SUBJECT

Mathematical Models in Biology

lecture

Master

Faculty of Science

Biology MSc

2

Semesters 1-4

Autumn/Spring semester

Textbooks and/or other information media in use or recommended:  

I. Lotka-Volterra model

1. Lotka-Volterra model, two-species   Historical ground of Lotka-Volterra model, two species models: predator-prey and competition models.  

2. Multi-species Lotka-Volterra model  Stability of rest points. Linearization and Ljapunov’s direct methods.  

3. Conservative and dissipative Lotka-Volterra models.  

4. Critics of Lotka-Volterra models and an overlook.  

II. Fisher-model

5. A Fisher’s selection model Historical ground of Fisher’s selection model. Two-allele cases.  

6. Multi-allele model Rest point, and its stability .

7. Maximality principle.  Lagrange’s method  

8. Fundamental theorem of natural selection

III. Evolutionary Game dynamics  

9. Evolutionary matrix games   Frequency dependent selection. Nash-equilibrium, evolutionary stable strategy.(ESS)

10. Examples Hawk-Dove games. Prisoner’s dilemma  

11. ESS and its equivalent reformulations  Dynamical and static equivalent reformulations (with average fitness, fitness variance and covariance)  

12. Replicator dynamics  Replicator dynamics, Ess asymptotically stable rest point of replicator dynamics.  

13. Density dependent evolutionary games for single species

14. Multi-species ESS and density dependent game  Coevolutionary dynamics and evolutionary stability  

15. Two-species cases  

• Elizabeth S. Allman, John A. Rhodes: Mathematical Models in Biology: An Introduction, Cambridge University Press, 2004