MATHEMATICAL MODELS APPLIED TO THE ENVIRONMENT

Knowing how to construct and understand mathematical models that describe qualitatively and quantitatively some phenomena related to the environment. Knowing how to use the main concepts of differential equation theory for application in the biological, geological and environmental fields. Knowing how to predict and justify the evolution of simple phenomena, described by ordinary differential equations, related to the biological, geological and environmental sciences.

In particular, the learning objectives of the course of Mathematical models applied to the Environment, according to the Dublin descriptors, are:

• Knowledge and understanding: The student will learn some basic concepts related to mathematical modelling and will develop both computing ability and the capacity of manipulating some common mathematical models.
• Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Biology and Environmental Sciences.
• Makin​g judgements: The student will be stimulated to autonomously deepen his/her knowledge. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
• Communication skills: The frequency of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for describing in details some phenomena that arise from Applied Sciences.
• Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided investigations, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

Direct Instruction.

The results presented during the course will be analyzed and discussed also using appropriate software.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the program planned and outlined in the Syllabus.

1. Review and complements. Reviews of Differential Calculus and Integral Calculus for real functions of a real variable. General information on ordinary differential equations: definitions and theorems of existence and uniqueness. General information on systems of ordinary differential equations. Stability. Introduction to mathematical models. Computer applications. 
2. Mathematical models for climate change. Climate systems. The energy balance of the Earth. General circulation: general information and energy analysis. Climate change: general information, radiative forcing and climate sensitivity parameter. General circulation models GMC (General Circulation Model). 
3. Mathematical models for urban development. Generality. Elements of cellular automata theory: notion of cellular automaton, grids or lattices, neighborhoods, set of states, evolution or transition rules. The Greenberg-Hastings model. The White-Engelen model.
4.  Mathematical models in population dynamics. Discrete Malthus model. Continuous Malthus model. Discrete Verhulst model. Continuous Verhulst model and its generalizations. Study of the stability of the equilibrium solutions of the Malthus and Verhulst models. Von Bertalanffy model. 
5. Mathematical models for environmental systems. Model for evaluating the quality of greenery. Model for the evolution of metastability in an environmental system. Model for the evaluation of the production and diffusivity of biological energy in an environmental system. Determination of parameters: determination of territorial biopotentiality, description of the environmental impact between built-up areas and natural areas with medium and high biopotentiality, determination of the percentage of residential surface area, determination of sector and system metastability, determination of the connectivity parameter, determination of the ratio between the sum of the lengths of the impermeable barriers present in the environmental system and its external perimeter, determination of the ratio between the sum of the biologically inactive surfaces of the sectors and the total surface of the ecomosaic. 
6. Mathematical models for territorial sciences. Lotka-Volterra model. Duffing model. Study of the stability of the equilibrium solutions of the Lotka-Volterra and Duffing models. Lotka-Volterra type models for the study of interactions between social groups: cooperation models, competition models, prey-predator models.

1. R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).
2. K.-P. Hadler, J. Müller, Cellular Automata: Analysis and Applications, Springer (2017).
   
3. N. Hritonenko, Y. Yatsenko, Mathematical Modeling in Economics, Ecology and the Environment. Second edition,  Springer (2013).
4. H. Kaper, H. Engler, Mathematics & Climate, SIAM (2013).
   
5. A. Fowler, Mathematical Geoscience, Springer (2011).
6. Teaching notes.
   

The exam consists of an interview on the topics that have been the subject of the lessons. The evaluation will be expressed in thirtieths.

The student, at his/her choice, may decide to present a paper concerning a topic not covered in the course and pertinent to the topics of the same. This work will affect the final evaluation.

It should be noted that the above document is not mandatory.

Note. Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of the Department of Biological, Geological and Environmental Sciences.