# Adaptive dynamics, fall 2010

### Lecturer

### News

- Tuesday, December 7, was the last lecture of this course.

**Scope**

10 cu.

**Type**

Advanced studies

### Prerequisites

Some acquaintance with (systems of) ordinary differential equations would come in handy.

### Lectures

Weeks 37-42 and 44-50, Tue 14-16 in room B321, Thursday 14-16 in room B322.

**Lecture notes**

**1 INTRODUCTION:**

1.1 Introduction;

1.2 Introduction (recap);

1.3 Lotka-Volterra competition model;

1.4 Lotka-Volterra competition model (continued);

**2 A MORE GENERAL THEORY OF ADAPTIVE DYNAMICS:**

*2.1 The ecological timescale: *

2.1.1 Invasion and invasion fitness;

2.1.2 The outcome of an invasion event;

2.1.3 The outcome of an invasion event (continued);

*2.2. The evolutionary timescale:*

2.2.1 Classification of the local configuration of the pairwise invadability plot;

2.2.2 Evolution in a dimorphic population; First example; Second example;

2.2.3 Differential inclusions and total stability;

2.2.4 Master equation, canonical equation and diffusion approximation; Example;

2.2.5 Comparison of different stability concepts; Example;

**3. FURTHER GENERALIZATIONS:**

3.1 Invasion fitness for structured populations in continuous or discrete time;

**4. SPECIAL TOPICS AND CASE STUDIES:**

4.1 Evolution of seed size and seedling competitive ability;

4.2 Predator-prey coevolution and critical function analysis;

4.2.1 Invasion fitness for a variable resident environment;

4.2.1 Critical function analysis;

**APPENDIX:**

A.1 Local stability analysis of ODEs;

A.2 Elements of the theory of Poincare and Bendixon;

A.3 Example: the resource-consumer model of Gause;

A.4 The theorem of Perron and Frobenius;

**REFERENCES:**

Dieckmann & Law (1996) *Journal of Mathematical Biology* **34**: 579-612 (About: derivation of the 'canonical equation' of adaptive dynamics),

Geritz *et al.* (1998) *Evolutionary Ecology* **12**: 35-57 (About: the basic AD framework: classification of singularities and isoclines),

Geritz *et al.* (1999) *Theoretical Population Biology* **55**: 324-343 (About, in particular in the Appendix: the connection of the isoclines to the boundary of the coexistence set),

Geritz *et al.* (2002) *Journal of Mathematical Biology* **44**: 548-560 (About: resident-invader dynamics for similar strategies; the case of multiple resident population attractors),

Geritz (2005) *Journal of Mathematical Biology* **50**: 67-82 (About: resident-invader dynamics for similar strategies; four basic kinds of outcomes),

Gyllenberg & Parvinen (2001) *Bulletin of Mathematical Biology* **63**: 981-993 (About: evolutionary suicide),

Meszena *et al.* (2005) *Physical Review Letters* **95**: 078105 (About:resident-invader dynamics for similar strategies; the time-scale separation argument),

**Exercises**

Exercises 1-3 --> Solution 1-3 Solution 1-2

Exercises 4-7 --> Solution 4-7

Exercises 8-9 --> Solution 8 (Maple / *Mathematica*), Solution 9 (Maple / *Mathematica*)

Exercises 10-11 --> Solution 10 (Maple), Solution 11 (Maple / *Mathematica*)

Exercises 12-13 --> Solution 12-13 (MatLab & Maple / *Mathematica*)

These are all the exercises. The remaining time of the course you are supposed to work on one of projects listed see below (see under the heading "Projects").

### Exam

The exam will be in the form of a project (= advanced exercise) or article reading plus discussion. Please, contact the lecturer to make your preference clear.

**Projects**

Predator-prey; Evolutionary cycles; Evolutionary arms-race; Cannibalism; Cooperation; Virulence 1; Virulence 2; Two-patch model; Prey evolution; Resistance; Resource use;

- Choose one project and hand in the completed project before the end of January 2011. When you hand in the project we make an appointment to discuss the results.

- We have reserved computer room C128 on Fridays from 14-16 for you to work on the projects. The assistant will be there for at least 15 minutes to help you if need be. If nobody shows up during that time, I can find him in his office B425.

- Here are some worked out examples (in Mathematica) to show what you are supposed to be able to do. We've seen these examples already during the lectures or the exercises:

Predator-prey (different from the project above); LV asymmetric competition (same as during the exercises); Cannibalism (same as during the lectures);

### Exercise groups

Group |
Day |
Time |
Place |
Instructor |
---|---|---|---|---|

1. |
Fri |
14-16 |
B321 |
Jaakko Toivonen & Paolo Muratore Ginanneschi |