A connection between discrete individual-based and continuous population-based models: A forest modelling case study

To represent, analyse and discuss aspects or ideas related to biological systems from a general perspective modelling is a necessary tool. With this in mind, teaching science and doing research based on the elaboration of models is widely accepted in the academic community.

The discussion among theoretical biologists on reductionism versus holism has been featured in many publications. Reductionists break systems down to their smallest blocks of interest, trying to recover large-scale phenomena from small- scale mechanisms. Those with a holistic view prefer to deal with aggregated properties of systems first, and then add complexity by considering the inner structure of the system. These different views of theory construction are mirrored in any research field; there are those who would rather build highly simplified models first, adding later new variables in order to increase the realism, while others prefer to start by including every aspect of the system, perhaps eventually discarding many of them as the modelling process proceeds [1-4]. The alternative views may thus converge towards similar models but not necessarily.

For instance, most of the population biology theory arose from very simple differential equations where a single variable represents population density; solutions of these are analyzed mathematically, and potentially compared to abundance estimates from field or lab observations [5, 6]. Although these models have had a great influence on ecological theory, their aggregated form is particularly difficult to relate to observational biology. As this aggregated view of a population is highly simplified, many extensions have been made to incorporate (1) size, age or physiological structure (leading to coupled systems of differential equations); (2) space (leading to metapopulation models in which a population is broken down into distinct patches, or in the case of continuous space to partial differential equations); (3) discrete generations (leading to difference equation models and matrix models); (4) stochastic effects (leading to birth and death processes and stochastic differential equations).

As extensions are added, the models become less analytically tractable and considerably harder to analyze except numerically, and not always with enough confidence and success [7, 8]. However, the use of such extensions may well be required, to ensure the models are sufficiently realistic to be applicable to specific problems in managing natural systems. Modern theory construction should not be bound by the limits of analytical mathematics. The new field of computational ecology is an attempt to combine more realistic models of ecological systems with the often large data sets available to aid in managing these systems, utilizing techniques of modern computational science to manage the data, visualize model behaviour, and statistically analyze the complex dynamics which arise [9-12]. This often involves the use of Geographic Information Systems to provide underlying static or dynamic maps of abiotic and biotic factors, which are of importance in the natural system for specific interest [13]. Computational ecology will further intensify as a necessary way to analyze complex problems of natural system management involving the coupling of detailed, spatially- explicit ecological models with physical models for abiotic components and the attendant effects on the system of human actions.

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(Author: Pablo Gómez-Mourelo, Marta Ginovart

Published by Sciedu Press)