TECNICHE MATEMATICHE DI MODELLIZZAZIONE

Knowing how to construct and understand mathematical models that describe qualitatively and
quantitatively some phenomena related to the environment. Knowing how to use mathematical
tools and techniques in order to realize some mathematical related to Biology and Environmental
Sciences.

General generalities on mathematical models.
Populations, samples and processes. Pictorial and Tabular Methods in Descriptive Statistics.
Measures of Location. Measures of Variability. Probability: sample spaces and events, axioms,
interpretations and properties of Probability, counting techniques, conditional probability,
independence. Interpolation techniques and Pearson coefficient. Applications with GeoGebra.
Discrete random variables and probability distributions. Expected values. The Binomial probability
distribution. The Poisson probability distribution. Continuous random variables and probability
distributions. Probability density functions. Cumulative distribution functions and expected values. The
Normal distribution. The exponential distribution. Probability plots. Expected values, covariance and
correlation. Statistics and their distributions. Basic properties of confidence intervals. Large-Sample
confidence intervals for a population mean and proportion. Interval based on a normal population
distribution. Confidence intervals for the variance and standard deviation of a normal population.
Hypotheses and Test procedures. Z tests for hypotheses about a population mean. The One-Sample
T test. Tests concerning a population proportion. Further aspects of hypothesis testing. Z tests for
confidence intervals for a difference between two population means. The Two-Sample T test and
confidence interval. Analysis of paired data. Inferences concerning a difference between population
proportions. Single-Factor ANOVA. Multiple comparisons in ANOVA. More on Single-Factor ANOVA.
Generalities on ordinary differential equations. First order ordinary differential equations. Maximal
solutions. Separable equations. Linear first order differential equations. Theorem of existence and
uniqueness of a solution. Second order linear ordinary differential equations with constant coefficients.
Malthus and Verhulst models. Analysis of scenarios for an isolated population and for a not-isolated
population.
Equilibrium solutions and stability analysis. Stability criteria. Models of landscape ecology in the
absence of residential areas and in the presence of residential areas.

1. Jay L. Devore – Probability and Statistics for Engineering and the Sciences - Ninth Edition,
Cengage Learning (2014).
2. J.R. Brannan, W.E. Boyce – Differential Equations. An introduction to Modern Methods and
Applications – Second edition, John Wiley & Sons, Inc. (2011)
3. N. Hritonenko, Y. Yatsenko - Mathematical Modeling in Economics, Ecology and the
Environment. Second edition – Springer (2013)
4. Lecture notes.