On the
Canonical Equation
of
(directional) Adaptive Dynamics
for
Physiologically Structured Populations
Hans Metz
ADN, IIASA
Michel Durinx
a few words about
- adaptive dynamics
- (physiologically) structured populations
the canonical equation
- derivation for structured populations
some final methodological comments
Adaptive Dynamics
• Example illustrating the individual-based justification.
• Where in the hierarchy of abstractions? Scope.
• The fitness landscape picture of evolution: it is a seascape!
reproduction
faithful
mutated
death
(The agent-based version of a so-called Lotka-Volterra model, with mutation added.)
P
P
Simplifying assumptions
1. mutation limited evolution
2. clonal reproduction
3. good local mixing
4. largish system sizes
consequence:
sX(X) = 0
5. "good" c-attractors
6. interior c-attractors unique
these determine the
resident environment
7. smoothness of sX(Y)
8. small mutational steps
i.e.toseparated
guaranteepopulation
near deterministic
dynamical
and
population
mutational
dynamics
time scales
Simplifying assumptions
1. mutation limited evolution
2. clonal reproduction
3. good local mixing
4. largish system sizes
5. "good" c-attractors
6. interior c-attractors unique
7. smoothness of sX(Y)
8. small mutational steps
essential, can be relaxed
The fitness landscape picture of evolution:
Evolution proceeds through uphill movements
in a fitness landscape that keeps changing so as to
keep the fitness of the resident types at exactly zero.
fitness landscape:
sX ...X (Y)
1
k
evolutionary time
0
0
0
0
0
resident trait value(s) x
mutant trait value y
Physiologically Structured Populations
• Population dynamical equilibria: demography with density dependence
• Invasion by mutants: back to standard classical demography again
Calculating equilibria for
physiologically structured populations
After Diekmann, Gyllenberg & Metz (2003) Theor Pop Biol 63: 309-338 :
b = L(I)b,
equilibrium
birth rate per
unit of area
I = F(O),
environmental input,
i.e., the environment as
perceived by the individuals
next generation operator
(i.e., Lij is the lifetime number of births
in state i expected from a newborn in state j)
O = G(I) b
per capita lifetime
impingement on
the environment
population output,
i.e., impingement
on the environment
Start with the case where b is a scalar
i.e., everybody is born equal, but for the inherited traits.
Individuals do not inherit state variables
like fat reserves, or social status
Calculating mutant invasion rates for
physiologically structured populations
Invasion fitness:
sX (Y)  rY (IX )
with rY(IX) the (unique) solution of
˜ (I , r) = 1

Y X
with
Laplace transform
of birth kernel
(Lotka’s equation,
scalar version)

˜ (I , r)   e ra  (I ,da)

Y X
Y X
0
 the so-called birth kernel of the invader Y in environment IX
[i.e., Y(IX,a) is the expected number of invaders produced
LY(IX) = Y(IX,∞)
up to age a by a newborn invader].
the
Canonical Equation
To first order of approximation for small mutational steps:
dX =
dt
1
2
a e N(X)
C
sX(Y)
Y Y=X
T
where e is the probability of a mutation per birth event,
C
is the mutational covariance matrix,
and
depends i.a. on the reproductive variability.
a
When the community attractor is just a point:
a = 2 Tf (Ts 2)
Tf = average age of giving birth,
Ts = average lifetime
2 = Var[lifetime offspring number] .
/
(of resident)
For ODE population models:
Dieckmann & Law (1996) J Math Biol 34: 579-612
Champagnat, Ferrière & Ben Arrous (2001) Selection 2
derivation for general
Physiologically Structured Populations
mean evolutionary step per time unit:
(average)
population
size
(average)
pro capita
birth rate
probability
of mutation
per birth event
 Ne
probability density
of mutation steps

 (Y  X)
P{Y invades} f (Y  X)d(Y  X)
Mutants arrive singly. Therefore we have to account
for the intrinsic stochasticity of the invasion process.
In an ergodic environment:
a population starting from a single individual
either goes extinct, with probability
Q,
or "grows exponentially" at a relative rate
r(y,Ix).
*
Under very general conditions
r(I) > 0.
P > 0 if and only if
*
For constant
(i)
I and small r(I) > 0
P - 2 ln [ R0(I)]
/ 2
with 2 a measure for the variability in the life-time
offspring production; when everybody is born equal
2 = Variance [life-time offspring production]
lifetime number of kids

Q =  pm Q
m
 g(Q)
m 0
g(0)  p0 ,

with g(x) :  pm x m
m0
g(1)  1,
g(1)  Em  R0 ,
R0 < 1
g
R0 > 1
g(1)
  Em(m  1)  Var m  R02  R0
with e := R0 - 1 ≈ ln(R0) :
Q
1
1  P  1  (1 e)P  (Var m  e  e2 )P2  [h.o.t (< 0)]
2
2e
P =
 h.o.t.

