Mathematical Modelling in Population Dynamics

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Dr. Horst Malchow Professor of Applied Systems Science Institute of Environmental Systems Research Department of Mathematics and Computer Science University of Osnabr¨uck

2 February 2012 Tel/Fax E-Mail Internet

+49-541-969-2499/2599 [email protected] www.usf.uos.de/∼malchow

¨ 49069 Osnabruck, Germany

Mathematical Modelling in Population Dynamics Compact Course and Exercises

I. Lecture Programme 1. Introduction 1.1 Why mathematical modelling? 1.2 From systems analysis to model schemes and mathematics 1.3 Model classes, analytical and numerical methods 1.4 Recommended textbooks, cf. chap. (II.)

2. Environmental systems and compartments 2.1 Linear differential equations 2.2 Explicit solution of selected problems

3. From chemical kinetics to biological dynamics 3.1 Order of chemical reactions and formal kinetics 3.2 Nonlinear differential equations 3.3 Stationary solutions and their stablity 3.4 Models of enzyme kinetics 3.5 Translation of chemical into biological language

4. Population growth 4.1 Continuous vs. discrete time 4.2 Linear, exponential and logistic growth 4.3 Logistic growth and harvesting with maximum sustainable yield 4.4 Allee effect 1

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5. Population growth and interactions 5.1 Mutualism, predation and competition 5.2 Functional response of predators to prey abundance, Holling types 5.3 Holling-type II: Oscillations 5.4 Holling-type III: Oscillations and excitability 5.5 Periodic environments and deterministic chaos 5.6 Aperiodic dynamics of multi-species systems 5.7 Intra- and interspecific competition 5.8 Transmission of infectious diseases

6. Population growth and movement 6.1 Continuous vs. discrete space 6.2 Interaction, diffusion and advection 6.3 Exponential and logistic growth as well as bistable systems with diffusion 6.4 Diffusive front speeds

7. Population growth, interactions and movement 7.1 Diffusive and advective instabilites of spatially uniform population distributions 7.2 Turing structures 7.3 Branching and net structures 7.4 Target patters and spiral waves 7.5 Heterogeneous environments, spatiotemporal chaos 7.6 Biological invasions and spread of infectious diseases

II. Recommended textbooks A LLEN , L. J. S. (2007). An introduction to mathematical biology. Upper Saddle River NJ: Pearson Education. AUGER , P., L ETT, C. & P OGGIALE , J.-C. (2010). Mod´elisation math´ematique en e´ cologie. IRD ´ Editions. Paris: Dunod. B RITTON , N. F. (2003). Essential mathematical biology. Berlin: Springer. E DELSTEIN -K ESHET, L. (2005). Mathematical models in biology, vol. 46 of Classics in Applied Mathematics. Philadelphia: The Society for Industrial and Applied Mathematics. M ALCHOW, H., P ETROVSKII , S. V. & V ENTURINO , E. (2008). Spatiotemporal patterns in ecology and epidemiology: Theory, models, simulations. CRC Mathematical and Computational Biology Series. Boca Raton: CRC Press. 2

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M URRAY, J. D. (2003). Mathematical biology. II. Spatial models and biomedical applications, vol. 18 of Interdisciplinary Applied Mathematics. Berlin: Springer. O KUBO , A. & L EVIN , S. (2001). Diffusion and ecological problems: Modern perspectives, vol. 14 of Interdisciplinary Applied Mathematics. New York: Springer. ¨ ¨ V RIES , G., H ILLEN , T., L EWIS , M., M ULLER , J. & S CH ONFISCH , B. (2006). A course in mathematical biology: Quantitative modeling with mathematical and computational methods. Mathematical Modeling and Computation. Philadelphia: Society for Industrial and Applied Mathematics.

Osnabr¨uck, 2 February 2012

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