Geometric Numerical Integration in Ecological Modelling

Abstract

A major neglected weakness of many ecological models is the numerical method used to solve the governing systems of differential equations. Indeed, the discrete dynamics described by numerical integrators can provide spurious solution of the corresponding continuous model. The approach represented by the geometric numerical integration, by preserving qualitative properties of the solution, leads to improved numerical behaviour expecially in the long-time integration. Positivity of the phase space, Poisson structure of the flows, conservation of invariants that characterize the continuous ecological models are some of the qualitative characteristics well reproduced by geometric numerical integrators. In this paper we review the benefits induced by the use of geometric numerical integrators for some ecological differential models.

Using awareness to Z-control a SEIR model with overexposure: Insights on Covid-19 pandemic

“…In fact, as extensively reported in the literature, the use of positive integrators much improve the results whenever positive quantities (i.e. density, concentration) need to be numerically simulated [52] , [53] , [54] , [55] , [56] , [57] .…”

Making spatial-temporal marine ecosystem modelling better – A perspective

“…The literature provides many suggestions for metrics and their application to assess the skill of models to reproduce observations, as reviewed by Bennett et al (2013) and Hipsey et al (2020) . Traditional skill metrics encompass univariate and multivariate statistical approaches ( Diele and Marangi, 2020 ; Matott et al, 2009 ; Stow et al, 2009 ), but their focus on quantifying adherence to observations makes statistical approaches less useful to explain ecological behaviour ( Olsen et al, 2016 ; Pennekamp et al, 2017 ). Therefore, to obtain deeper insights into the ecological patterns and processes within MEM output, system-wide metrics should be included which assess emergent properties such as ecological patterns in marine food webs, network structures ( Fath et al, 2019 ), and commonly accepted ecological indicators ( Coll and Steenbeek, 2017 ; Olsen et al, 2016 ).…”

“…Assessing ecological realism of state variables within a running MEM is much trickier, and leaves open the debate whether a model's ability to reproduce observed trends is ecologically realistic, or a merely numerical artefact ( Arhonditsis and Brett, 2004 ), exacerbated by the general equifinal ( Hipsey et al, 2020 ) and underdetermined ( Anderson et al, 2010 ) nature of ecosystem models. Safeguarding internal ecological realism can be improved through careful selection of the geometric numerical integrations used within a MEM ( Diele and Marangi, 2020 ). For MEMs that are open to code modifications, internal state variables (for instance, related to consumption, displacement, recruitment, niches, mortalities, etc.)…”

Locally Exact Integrators for the Duffing Equation

“…Remark 1. The discrete gradient method and its modifications can be also directly applied to the so-called M-systems popular in ecological modelling [26]. The obtained integrators preserve exactly all trajectories in the two-dimensional phase space and the locally exact modification yields more accurate time dependence along the trajectories, especially in the neighbourhood of the stable equilibrium.…”