Math. Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362. [16] A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.  doi: 10.1080/13873950701742754. [17] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007/08), 1023-1069.  doi: 10.1137/060659211. [18] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), 277-318.  doi: 10.1007/s00285-009-0299-y. [19] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [20] E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284. [21] E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems, Understanding Complex Systems, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1. [22] J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26.  doi: 10.1016/0771-050X(80)90013-3. [23] P. Getto, M. Gyllenberg, Y. Nakata and F. Scarabel, Stability analysis of a state-dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0. [24] W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543. [25] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [26] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [27] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python, (2001), URL http://www.scipy.org/. [28] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [29] L. Petzold, Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM J. Sci. Stat. Comput., 4 (1983), 137-148.  doi: 10.1137/0904010. [30] M. J. D. Powell, A hybrid method for nonlinear equations, Numerical Methods for Nonlinear Algebraic Equations, Gordon and Breach, London, (1970), 87–114. [31] W. C. Rheinboldt and J. V. Burkardt, Algorithm 596: A program for a locally parameterized continuation process, ACM Trans. Math. Softw., 9 (1983), 236-241. [32] J. Sánchez Sanz and P. Getto, Numerical bifurcation analysis of physiologically structured populations: Consumer-resource, cannibalistic and trophic models, B. Math. Biol., 78 (2016), 1546-1584.  doi: 10.1007/s11538-016-0194-9. [33] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [34] L. N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. [35] L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. [36] L. T. Watson, S. C. Billups and A. P. Morgan, Algorithm 652: Homepack: A suite of codes for globally convergent homotopy algorithms, ACM Trans. Math. Softw., 13 (1987), 281-310.  doi: 10.1145/29380.214343.

show all references

##### References:
 [1] AUTO, URL http://indy.cs.concordia.ca/auto/. [2] MatCont, URL https://sourceforge.net/projects/matcont/. [3] E. L. Allgower and K. Georg, Introduction to Numerical Continuation Methods, Classics in Applied Mathematics, 45. SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898719154. [4] A. Andò and D. Breda, Collocation techniques for structured populations modeled by delay equations, SEPA SIMAI series, Springer. [5] D. Breda, O. Diekmann, W. de Graaf, A. Pugliese and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), 103-117.  doi: 10.1080/17513758.2012.716454. [6] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel and R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Sys., 15 (2016), 1-23.  doi: 10.1137/15M1040931. [7] D. Breda, P. Getto, J. Sánchez Sanz and R. Vermiglio, Computing the eigenvalues of realistic Daphnia models by pseudospectral methods, SIAM J. Sci. Comput., 37 (2015), A2607–A2629. doi: 10.1137/15M1016710. [8] C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp., 19 (1965), 577-593.  doi: 10.2307/2003941. [9] H. Dankowicz and F. Schilder, Recipes for Continuation, Computational Science & Engineering, 11. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. doi: 10.1137/1.9781611972573. [10] A. M. de Roos, PSPManalysis, URL https://cran.r-project.org/package=PSPManalysis. [11] A. M. de Roos, A gentle introduction to models of physiologically structured populations, in Structured-Population Models in Marine, Terrestrial and Freshwater Systems, Chapman and Hall, New York, (1997), 119–204. [12] A. M. de Roos, O. Diekmann, P. Getto and M. A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, B. Math. Biol., 72 (2010), 259-297.  doi: 10.1007/s11538-009-9445-3. [13] A. M. de Roos, J. A. J. Metz, E. Evers and A. Leipoldt, A size-dependent predator prey interaction: Who pursues whom?, J. Math. Biol., 28 (1990), 609-643.  doi: 10.1007/BF00160229. [14] P. Deuflhard, B. Fiedler and P. Kunkel, Efficient numerical pathfollowing beyond critical points, SIAM J. Numer. Anal., 24 (1987), 912-927.  doi: 10.1137/0724059. [15] A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MatCont: A MATLAB package for numerical bifurcation analysis of ODEs, ACM T. Math. Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362. [16] A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.  doi: 10.1080/13873950701742754. [17] O. Diekmann, P. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007/08), 1023-1069.  doi: 10.1137/060659211. [18] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka and A. M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), 277-318.  doi: 10.1007/s00285-009-0299-y. [19] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. [20] E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284. [21] E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, Numerical Continuation Methods for Dynamical Systems, Understanding Complex Systems, Dordrecht, (2007), 1–49. doi: 10.1007/978-1-4020-6356-5_1. [22] J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26.  doi: 10.1016/0771-050X(80)90013-3. [23] P. Getto, M. Gyllenberg, Y. Nakata and F. Scarabel, Stability analysis of a state-dependent delay differential equation for cell maturation: Analytical and numerical methods, J. Math. Biol., 79 (2019), 281-328.  doi: 10.1007/s00285-019-01357-0. [24] W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543. [25] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [26] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8. [27] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python, (2001), URL http://www.scipy.org/. [28] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [29] L. Petzold, Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM J. Sci. Stat. Comput., 4 (1983), 137-148.  doi: 10.1137/0904010. [30] M. J. D. Powell, A hybrid method for nonlinear equations, Numerical Methods for Nonlinear Algebraic Equations, Gordon and Breach, London, (1970), 87–114. [31] W. C. Rheinboldt and J. V. Burkardt, Algorithm 596: A program for a locally parameterized continuation process, ACM Trans. Math. Softw., 9 (1983), 236-241. [32] J. Sánchez Sanz and P. Getto, Numerical bifurcation analysis of physiologically structured populations: Consumer-resource, cannibalistic and trophic models, B. Math. Biol., 78 (2016), 1546-1584.  doi: 10.1007/s11538-016-0194-9. [33] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics, 57. Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [34] L. N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. [35] L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. [36] L. T. Watson, S. C. Billups and A. P. Morgan, Algorithm 652: Homepack: A suite of codes for globally convergent homotopy algorithms, ACM Trans. Math. Softw., 13 (1987), 281-310.  doi: 10.1145/29380.214343.
