Theoretical Modelling in Biology (G0G41A ) Pt I. Analytical Models I. Introduction

Tom Wenseleers Dept. of Biology, K.U.Leuven

7 October 2008

Course instructors

Tom Wenseleers Lab. of Entomology Pt. I. Analytical models (4x2h) [email protected]

Joost Vanoverbeke Lab. of Aquatic Ecology and Evolutionary Biology Pt. II. Simulation-based models (3x2h) [email protected]

Goals

• Learn to appreciate the utility of biological modelling in evolution & ecology • Get conceptual insight into interesting biological questions • Learn to use two modelling platforms:

Mathematica

for analytical models and

Simile

for simulation-based models • Demonstrate that with the help of these programs you do not need to be a mathematician to make models!!

Goals

• Ensure that if you come across models in the literature you can easily check the results or further extend them.

• 35% of all

Evolution

and

Ecology

articles use mathematical models, 60% of all of those in

American Naturalist

• Modelling is also a very valued skill in industry

Outline

• What is theoretical modelling?

• Why model?

• The modelling cycle • Modelling approaches • Course schedule & some practical things • Introduction to

Mathematica

What is theoretical modelling?

• “The activity of translating a real problem into mathematics for subsequent analysis” Edwards and Hamson, 1996

Why model?

Why model?

Darwin: "

Mathematics seems to endow one with something like a new sense"

Why model?

• One of the benefits of formal mathematical models is that they can show whether proposed mechanisms or verbal ideas can actually work . • Theoretical models may also 1) make useful quantitative predictions 2) generate new insights , e.g. explain counterintuitive phenomena 3) suggest further experiments that might help to discriminate between alternative theories

E.g. handicap principle

• Amotz Zahavi (1975): peacock's tail is a signal of the male that it is of good quality, since only good quality males can afford such an expensive display • Idea was widely ridiculed and criticised.

Traditional idea was that the tail just helps attract mates by making them more visible.

• In 1990 Alan Grafen then showed theoretically under what circumstances the idea can work . Later empirical work showed peacock's tail is indeed a Zahavian handicap.

The modelling process

The modelling process

• Can be formulated as a “modelling cycle” • The starting point is biology not mathematics • Usually the hardest part in modelling is identifying an interesting question , not solving the equations!!

The modelling cycle

Step 1: Identify the problem

The modelling cycle

1. Identify the problem

• The problem needs to be biologically interesting and non-trivial • But it also needs to be tractable : as the Nobel laureate Peter Medawar put it, science is the ‘art of the soluble’, and part of that art is choosing a problem which will turn out to be soluble.

The modelling cycle

1. Identify the problem

One common use for models is to try to explain puzzling phenomena E.g. why are as much as 30% of all broods deserted by both parents in penduline tits?

Every parent benefits from letting the other parent take care of the young, cost is that parents sometimes both leave.

At equilibrium benefit=cost .

With model we could explain why 30% of the nests are abandoned and why exactly this species has this system.

The modelling cycle

Step 2: List the factors and assumptions

Listing the assumptions and key factors is not trivial or easy!

Abstract your problem!

• Experiments look at the world in a simple way Us rats don't usually live these conditions...

• Models also look at the world given certain

assumptions

and taking into account the effects of a limited set of

parameters

Don’t create a Rube Goldberg model!

Always start from the simplest model possible.

Modelling cycle

The modelling cycle

1. Identify the real problem 2. List the factors and assumptions Abstraction:

involves making simplifying assumptions which will make a problem tractable, at the risk of course, of oversimplifying and thus making the solution of less value. Can also involve discarding parameters which on an a-priori basis you don't expect to matter.

Model formulation:

define the change) and parameters variables (entities that (quantities that are fixed) in the model, define how they are bounded and how they interact , choose a time scale as discrete or continuous and whether to treat time

Ockham’s razor

14th-century English logician William of Ockham “If two explanations can both explain the observations, we should prefer the explanation that postulates fewer entities or processes or that makes the smallest number of independent assumptions.”

The simplest solution is the best!

The modelling cycle Step 3: Formulate and solve the mathematical problem

The modelling cycle

1. Identify the real problem 2. List the factors and assumptions 3. Formulate and solve the mathematical problem

This is often the easiest part!

3. Formulate and solve the mathematical problem

•

Formulation:

several ways of proceeding, i.e. different model types and methods, each of which has its pros and cons (detailed later) • •

Two main approaches:

analytical models or numerical simulations

Solving the model:

generally one of the easiest parts of the process. Number of computer packages to do this.

