1. 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.

2. 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

3. Top row, Heliconius erato and bottom row, Heliconius melpomene
Müllerian mimicry
Top row, Heliconius erato and bottom row, Heliconius melpomene
- Mimicry between unpalatable species
Shared signal dilutes predation risk

4. Fritz Müller 1891 Survival gain when mimicry is attained
Number of attacks (for separate populations)
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

5. How does Müllerian mimicry evolve?
Saltational evolution
Purifying selection prevents most mutations from invading
Fisher 1930
Peak shift produces gradual evolution towards mimicry

6. Coevolutionary convergence
Advergence or coevolutionary convergence ?
Advergence
Coevolutionary convergence
Saltational evolution gives only advergence
If evolution is gradual, both advergence and coevolutionary convergence seem possible

7. 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
Limenitis archippus
Mimic and model in Batesian mimicry
Müllerian mimics

8. 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

9. Predators Prey Predators accumulate inhibition
Next encounter: altered probability of attack q (h) =
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

10. 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

11. 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

12. 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.

13. 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

14. 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.