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Avoiding AttackThe Evolutionary Ecology of Crypsis, Warning Signals and Mimicry$

Graeme D. Ruxton, Tom N. Sherratt, and Michael P. Speed

Print publication date: 2004

Print ISBN-13: 9780198528609

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198528609.001.0001

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(p.202) Appendices

(p.202) Appendices

Avoiding Attack
Oxford University Press

Appendix A A summary of mathematical and computer models that deal with Müllerian mimicry


Structure of model

Example conclusions


Müller (1879)

Predators in an area need to sample n items of a given morph before learning that it is distasteful.

Mimetic prey should benefit according to the reciprocal of their squared abundances

A “truly epoch-making paper” Dixey (1908)

Blakiston & Alexander (1884)

As Müller (1879)

Mimetic prey should benefit only approximately according to the reciprocal of their squared abundances

A question of how one measures the per capita success of a species (indeed a hypergeometric approach with reducing densities of prey as they are sampled may provide an even better approximation, especially for prey at low densities).

Marshall (1908)

A simple arithmetic case (not a detailed model) along the lines of Müller (1879)

If two unequally unpalatable species inhabit the same region then predation will lead the less numerous to resemble the more numerous but not vice-versa.

The first of many controversial theory papers on Müllerian mimicry In fact, recent research has increasingly considered the evolutionary dynamic as one of advergence rather than convergence (Mallet 1999 [2001a])

Fisher (1927)

A largely verbal response to Marshall's theory, emphasizing the importance of considering intermediate states.

A mutant of a common species which resembled a rare one, may not necessarily lose all of its advantage if it retains some resemblance to the common species.

Largely repeated in Fisher's classic (1930) text. Dixey (1908) had a similar argument, suggesting that an intermediate mutant may gain the benefits of both worlds (cf the multi-model approach of Edmunds 2000 applied to Batesian mimicry).

Holling (1965)

A detailed mechanistic model involving attack thresholds and learning and predicting domed functional responses for unpalatable prey (changes in predator attack rate with prey density)

Supports Müller's theory but suggests that the phenomenon is more complex than that envisaged by Müller, as a result of arguably more realistic predatory assumptions

Hunger (mediated by the absence of alternative prey) “destroys the advantages of Müllerian mimicry for prey” (page 39). These arguments have been rekindled 40 years later (Kokko et al. 2003; Sherratt et al. 2004b).

Huheey (1976)

An extension of Huheey (1964) in which attacking unpalatable species causes both species to be avoided for ni subsequent encounters

When ni ≠ nj then the less unpalatable species will always act as a Batesian mimic (parasite) on the more unpalatable species, increasing the attack rate on the species

In essence, a forgetting rather than a learning model. The model was quickly criticized by Sheppard & Turner (1977) and Benson (1977) who questioned its radical prediction (effectively no Müllerian mimicry), pointing out that the theory was based on relative frequencies, and that time was not explicit.

Owen & Owen (1984)

A “recurrent sampling” model (temporary deterrence from attacking models/mimics after attacking an unpalatable prey item). Abundance-Regulated-Anamnesis (ARA) approach allows one to consider absolute rather than relative abundance (Huheey 1964, 1976).

Classical Batesian and Müllerian mimicry can arise, but in the case of unequal unpalatability of two distasteful models/mimics, the exact nature of the interaction depends on the abundances of the species involved.

The predator has only a short-term memory so that it always samples at an encounter immediately subsequent to a non-sampling encounter. This amnesia appears unrealistic, and results in a minimum attack rate of 50%.

Turner et al. (1984)

First of the published “Monte-Carlo” simulation approaches (long considered by JRG Turner & PM Sheppard) with predators stochastically encountering “Models”, “Mimics” (and “Solo” and “Nasty” as controls). Probability of future attack of a phenotype (P) altered to P = P * a (if “unpleasant”) or P = 1 − (P * b) (if “pleasant”), with a and b as pleasantness constants The predator also forgets what it has learnt over time.

Predictions are consistent with traditional differences between Müllerian (mutualistic) and Batesian (parasitic) mimicry

Asymptotic attack rates on models and mimics in the absence of forgetting are either 0 or 1.

Hadeler et al. (1982)

A population-dynamical model for prey (predator population held constant) with predators varying in their experience (learning and recruitment of inexperienced predators).