Pseudo-arclength continuation (top-left) with natural parameterization (top-right) and natural continuation with tangent (bottom-left) and secant (bottom-right) prediction
Equilibrium branch $\bar S(\mu)$ with zoom (left) and relevant residual (right) of the Daphnia model, computed with [6] (solid line with circles) and [32] (dashed line with diamonds). See text for more details
An example of bordered block diagonal structure of the Jacobian matrix for $n = 10$ (determining the size of the diagonal blocks) and $N = 5$ (determining the number of the diagonal blocks)
Internal (lines with circles) versus external (horizontal lines) continuation for (21): error on the true curve (22) (top) and elapsed time (bottom, $\text{s}$) using $n$ collocation points and $N = 10$ quadrature nodes. See text for more details
Internal (lines with circles) versus external continuation (lines with squares) for (21): error on the true curve (22) (top) and elapsed time (bottom, $\text{s}$) using $n = 12$ collocation points and $N$ quadrature nodes. See text for more details
Internal (lines with circles) versus external (horizontal lines) continuation for (23) and (24): error on the true curve (25) (top) and elapsed time (bottom, $\text{s}$) using $n$ collocation points and $N = 10$ quadrature nodes. See text for more details
Internal (lines with circles) versus external (horizontal lines) continuation for (26), (27) and (28): error on the true curve (29) (top) and elapsed time (bottom, $\text{s}$) using $n$ collocation points and $N = 10$ quadrature nodes. See text for more details
Internal (lines with circles) versus external (horizontal lines) continuation for (30) and (31): error on the true curve (32) (top) and elapsed time (bottom, $\text{s}$) using $n$ collocation points and $N = 10$ quadrature nodes. See text for more details
Equilibrium branch $\bar S(\mu)$ with zoom (left) and relevant residual (right) of the Daphnia model, computed with the internal continuation (dash-dot line with stars), superposed to Figure 2 for comparison. See text for more details
Rates (top) and parameters (bottom) of the considered Daphnia model
 resource intrinsic rate of change $f(S)=a_{1}S(1-S/C)$ consumer growth rate $g(\xi, S)=\gamma_{g}\left(\xi_{m}f_{r}(S)-\xi\right)$ consumer mortality rate $\mu(\xi, S)=\mu$ consumer adults reproduction rate $\beta(\xi, S)=r_{m}f_{r}(S)\xi^{2}$ consumer ingestion rate $\gamma(\xi, S)=\nu_{S}f_{r}(S)\xi^{2}$ Holling type Ⅱ functional response $f_{r}(S):=\sigma S/(1+\sigma S)$ size at birth $\xi_b=0.8$ size at maturation $\xi_{A}=2.5$ maximum size $\xi_{m}=6.0$ growth time constant $\gamma_{g}=0.15$ functional response shape parameter $\sigma=7.0$ maximum feeding rate $\nu_{S}=1.8$ maximum reproduction rate $r_{m}=0.1$ mortality rate parameter $\mu=\$varying environment carrying capacity $C=25$ flow-through rate $a_{1}=0.5$ maximum age $a_{\max}=70$
 resource intrinsic rate of change $f(S)=a_{1}S(1-S/C)$ consumer growth rate $g(\xi, S)=\gamma_{g}\left(\xi_{m}f_{r}(S)-\xi\right)$ consumer mortality rate $\mu(\xi, S)=\mu$ consumer adults reproduction rate $\beta(\xi, S)=r_{m}f_{r}(S)\xi^{2}$ consumer ingestion rate $\gamma(\xi, S)=\nu_{S}f_{r}(S)\xi^{2}$ Holling type Ⅱ functional response $f_{r}(S):=\sigma S/(1+\sigma S)$ size at birth $\xi_b=0.8$ size at maturation $\xi_{A}=2.5$ maximum size $\xi_{m}=6.0$ growth time constant $\gamma_{g}=0.15$ functional response shape parameter $\sigma=7.0$ maximum feeding rate $\nu_{S}=1.8$ maximum reproduction rate $r_{m}=0.1$ mortality rate parameter $\mu=\$varying environment carrying capacity $C=25$ flow-through rate $a_{1}=0.5$ maximum age $a_{\max}=70$
Computational time and maximal residual for the continuation of the Daphnia model
 method computational time maximal residual [6] ${155.88}\ \text{s}$ $7.6652\times10^{-4}$ [32] ${59.32}\ \text{s}$ $4.6768\times10^{-6}$ internal continuation with $n=N=10$ ${1.73}\ \text{s}$ $8.1723\times10^{-3}$ internal continuation with $n=N=15$ ${4.40}\ \text{s}$ $3.6517\times10^{-5}$ internal continuation with $n=N=20$ ${9.18}\ \text{s}$ $3.6854\times10^{-7}$
 method computational time maximal residual [6] ${155.88}\ \text{s}$ $7.6652\times10^{-4}$ [32] ${59.32}\ \text{s}$ $4.6768\times10^{-6}$ internal continuation with $n=N=10$ ${1.73}\ \text{s}$ $8.1723\times10^{-3}$ internal continuation with $n=N=15$ ${4.40}\ \text{s}$ $3.6517\times10^{-5}$ internal continuation with $n=N=20$ ${9.18}\ \text{s}$ $3.6854\times10^{-7}$
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