Evolutionary Game Theory with Gfunctions

1
Evolutionary Game Theory with G-functions
Summary of the Workshop presented by Tom Vincent
Adelaide Oct 05
John Hawkins
2
Darwins Three Postulates

• Lewontin 1974 Distils Darwin into 3 postulates.
• Like tends to beget like and there is heritable
  variation in traits associated with each type of
  organism
• Among organisms there is a struggle for
  existence.
• Heritable traits influence the struggle for
  existence.

3
Inner and Outer Game

• In Hutchinsons concept of the ecological
  theatre, evolution consists of two parts.
• An inner game in which organisms compete.
  Satisfying Darwins postulates 2 3.
• An outer game in which reproduction is
  coordinated with the results from the inner game.
  With allowance for random change. Satisfying
  Darwins postulate 1.

4
Evolution as a Game

• Iterated Game
• Players do not choose strategies, they inherit
  them.
• Reproduction is proportional to payoff.
• All players with the same strategy set experience
  the same payoff
• Essentially applying a mean field assumption
• Payoff to individuals of a species is averaged
• Evolution is simply the change of strategy
  distribution within the population

5
A Starting Point

• A simple model to illustrate evolutionary game
  theory and the G-function method.
• Consider the evolutionary dynamics of a number of
  very similar species in the same niche.
• All share a single important trait that can be
  quantified.
• They are evolving on a one dimensional continuous
  strategy space.

6
Starting with Population Dynamics

• Use Standard Population dynamical models as a
  starting point for developing and example.
• Difference equation form of a LV model is

7
Fitness Function H

• Create a generalised form of the population
  dynamics by introducing a fitness function Hi for
  each interacting species i.
• In difference equation form

• Where u is the vector of strategy choices, x is
  the vector of population numbers, and R is an
  optional parameter representing the available
  resources.

8
Multiple Species Model

• To change the Lotka-Volterra model so that it
  incorporates multiple interacting species, we
  need to include the following in the fitness
  calculation.
• The strategy choice of each species.
• The competitive interaction between species.

9
L-V Multiple Species Model

• For each species i between 1 and ns

• Intrinsic growth rate r is given for each species
• The carrying capacity Ki and interaction
  coefficients aij are defined as functions.

10
Carrying Capacity and Intrinsic Growth Rate

• Carrying capacity of species i given by

• Species interaction coefficients a given by

11
Exploring the Game

• We can explore this game by varying
• The values of the free parameters
• The number of species
• Initial conditions (population and strategy
  choice)
• In general the winning strategy depends on the
  number of species.
• There are different equilibrium situations
  depending on the collection of strategies.

12
Darwinian Dynamics

• To be an evolutionary model we require the
  capacity for species to change strategy over
  time.
• How to introduce the genesis of variation?
• We could try and simultaneously model all
  possible variants as separate populations.
• Not feasible or realistic.
• We could introduce a random spread of variants
  around a species mean.
• We want to model strategy dynamics

13
G-Function

• A function G(v,u,x,R) is a fitness generating
  function if for every i1ns

• Thus the population model becomes

14
G-function Adaptive Landscape

• Plot the G function versus v for fixed values of
  u and x.
• Shows the fitness outcome of an individual
  changing its strategy, if we assume that doing so
  has a infinitesimal affect on the distribution of
  strategies.
• Importantly this plot differs from the plot of
  the fitness function versus the strategy choice
  of a species ui
• This means that evolution on a G-function
  landscape will not necessarily maximise the
  fitness of the species.

15
Adaptive Landscape
16
Landscape Dynamics

• For the fitness function, if all other species
  remain at constant strategy and population
  levels, then the shape of the adaptive landscape
  remains the same.
• However, for the G-function as the species
  changes its strategy the adaptive landscape
  changes around it.
• This has the potential to complicate the
  evolutionary dynamics.

17
ESS Evolutionary Stable Strategies

• Maynard-Smith and Price (1973) defined an ESS as
  a strategy that is able to resist invasion from
  alternative strategies.
• Given a population in which the ESS strategy is
  nearly uniformly employed.
• Then no other strategy will provide a higher
  reproduction rate, hence will not be able to
  overcome the dominance of the ESS
• Maximum points on the Adaptive Landscape

18
Convergence Stability

• An ESS gives no guarantee that evolution can
  actually arrive at such a situation.
• Eshel (1983) added the notion of convergence
  stability
• A ESS has convergence stability if a population
  will evolve to that strategy when the strategy
  composition is near the ESS.
• An ESE Ecologically Stable Equilibrium is a fixed
  distribution of population sizes that occurs with
  an ESS.