The modelling cycle Step 4: Interpret the mathematical solution

The modelling cycle

1. Identify the real problem 2. List the factors and assumptions 3. Formulate and solve the mathematical problem 4. Interpret the mathematical solution

4. Interpret the mathematical solution

• Solution may be an equation or you may need to plot a number of graphs • How do your parameters affect your variables?

• Sensitivity analysis : how robust are your results?

• What do the results imply or suggest?

• What does it tell us that is new and that we did not understand before?

• What predictions can we make?

The modelling cycle Step 5: Compare with the real world

The modelling cycle

1. Identify the real problem 2. List the factors and assumptions 5. Compare with the real world 3. Formulate and solve the mathematical problem 4. Interpret the mathematical solution

The modelling cycle

• Do the model’s results match existing data?

= model validation

• Frequently, a formal model may also suggest important parameters that would be desirable to be measured empirically • Full model validation may only be possible after such additional empirical research

If all is well...

The modelling cycle Step 6: Publish!

Otherwise...

The modelling cycle

1. Identify the real problem 2. List the factors and assumptions

Go back to beginning: • What factors have I missed?

• What processes have I oversimplified?

• In short, what is wrong?

1. Identify the real problem

Recap

2. List the factors and assumptions 5. Compare with the real world 6. Publish and make testable predictions 3. Formulate and solve the mathematical problem 4. Interpret the mathematical solution

Modelling approaches

Two main approaches to modelling

• Analytical - using only maths - usually deterministic • Numerical simulation - problem numerically solved or simulated on a computer - stochasticity automatically covered

Analytical models

• often give nice and elegant results • results simpler to interpret than those stemming from simulation models • requires a greater level of abstraction most simulation models than • downside is that more simplifying assumptions are made, e.g. infinite population size, usually no stochasticity and not spatially explicit

Example of analytical model

• When should you fight for a resource?

• If both individuals (players) fight, each get half the resource (V/2) but they also pay a fighting cost (C) • If both don’t fight, they share the resource with no cost (each gets V/2)

Hawk-dove game

Maynard Smith & Price 1973

Game theoretic solution

• Frequency of hawks =

p

• Frequency of doves = (1-

p

) • Fitness of hawk genotype = baseline fitness + (1-p)*payoff of interacting with dove [i.e.

V

] + p*payoff of interacting with hawk [i.e. (

V

-

C

)/2] • Fitness of dove genotype = baseline fitness + (1-p)*payoff of interacting with dove [i.e. V/2] + p*payoff of interacting with hawk [i.e. 0]

Game theoretic solution

• evolutionary stable state (ESS): equilibrium mix • at ESS in hawk-dove game fitness hawk = fitness dove (1-

p

)*

V + p

*(

V

-

C

)/2 = (1-

p

)*

V

/2 which occurs when

p

=

V

/

C

= equilibrium frequency of hawk Maynard Smith

Other example: optimality models e.g. D. Lack's clutch size model

Why do some birds have large clutches and others small ones?

Hypothesis:

in each species parents should maximise the number of surviving young. This is determined by how many offspring the parent can feed.



s

/ 

c

 0

optimal clutch size

Clutch size

c

Probability of survival Clutch size

c

Recurrence, difference and differential equations

• most analytical models employ either recurrence equations or differential equations • recurrence equations: variable (n) in next time unit is written as a function of the variable in the current time unit n(t+1) = "some function of n(t)" or we can calculate the difference equation D n = n(t+1) - n(t) = "some function of n(t)" (discrete time steps) • differential equation: rate of change of variable over time d(n(t))/dt = "some function of n(t)" (continuous time)

Recurrence, difference and differential equations

• can be used to model the increase or decrease of a genotype or the increase or decrease of the abundance of a species over time • readily solved using computer algebra systems as

Mathematica

or

Maple

such • so don't be scared of the maths: the computer can do most of the hard work for you!!