Classical Müllerian and Batesian mimicry (interpreted as an increase or decrease in equilibrium abundance caused by mimicry) can arise, but a distasteful mimic can be parasitic on a heterospecific dependent on learning and population parameters

One of the few models to consider the population dynamical consequences of mimicry

Speed (1993a)

A “Pavlovian” predator. Similar Monte-Carlo simulation to Turner et al. (1984) but with (arguably) a more realistic learning algorithm. Here: ΔP = (0.5 + ∣λi − 0.5∣) *(λi − P) where λi is the “asymptotic attack parameter” on prey type i when encountered alone in the absence of forgetting.

Weakly unpalatable prey may act as parasites on more highly unpalatable prey (“quasi-Batesian” mimicry).

The equilibrium value of P (ΔP = 0) is λi. It is questionable whether learning behaviour with non-zero λ values for unpalatable prey would be selectively advantageous to predators

Gavrilets & Hastings (1998)

A haploid population genetic model involving 2 species each with two parallel morphs (A, a and B, b) and linear parameters characterizing both within-species and between-species interactions (Batesian and Müllerian situations)

The evolutionary dynamics are strongly affected by the relative strengths of within and between species interactions.

Non-equilibrium dynamics (a “chase”) is a general feature of these linear models. As the authors note, more work should be conducted to explore the possibility that co-mimics in the natural world are in dynamic fluctuation.

MacDougall & Dawkins (1998)

Müllerian mimicry may reduce the information load on predators and thereby reduce the number of discrimination errors.

When discrimination errors are made then unpalatable prey may suffer a greater change in mortality than palatable prey Thus, if mimicry reduces discrimination errors then it may be beneficial to all unpalatable prey whatever their unpalatability

This paper addresses an important phenomenon that had previously been overlooked in the study of Müllerian mimicry (Ruxton 1998). However, no explicit relationship between mimicry and discrimination errors was described.

Speed (1999a)

A re-examination of unusual features of the Owen & Owen (1984) model, supported by stochastic simulation.

If prey forget not to attack less defended prey (mimic) more rapidly than more defended prey (model) then it may generate a humped relationship between mimic density and attack rate on encounter, even in the original model of Turner et al. (1984).

Predictions may be highly sensitive to relatively subtle changes in assumptions about the way predators behave.

Speed (2001)

For two species (A & B), numbers of distinct unpalatable morphs consumed before learning is complete is nA and nB respectively Number attacked if they resemble one another (nAB) is the weighted mean (according to the abundances of the two species).

Quasi-Batesian mimicry may arise at low density levels

Although only numerical results are shown, this is an analytically tractable model in the same way as Box 1.

Speed & Turner (1999)

A comprehensive and systematic analysis of 29 different combinations of learning/forgetting algorithms

Most of the 29 models included classical Batesian and Müllerian mimicry but the vast majority also included “unconventional” forms of mimicry namely quasi-Batesian and quasi-Müllerian mimicry

Qualitative predictions are dependent on both forgetting and learning assumptions

Sasaki et al. (2002)

Reaction-diffusion model of the evolutionary dynamics of Müllerian mimicry in one and two dimensional habitats with positive frequency-dependent selection.

In heterogeneous habitats, areas of distinct co-mimics can be maintained due to the balance between frequency-dependent selection and biased gene flow.

Given the debate over polymorphisms in Müllerian mimicry spatially-explicit models which can characterize the spatial scale at which polymorphisms occur will be very valuable.

Franks & Noble (2004)

An individual-based model in which palatable and unpalatable prey could evolve in appearance. Predators generalized to a degree and tended to avoid attacking phenotypes which were on average unprofitable.

The presence of Batesian mimics can influence the mean sizes of Müllerian mimicry rings that form.

One of the few models to allow phenotypes to evolve, and to consider mimicry evolution in complex communities

Kokko et al. (2003)

A state dependent model (fit, starving, poisoned, dead) identifying optimal behaviours of predators on encountering models and mimics as well as alternative prey

Both mutualistic and parasitic relationship between unpalatable model and unpalatable mimic are possible, mediated by the abundance of alternative prey

If predators handle toxins relatively well but are likely to be limited by food intake, then “quasi-Batesian” (parasitic) mimicry becomes more likely between unpalatable species. The trade-off between food intake and toxin intake influences this relationship.

Sherratt et al. (2004b)

A state dependent model (different levels of energy and toxins) identifying optimal behaviours of predators on encountering models and mimics as well as alternative prey The two toxic species can contain the same poison or different poisons.

When two unpalatable species contain high levels of different toxins then they may have a mutualistic relationship-predators do not know what sort of poison they will consume.

A case of convergent thinking with Kokko et al. (2003). This study also emphasizes the great importance of alternative palatable prey in mediating the nature of the mimetic relationship.

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