Var m
*
For constant
(i)
I and small r(I) > 0
P - 2 ln [ R0(I)]
/ 2
with 2 a measure for the variability in the life-time
offspring production; when everybody is born equal
2 = Variance [life-time offspring production]
(ii)
r(I) - ln [R0(I)] / Tf
with Tf the mean age at offspring production.
characteristic equation:

˜ ( r ) :  era (da)  1

0
take logarithms:
˜ (r )  0
(r ) : ln 
1

-ra
(r)
  
 -ae (da)
-ra
 e (da) 0
0
(0)  ln R0 ,

(0)
   Tf , with Tf :  a(da) /R0
0
to first order of approximation:
ln R0  r Tf  h.o.t  0
ln R0
 r
 h.o.t
Tf
From t he t heory of branching processes II:
Let P be probability of invading:
*
P = 1- Q.
Under very general conditions
r(I) > 0.
P > 0 if and only if
*
For constant
(i)
I and small r(I) > 0
P - 2 ln [ R0(I)]
/ 2
with 2 a measure for the variability in the life-time
offspring production; when everybody is born equal
2 = Variance [life-time offspring production]
(ii)
r(I) - ln [R0(I)] / Tf
with Tf the mean age at offspring production.

P - [2 Tf
/ 2 ]
r(I)
mean evolutionary step per time unit:
probability
of mutation
per birth event
(average)
population
size

(average)
pro capita
birth rate
 Ne
(Y  X) P{Y invades} f (Y  X)d(Y  X)
2Tf
1
Ts
Ne
probability density
of mutation steps
2Tf
Ts
2


(Y 
dX

dt
2
rY (IX )

X)(Y 

2Tf

2
T s X (Y)
X)

 Y
sX (Y)
 
 
Y  X  

T
f (Y  X)d(Y  X)
s (Y)
T
Y
Ne
C(X)


2 2
Ts
 Y Y X 
2Tf
1
For ODE population models:
2 = 2
Tf = Ts

2Tf
Ts
2
1
Finitely many different birth states:
(e.g. multiregional models)
The previous results immediately extend to this case
with appropriate definitions for
2 , Tf , and Ts
(although it was a bit of a hassle to get the formulas right).
For the case of infinitely many birth states the required
branching process results have not yet been proven.
Finitely many different birth states:
A second application is to periodic environments:
Take the phase (of the oscillation) at which an individual is born
as its birth state.
(The proof still has to be worked out in all its gory technical detail.)
NB The idea of a proof works only for discrete time models
due to the lack of hard results for continua of birth states.
(Ulf Dieckmann has additional heuristic results for simple ODE community
models that apply to mutant invasion in any type of ergodic attractor.)
To round of: What happens near branching points?
This depends on the local geometrical
structure of the function sX(Y).
For one-dimensional trait spaces
we have a full classification of what happens
around any type of “singular point”,
like Evolutionarily Stable Strategies or
branching points.
?
For higher dimensional trait spaces
we have so far only proved that:
Any model in the class treated in this talk
locally around a singular point
has the same evolutionary properties as some
appropriately chosen so-called (simple!) Lotka-Volterra model.
Some final methodological remarks:
• In this case it was possible to do an analytical model reduction
for a large class of eco-evolutionary models.
• Therefore simple members of that class give a good indication of
particular dynamical options within the larger class.
• By embedding those special models into a (vastly) larger class,
and working through the reduction procedure,
they can be connected to the “real world”.
(Therefore, in this case, toy models are good metaphors.)
• However, to see which properties survive as universal within an
appropriate larger class, one has to do the embedding-reduction exercise.
• Sobering aside: Within evolutionary biology we have by now had
close to a century to hone our mathematical tools.
Yet, the results that I just discussed are so new that they are still
unpublished.
The End
(for today)