Download Müllerian mimicry

Alexandra Balogh and Olof Leimar
Department of Zoology
Stockholm University
Sweden
Müllerian mimicry:
An examination of Fisher’s theory
of gradual evolutionary change
Top row, Heliconius erato and bottom row, Heliconius melpomene, Müllerian co-mimics.
All things are not equally nice to eat…
Blue jay (Cyanocitta cristata) while eating monarch butterflies
(Danaus plexippus) (left) and a few minutes later (right)
- Aposematism is a way of signalling unprofitability to predators
- Avoidance learning reduces predation on aposematic populations
Müllerian mimicry
Top row, Heliconius erato and bottom row, Heliconius melpomene
- Mimicry between unpalatable species
- Shared signal dilutes predation risk
Fritz Müller 1891
a1n ,
a1  a2 n
n
a2 n
a1  a2
Number of attacks (for
separate populations)
n  a1n  a2 n , n  a2 n  a1 n
a1  a2 a1  a2
a1  a2 a1  a2
g1 
a2n , g  a1n
2
a1 (a1  a2 )
a2 (a1  a2 )
g1 a22

g2 a12
Survival gain when
mimicry is attained
Advantage from
cooperation per capita
The relative advantage is proportional
to the relative population size squared
Resulting appearance will depend on the relative initial protection of the
participants of the cooperation, like abundance and unpalatability
How does Müllerian
mimicry evolve?
Saltational evolution
Fisher 1930
Purifying selection prevents
most mutations from invading
Peak shift produces gradual
evolution towards mimicry
Advergence or coevolutionary convergence ?
Advergence
Coevolutionary
convergence
Saltational evolution gives only advergence
If evolution is gradual, both advergence and
coevolutionary convergence seem possible
No empirical evidence for coevolution
• There seems to be a model and a mimic (Mallet 1999)
• Typical model characters: higher abundance, larger
geographical distribution, higher unpalatability, more
”original” appearance
• Because of this, Müllerian mimicry is often believed
have come about through saltations
fake character
original character
Danaus plexippus
Mimic and model in Batesian mimicry
Limenitis archippus
Müllerian mimics
Testing Fisher’s process: Model
• Individual-based simulations of a community of two prey
types and predators
• Predator avoidance learning and generalization
• Prey appearance is a one-dimensional quantitative trait
• Given that a gradual process is possible, assess
advergence through individual-based simulations and by
solving the canonical equation
d
dt
d
dt
x  m N  W
a
a a a
x  m N  W
b
b b b
Predators
• Predators accumulate
n
inhibition h  x, t    e (t t ) g  x, xi  yi
i 1
• Next encounter: altered
e s ( h h )
probability of attack q (h) =  s ( hh )
i

e
1
2 2
( x  xi )2
0
e
0
1
Probability of attack on a discovered prey depends on
predator experience and on the trait of the encountered
prey (predator generalization).
Prey
• Individual-based: survivors reproduce, mutations with a
given distribution of effect sizes are produced
• Canonical equation: invasion fitness of mutants computed,
canonical equation integrated
Survival and invasion fitness
Initially similar prey types
Initially more distinct
Two types of predators
Invasion fitness
Na = 1000, Nb = 5000, Np = 100
Predator generalization
Fisher’s process is possible
(individual-based simulations)
Fisher’s process is posible for traits sufficiently similar
for predator generalization
It is also possible for large trait differences when a
predator spectrum is used
The degree of advergence depends on the
range of mutational increments
Curves 1-3 correspond to succesively smaller ranges
of mutational increments, 3 computed by solving the
canonical equation.
Conclusions
• Gradual evolution by peak shift towards Müllerian mimicry is possible
also for large initial trait differences when a proportion of predators
generalize broadly
• The range of mutational increments affects the degree of advergence
– The canonical equation approximates the evolutionary trajectory for very
small mutational effects
– For somewhat larger ranges of mutational effects, there is gradual
evolution and more advergence than predicted by the canonical equation
– The deviation from the canonical equation is related to the curvature of
invasion fitness
• Gradual evolution through Fisher’s process seems consistent with
observations of advergence in Müllerian mimicry in nature
Alexandra Balogh and Olof Leimar
Department of Zoology
Stockholm University
Sweden
Müllerian mimicry:
An examination of Fisher’s theory
of gradual evolutionary change
Top row, Heliconius erato and bottom row, Heliconius melpomene, Müllerian co-mimics.