Mathematical Models in Biology
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.
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