Computer algebra systems

http://www.wolfram.com/ http://www.maplesoft.com/ Many other less powerful systems: http://en.wikipedia.org/wiki/Comparison_of_computer_algebra_systems

Simulation-based models

• more complex models solved numerically can often only be • we can then either solve our equations numerically using packages such as

Mathematica, Matlab

or graphical environments such as

Stella

or

Simile

• or we can simulate what happens to the population by explicitly modelling every single individual ( individual-based models ), this automatically takes into account stochasticity

Typical usage of simulation-based models

• Complex , highly realistic models • Spatially explicit models • Finite population models ( stochastic models ) • Models with many interacting species • Self-organised behaviour : how simple individual behaviour can lead to complex overall patterns

Example: spatially explicit model

• cooperators: invest in cooperation • defectors: benefit do not invest in cooperation but get • interactions are local on a lattice (cellular automaton) • birth is stochastic • space can sometimes help to stabilize cooperation (Nowak & May 1992)

Self-organised behaviour

System composed of many “sub-units” or “agents” workers in an ant colony cells in a multicellular organism birds in a flock Individual agents react to local conditions Global pattern arises from agent behaviour No one is in overall charge, no "leader" or "master plan" No one “knows” the overall state of the system

Camazine, S., Deneubourg, J. L., Franks, N. R., Sneyd, J., Theraulaz, G., Bonabeau, E. 2001. Self-organization in biological systems. Princeton University Press.

Simulation platforms

• Traditional low-level implementation: e.g.

Basic, Fortran, C++

• Graphical modelling interface ("Systems dynamics"):

Stella, Modelmaker, Powersim, Vensim, Matlab/Simulink, Simile

• Individual-based models:

Swarm, Echo, XRaptor, Matlab/Simulink, Gecko, StarLogo, Simile

Graphical modelling environments Simile, Stella, Modelmaker etc...

Graphical interface to construct more complex difference or differential equations models: systems dynamics approach

Simile

also allows individual based models to be constructed.

When to use different models?

• Different models give different insight: • Analytical models - Quick and easy to understand - Sometimes overly simplistic - But good to test whether some idea can work conceptually

When to use different models?

• Different models give different insight: • Simulations - More complicated to interpret and harder to plot results for all possible ranges of parameters - More appropriate when a model needs to be highly realistic, e.g. in fisheries where models are used for policy making - Can also be used to test analytical models

What have we learned?

Modelling isn't so hard Different models can give different insight

Course schedule

Place: Dekenstraat 2, VHI - 02.09 (LUDIT PC-KLAS H2), every Tuesday 10h30-12h30 7/10 Introduction, how to build a model, modelling approaches, introduction to the Mathematica platform for solving analytical models 14/10 Difference and differential equation models, with examples from population ecology (population growth, competititve interactions between two species, functional responses) 21/10 Game theory and inclusive fitness: evolutionarily stable strategies, applications in the areas of social evolution and the evolution of mating strategies 28/10 Explicit genetic models, applications from evolutionary genetics and behavioural ecology 04/11 Introduction to numerical simulation models using the Simile platform, examples from population ecology (population dynamic models with interactions between more than two species and with different age classes) 11/11 Public holiday, no course 18/11 25/11 Individual-based and spatially explicit models Modelling of food webs

Practicals: Do your own modelling project!

• Check the results of a

Mathematica

or

Simile

model from the literature and make a using minor extension (e.g. add one more parameter), or develop a simple new model • A list of possible topics also suggest a topic will be provided by us but you can of your own • Best way to get hands-on experience in modelling • One project per group of 2 students • Groups and topics should be decided before the end of October , please already indicate your interests on sheet provided

Practicals: Do your own modelling project!

• Projects will be supervised on a semi-individual basis me (analytical models) and Joost (simulation models) by • Three stages: (1) determine best modelling approach (2) develop model (3) validate model, perform sensitivity analysis • Assessment: based on final Powerpoint presentation plus a short 5 page paper with the key results • Counts as the exam (no formal exam will be organised)

Course material

• TOLEDO: http://toledo.kuleuven.be/ - Powerpoint slides -

Mathematica

notebooks and

Simile

exercises and problem solutions files with - useful papers • Installation CD with

Mathematica 6, Simile 5.3

and

Populus

- check file "installation instructions.doc"

Recommended books Pt. I

M. Bulmer (1994) Theoretical Evolutionary Ecology.

H. Kokko (2008) Modelling for Field Biologists.

Recommended books Pt. I

J.H. Vandermeer & D.E. Goldberg (2003) Population Ecology: First Principles.

D. Alstad (2000) Basic Populus Models of Ecology.

(see also Populus help file on CD)

Recommended books Pt. I

S.P. Otto & T. Day (2007) A Biologist's Guide to Mathematical Modeling in Ecology and Evolution.

Feedback

This course is organised for the first time. Everything can be changed flexibly, and we appreciate your feedback!