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Collocation techniques for structured populations modeled by delay equations. (English) Zbl 1447.37077
Aguiar, Maira (ed.) et al., Current trends in dynamical systems in biology and natural sciences. Selected contributions presented at the ninth international workshop of dynamical systems applied to biology and natural sciences, DSABNS, Turin, Italy, February 7–9, 2018. Cham: Springer. SEMA SIMAI Springer Ser. 21, 43-62 (2020).
The authors in this paper study coupled equations such as the Daphnia model [O. Diekmann et al., J. Math. Biol. 61, No. 2, 277–318 (2010; Zbl 1208.92082)], which is an age-structured population model competing for an unstructured resource. This model, named after the organisms in the genus Daphnia, can be written as a coupled system of integral equations whose function terms are themselves solutions to an initial value problem ordinary differential equations.
The perspective is through continuation or homotopy methods. The primary problem stated by the authors is that in real-world models, solving the ordinary differential equations defining the terms in the coupled system is often a computationally inefficient procedure, since the ordinary differential equations are numerically resolved over each new iteration of the continuation parameter.
The authors combine this with polynomial collocation to quickly find solutions. Collocation is a method for solving ODEs which looks at a restricted space of functions such as polynomial (in polynomial collocation) that satisfy the ODE system at certain collocation points.
The main thrust of this paper is a series of demonstrations, starting with a simplified Daphnia model, on how one would implement polynomial collocation and continuation. They explore a variety of problems, including computing periodic solutions of models like the Daphnia model. In the penultimate section, explicit examples are demonstrated using the delayed logistic equation and a renewal equation for which an exact period solution is known.
In this work, the authors have not yet applied their method to the full Daphnia model but state that they plan to do this in a future work. They say they also plan to explore the error and convergence analysis when it comes to computing periodic solutions. In this realm, and if their method can be implemented in various popular software packages, it may provide a welcomed improvement to numerical simulation of differential equations. Personally, I would also like to see some example comparisons in terms of actual time taken per computation with various methods, to gain an intuitive understanding of the efficiency gain with a practical implementation.
For the entire collection see [Zbl 1445.92001].
MSC:
 37N25 Dynamical systems in biology 92D25 Population dynamics (general) 92B05 General biology and biomathematics 34K13 Periodic solutions to functional-differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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