Division of Mathematical Ecology
Modeling is a principal tool - perhaps the primary tool - for studying the behavior of large, complex systems such as environmental or social systems. Investigations of environmental change require analyses of processes involved on a detailed level, integrated models that can predict environmental response over a wide range of space and time scales, and the capacity to translate those predictions into a format from which people can make decisions. The objectives of any proposed models can be divided into two categories: prediction (and control) and explanation (this includes simulation of the behavior of the true system under different conditions to those which prevailed during the data collection exercise, the estimation of parameters which have a physical interpretation, and the evaluation of different control strategies). Environmental systems present considerable difficulties for modelers. Compared with engineered systems, for example, these difficulties arise from several factors. One is the sheer complexity of internal system behavior arising from dynamic and multidimensional interactions as well as external actions that are physical, and possibly chemical and biological. Another factor is that interactions may occur in more than one medium, each of which is heterogeneous. In general, knowledge of the subsystems and the interrelations of the components of an ecosystem under investigation is often not sufficient for an adequate model simulation. On the other hand, ecosystems are almost always hierarchical. The important property, for our purposes, of such hierarchical systems is that the behavior of the units at any specific level can be described and explained without a need for a detailed picture of the structures and behavior at the levels below. Only gross, aggregated features of the layer below show up in the next layer above. The members of the Division see their major task in providing methods to integrate the given knowledge of the different subsystems relevant to Ecosystems Dynamic into well-founded and endurable results to support the formulation of policy options for global and local environmental management. The following types of environmental models have been developed by the members of Division - Models of genetic structure of population
- Model of population dynamics
- Models of ontogenesis of higher plants
- Models of pollutants' dynamics in the ecosystems
- Models of soil carbon dynamics in natural and land-used ecosystems.
The major environmental problems are dealing with the systems of ecological "macroworld" represented with "emergent properties" that are of ultimate biological nature and that are impossible to be described with the equations of physics and chemistry. The possible approach to modeling of "macroworld" environmental systems is the integrated modeling approach. Integration means capturing as much as possible of cause-effect relationships and describing them with an operator of transition, or "input-output" function. In our study we propose the method of response functions as a method of the construction of purposeful, credible integrated models from data and prior knowledge or information. The data are usually time series observations of system inputs and outputs, and sometimes of internal states. Prior knowledge available may include conservation laws, idealized physical equations, model parameters and noise values, the nature of the system response and hence possible parametrizations. The method of response functions implies credible models in the sense that they are identifiable, and, hopefully, explain system output behavior satisfactorily. This approach gives us a possibility to take into account all essential features of ecosystem processes: complexity, multidimensioness, uncertainty, irreducability, and so on.
Last book:
In the mathematical modelling of a process, observation of data is used both to specify or refine the mathematical model and to analyse this by comparison with existing or new mathematical tools. The model will ultimately be used to predict or prescribe the future dynamics of the desired process and it is therefore essential to verify its performance over an actual course of events.
This volume presents a new approach to the construction of ecological models based on a generalised version of the response function method. Using this, all essential features of ecosystem processes, such as complexity, unknown mechanisms, multidimensionality, uncertainty, and irreducibility, can be taken into account. The authors apply the method to a variety of environmental problems and a CD-ROM containing demonstration versions of four models discussed is included. Last paper:
Abstract: The biggest advantage of Lyapunov's function method is that it is "direct", as
"increase along orbits", hence one does not need to solve the equation explicitly. The main
goal of this paper is to give the short overview of the existing two types: fitness-like and
entropy-like Lyapunov functions for generalized Lotka-Volterra systems and to discuss
correspondence between the structure of the interaction matrix and these Lyapunov
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