19
Insights from G-function

• Darwinian dynamics does not necessarily result in
  an ESS

sk lt sa
sk gt sa
20
Evolution with G-functions

• One method of modelling evolution with a
  G-function involves modelling strategy dynamics
  from a distribution of phenotypes.
• Assume that for each species xi a variation in
  strategies exist and ui now represents a mean.
• Evolution is change in mean strategy over time.
• G-function allows the evolutionary process to
  include intra-species competition.
• Each phenotype experiences a different payoff,
  despite being considered a member of the one
  species in the definition of the vectors u and x.

21
Strategy Dynamics

• For each species we assume the existence of a
  distribution of variants.
• For sub-group j of species i we have

• We can evolve a species by modelling it as a set
  of phenotypes defined by the set of duij
• The population dynamics drives a change in ui
  which in turn moves the entire distribution of
  phenotypes, thus changing each uij

22
Strategy Dynamics

• The following expression for the change in mean
  strategy is given (Derivation in the text).

Where
23
Fishers Fundamental Theorem

• The rate of increase in fitness of any organism
  at any time is equal to its genetic variance in
  fitness at that time.
• In terms of an adaptive landscape this means that
  the populations mean strategy will change in the
  direction of the upward slope of the adaptive
  landscape at a rate that is proportional to the
  slope and the amount of heritable variation
  within the population.
• Hence the greater the genetic variance the faster
  the evolution.
• si2 is sometimes called the speed term

24
Total System Dynamics

• The population dynamics given by

• And the strategy dynamics given by

25
Time Scales Fast / Slow

• The time scales of population dynamics and
  strategy dynamics may vary considerably from each
  other.
• An ecosystem may speed the majority of its time
  near a slowly changing population equilibrium.
• The majority of the time the G-function will have
  a value of zero
• We analyse the system by setting the G-function
  to zero and solving for the initial equilibrium
  and then updating strategy.

26
Modelling Fast / Slow Dynamics

• The simulations are performed thus
• 1 Given the initial conditions
• 2 Set the G-function to zero
• 3 Solve for the equilibrium populations
• 4 Use strategy dynamics to update the mean
  strategy
• 5 Repeat steps 2-4
• The long term behaviour can vary considerably
  from models in which population and strategy
  dynamics are updated on the same time scale.

27
Optimal Design Solutions

• An alternative method for investigating the
  equilibrium points in the strategy dynamics is to
  look for points of optimal design.
• An equilibrium where the G-function and the
  H-function have equal gradients.

• The optimal design equilibrium occurs when
  cooperative and disruptive strategy dynamics are
  balanced.

28
ESS Coalitions

• An ESS for an open ended number of species is
  more difficult to define.
• A coalition vector uc is said to be an
  evolutionary stable strategy for the equilibrium
  x if, we find for all ns gt ns and all
  strategies choices of species not in the
  coalition um , the equilibrium point x is a
  ecological stable equilibrium.
• Meaning, that no matter how many other species
  there are and what strategies they choose, their
  populations go to zero in the limit.

29
Non-Equilibrium Dynamics
30
Ecologically Stable Cycles

• Given a coalition vector uc of size ns the vector
  x is said to be an ecological cycle (periodic
  orbit, limit or n-cycle) provided that
• xi gt 0 for i1,,ns
• xi 0 for ins1,,ns
• An ecological cycle is said to be an ecologically
  stable cycle (ESC) if for a set of radii r from
  x for any x(0) within r from x the population
  dynamics equations asymptotically approach x as
  t goes to infinity.

31
Compound Fitness Function

• The compound fitness function for a cycle x in
  difference equation form is defined

• Where C is the length of the cycle
• Since an ecological cycle must return to its
  initial state, the compound fitness function must
  be equal to zero for an ecological cycle.

32
Non-Equilibrium ESS

• A coalition vector uc is said to be an
  evolutionary stable strategy for the ecological
  cycle x if
• for all ns gt ns and
• all strategies um
• The ecological cycle x is an ecologically stable
  cycle.

33
Extensions

• The general approach can be extended into
  multi-dimensional strategy spaces. In which case
  the G-function must handle a matrix of
  strategies.
• The G-function only models evolution of species
  that share a general strategy space. Interactions
  between very different organisms (eg
  predator-prey) must be modelled with multiple
  G-functions.
• G-functions must be coupled somehow

34
Applications

• Modelling Speciation and extinction
• Modelling cancer as somatic evolution of cells.
  Populations of cancer cells compete with healthy
  cells for resources and modify their environment,
  thus altering their evolution.
• Investigation of Relative Abundance

35
The End

• Questions?