ENVIRONMENTAL MODELLING

The course is included in the Environmental Sciences master programme, curriculum on ""Global Environmental Change",GEC. The main objective of the programme is to train highly skilled professionals, who could put into use their interdisciplinary knowledge for identifying and solving complex environmental problems. In particular, the GEC curriculum aims at providing an integrated and systemic view of economic and environmental dynamics, in order to enable master graduates to deal with global environmental challenges, in the context of sustainable development. To this regard, the challenges connected to adaptation and mitigation to Climate Changes will be one of the major focus of the curriculum. In this context, this course provides the basic knowledge for describing, using quantitative tools, and simulating the evolution of dynamic systems, such as the economic and environmental ones, by means of mechanistic models. These tools allow one to turn conceptual models into mathematical ones, whose output can be compared with field observations, thus enabling the elucidation of the mechanisms which generate the system output, on the basis of the flow of energy and matter at its boundaries and of the interactions among its main variables. This systemic approach is of paramount importance for developing predictive models, which could orient the implementation of environmental policies, including those concerning the adaptation to and the mitigation of Climate Changes.

Attending the classes, thoroughly studying the course notes and reading materials, carrying out homework and assignments will enable students to achieve the following learning outcomes.
1) Knowledge and comprehension
Knowledge of the terminology and of the main concepts of dynamic system theory. In line with the learning objectives of the programme, this knowledge will enable students to characterize and model the temporal evolution of environmental and economic data, thus providing a back ground for science -based policy making and management actions.
Understanding the relevance of the systemic approach in the investigation of complex systems, in the description of their dynamics sing process-based models, in the forecasting of their likely evolution.
2) Capacity of applying knowledge and comprehension
To be able to apply dynamic system theory to develop a range of models for simulating, e.g. the evolution of exploited renewable resources, marine food web dynamics, impacts of pollutants on biogeochemical cycles ....
To be able to plan management actions, aimed at mitigating the impacts of local and global antropic pressures.
3) Assessment capacity
To be able of assessing the environmental benefits brought about by the implementation of alternative scenarios of management actions, such as wildlife restoking, reduction of the loads of pollutants and nutrients, limitation of fishing effort .

Basic knowledge of calculus: function, most common functions of real variables (exponential, logarithm, trigonometric functions), derivative, derivation and differentiation rules, antiderivative, integrals, basic integrals and integration rules.

1) Ordinary Differential Equations (ODE). Order of an ODE. 1st order ODE. Initial value problem. Existence and uniqueness theorem. Particular and general solution. Autonomous equation: stationary points and qualitative analysis. General solution of a 1st order separable ODE. Radioactive decay and radionuclide half-life. Penetration of light in a water body. Micropollutant contamination: half-life of pollutants in environmental media. Population dynamics: Malthus equation, the logistic equation, general analytical solution of the logistic equation, carrying capacity.
2) First order linear ODE. General solution of homogeneous linear ODE. Modelling the evolution of a pollutant in a waterbody. Non-homogeneous linear ODE as a result of a mass balance equation. Solution of the non-homogeneous equation at constant pollutant input. General solution of a non homogeneous linear ODE: the variation of constant method . Solution of a linear first order ODE for three typical cases of time dependent forcing functions: linear , periodic , exponential. Application to the modelling of a pollutant in a water body. Input-output relationships. Inverse problem and its relevance for the implementation of the environmental legislation. Linear combination of forcings. The superposition principle and its application to the pollutant model. Exercises on linear ODE. Examples of application of ODE. Individual based growth models: the Von Bertallanfy equation.
3) 1D Dynamic systems State variables. Definition of a dynamic system. 1D autonomous systems. Direction field and phase portrait of autonomous equation. Orbits and trajectories. Phase portraits: the logistic equation. Stationary (equilibrium) points and their stability. Local stability analysis. Renewable resources. Open access resources. Management policies: quotas and effort control. Controlling quotas: pros and cons discussed using the logistic model with constant harvest. Controlling effort: pros and cons discussed using the logistic model with harvest proportional to the stock. Examples from fishery.
4) 2D linear dynamic systems Systems of ODE: the Streeter-Phelps model for simulating the dynamic of Dissolved Oxygen in a waterbody. 2D dynamic systems: state vector and state space. Autonomous 2D systems. Vector field. Existence and uniqueness Theorem in 2D. 2D linear systems. Particular solutions of 2D Linear dynamic system. General solution. Properties of the general solution. Trajectories and orbits for real eigenvalues. Numerical examples. Numerical solution of dynamic systems using ExCel and "R" programming environment.Constructing phase portraits using numerical solutions. Numerical exercises. Classification of orbits of 2D systems: typical phase portraits for complex eigenvalues. Periodic orbits. Summing up: classification of orbits of linear systems. Applications of 2D linear systems: the Streeter-Phelps model. Multimedia environmental models for fate and transport of organic micropollutants. Organic micropollutant bioaccumulation models.
5) 2D non-linear dynamic systems. Interactions among population in ecosystems. Predator-prey interactions. The Lotka-Volterra model. Stability of an equilibrium point. Stability analysis of 2D linear systems. Local stability analysis of non-linear systems. Stability analysis of the Lotka-Volterra model. More complex dynamics: predator functional response Holling I,and II. Emerging of periodical oscillation in predator-prey systems.
6) Guidelines for model building. Steps in model building: identification of model structure, a priori parameter estimation, calibration, validation, assessment of model performance: Goodness of Fit (GoF) indices and graphical methods.

Course notes and reading materials provided by the lecturer.

The achievement of the learning objectives if verified by means of a written test and of a report. The written test account for 24/30 to the final mark. The test includes 5 questions, aimed at ascertaining that a student has understood the main theoretical background and is able to apply it. The report presents the results of the coding of a simple model and of the comparison of model output with a set of observations, provided by the lectures. This assignement contributes for 6/30 to the final mark.

1) Lectures, based on the notes which are provided weekly to the students, in which power point presentations will also be used. The relevance of all topics in understanding and modelling environmental processes is underlined and illustrated by means of examples. Simple numerical problems, similar to those proposed in the final test, are solved and discussed during classes.
2) Correction of homework.
3) Use of software tools for the numerical solution of dynamic systems: these tools allow students to check the analytical solutions and to explore the system dynamics in real world situation, in which the forcings are estimated from time series of observations.

English

written and oral