# 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 & Continuous Dynamical Systems - S, 2020, 13 (9) : 2619-2640. doi: 10.3934/dcdss.2020165
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##### References:
Pseudo-arclength continuation (top-left) with natural parameterization (top-right) and natural continuation with tangent (bottom-left) and secant (bottom-right) prediction
] (solid line with circles) and [32] (dashed line with diamonds). See text for more details">Figure 2.  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
for comparison. See text for more details">Figure 9.  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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