On the performance of four methods for the numerical solution of ecologically realistic size‐structured population models

Abstract

Abstract• Size-structured population models (SSPMs) are widely used in ecology to account for intraspecific variation in body size. Three characteristic features of size-structured populations are the dependence of life histories on the entire size distribution, intrinsic population renewal through the birth of new individuals, and the potential accumulation of individuals with similar body sizes due to determinate or stunted growth. Because of these three features, numerical methods that work well for structurally similar transport equations may fail for SSPMs and other transport-dominated models with high ecological realism, and thus their computational performance needs to be critically evaluated.• Here, we compare the performance of four numerical solution schemes, the fixed-mesh upwind (FMU) method, the moving-mesh upwind (MMU) method, the characteristic method (CM), and the Escalator Boxcar Train (EBT) method, in numerically solving three reference problems that are representative of ecological systems in the animal and plant kingdoms. The MMU method is here applied for the first time to SSPMs, whereas the three other methods have been employed by other authors.• Our results show that the EBT method performs best, except for one of the three reference problems, in which size-asymmetric competition affects individual growth rates. For that reference problem, the FMU method performs best, closely followed by the MMU method. Surprisingly, the CM method does not perform well for any of the three reference problems.• We conclude that life-history features should be considered when choosing numerical method.

libpspm: A feature-rich numerical package for solving physiologically structured population models

“…For the RED model, the CM and EBT vastly outperform the upwind methods, with an order-of-magnitude higher accuracy for the same execution times. The CM method has the worst, almost unacceptable, performance for the Daphnia model, as was also reported by (Zhang et al, 2017). Surprisingly, however, it emerges as the best performing method for the RED model.…”

“…Thus, we have a total of eight solver methods: (1-2) explicit and semiimplicit fixed-mesh upwind methods (FMU and IFMU), (3) semi-implicit linear upwind differencing method (ILUD), (4-5) explicit and semi-implicit characteristic methods (CM and ICM), (6-7) explicit and semi-implicit EBT methods (EBT and IEBT), and (8) agent-based method (ABM). A conceptual description of the FMU, EBT, and CM methods can be found in (Zhang et al, 2017). A full description and implementation details of all the methods can be found in the Supplementary Information.…”

On models of physiologically structured populations and their reduction to ordinary differential equations

“…But physiologically structured population models also have disadvantages. There are only few numerical tools for their study, see Brännström et al (2013), Carrillo et al (2014), Breda et al (2016), Aye and Carlsson (2017), Zhang et al (2017), de Roos (2018 and Scarabel (2018). In this respect the situation for population models formulated in terms of ODEs is infinitely better.…”

Disturbance-generated competitive coexistence

“…See Supplemental when species j, which is still governed by Lotka-Volterra Here instead we use a 2 nd order method (code available at [41]) which uses a Flux Limiter approach to limit said numerical diffusion [42,43]. This method is more accurate than the EBT and has been studied in the context of size-structured population models [43][44][45][46][47][48][49][50]. Numerical experiments suggest this scheme is 2 nd order convergent in a smooth setting and the scheme has been shown to behave intuitively in the presence of discontinuities and singularities [42].…”