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.
Highlights
Ecological modelling based on non linear differential systems of equations can be divided into two main categories [1]
Geometric numerical integration should play a crucial role in the analysis of ecosystem models described by systems of differential equations
Flow structures that more frequently arise in ecological modelling and that have to be preserved are Poisson maps, dynamics evolving in positive phase-space, preservation of invariants
Summary
Ecological modelling based on non linear differential systems of equations can be divided into two main categories [1]. Due to the large amount of uncertainty contained in ecological data, the approach of conceptual modelling is an effective alternative aimed to understand the main features of the ecological dynamics by making scenario analysis [2] In both approaches, the mathematical model is based on governing laws described by non linear differential systems of equations. The approximated solutions should be able to reproduce the main physical qualitative characteristics of the observed quantities (e.g., positive concentrations, mass/energy conservation) in order to make accurate previsions or outline realistic scenarios For this reason, geometric numerical integration plays a crucial role in the analysis of ecosystem models described by systems of differential equations. As future perspective, we will present some examples of ecological models featured by both Poisson and biochemical structure [17] and we will discuss about some open questions related to their numerical approximation
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