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Batesian mimicry is a form of biological resemblance in which a noxious, or dangerous, organism (the model), equipped with a warning system such as conspicuous colouration, is mimicked by a harmless organism (the mimic). An example would be unrelated species such as Myrmarachne plataleoides, the spider, and the weaver ant Oecophylla smaragdina.

Given that evolution is a random process that doesn't have any pre-defined targets the probability of such convergence in evolutionary history should be nearly zero. My argument is that the search space for a 'nearly optimal' Batesian mimic would be too large and hence any search algorithm would be unable to avoid the curse of dimensionality. Even the sampler, which would need to sample from a wide variety of valid phenotypes in the parameter space would run into serious problems.

The problem of the curse of dimensionality for optimal sampling can be intuitively explained in the following way:

  • Let's say you have a straight line 100 yards long and you dropped a penny somewhere on it. It wouldn't be too hard to find. You walk along the line and it takes two minutes.

  • Now let's say you have a square 100 yards on each side and you dropped a penny somewhere on it. It would be pretty hard, like searching across two football fields stuck together. It could take days.

  • Now a cube 100 yards across. That's like searching a 30-story building the size of a football stadium.

  • The difficulty of searching through the space gets a lot harder as you have more dimensions.

Here are additional points to clarify what I mean:

My question is whether the Batesian mimic would need to initially be quite similar in morphology to the different species that it's converging to in morphology. In mathematical terms, if the parameter space for the visible morphological features is $(x_1,x_2,...,x_n) \in \mathbb{R}^n$ then convergence is probable if and only if the initial condition lies in a small enough neighborhood of the solution $(q_1,...,q_n)$.

One of the features of the fitness function is that I think there should be exponential payoffs in terms of reproductive success near the solution $\vec{q}$. Further, I think there are probably other very important environmental factors that might lead to such convergence. An arboreal jumping spider of similar size that shares the same environment as leaf-cutter ants is a good example. They would already have similar color among other features in order to blend in the same environment.

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    $\begingroup$ Well it is not too hard for it to evolve, because it has evolved, many times. What is your question? See my answer here regarding (what some people suggest are implausible) rates of evolution and adaptation. biology.stackexchange.com/questions/45005/… $\endgroup$ – rg255 Apr 18 '16 at 17:59
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    $\begingroup$ One of the (several) problems with your conclusion is that you assume the initial adaption needs to be "nearly optimal" to start with. Assume instead that looking 1% more like a coral snake provides a harmless species with a 1% improvement in reproduction. Modeling that leads to efficient mimicry within a relatively short number of generations. $\endgroup$ – iayork Apr 18 '16 at 18:11
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    $\begingroup$ There are no question mark in your question :) It is a sign that the question is probably unclear. $\endgroup$ – Remi.b Apr 19 '16 at 1:07
  • $\begingroup$ @iayork What do you mean by 1% more? Let's just look at a silly example. Let's suppose you're just comparing two 3 dimensional shapes. How does a square become 1% more like a sphere? $\endgroup$ – Aidan Rocke Apr 19 '16 at 9:47
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    $\begingroup$ Mimicry and camouflage is an extremely efficient defense. If an insect gains a color fleck that makes it 1 percent less likely to be eaten, that color fleck will grow, diverge and change over generations. $\endgroup$ – com.prehensible Oct 13 '17 at 12:25
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While the attempt to consider this in a mathematical framework perhaps isn't so useful, it seems that the biological essence of your question is along the lines of:

Does a Batesian mimic initially need to be initially quite similar in morphology to the different species which it evolves to mimic?

The answer is no, morphology does not need to be similar, but it does help make mimicry likely. The only requisite to evolution is the presence of heritable genetic information with the potential for change, and this prerequisite is the same for any trait. Selection is not a prerequisite but mimicry is more likely to evolve by adaptive evolution than random evolution. Given that phenotype is often strongly affected by the DNA and that DNA has a potential to mutate, then any distance in phenotypic space can be covered so long as sufficient mutations occur. Sufficient mutations are more likely to occur when

a) the distance in phenotypic space between the two initial species is small (the ancestral form of the mimic species is similar to the species it will mimic)

b) a large amount of time is allowed to elapse (more mutations can occur)

The first of these is simply stating that there are fewer (or less extreme) mutations required to get the species to be perfect mimics. The latter simply states that time allows for a greater number of mutations to occurs, making it more likely that large phenotypic spaces can be bridged.

Consequently, allowing identical time, mutation rates, mutation effect sizes, and selection coefficients, two species that initially look similar are more likely to evolve mimicry than two species that initially look very different, because fewer steps need to be taken for convergence of phenotypes.

This is the same for any trait. One could ask can humans evolve to be ten feet tall, the answer is yes, as long as the genetic variance arises. One could ask can humans evolve to be 20 feet tall, and again the answer is yes. However, the latter is less likely because it requires a greater movement in phenotypic space. More time also makes both results more likely.

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  • $\begingroup$ This answer seems reasonable. but, would you mind expanding on point a)? I mean can we quantify how small this distance must be in order for evolution of Batesian mimicry to be probable within a few generations? $\endgroup$ – Aidan Rocke Apr 19 '16 at 18:41
  • $\begingroup$ It would involve too many assumptions that would be violated in nature, there's a major problem with trying to reduce some evolutionary problems to equations - nature is just too complex $\endgroup$ – rg255 Apr 19 '16 at 19:38
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    $\begingroup$ I can take a look at point a tomorrow - it's explained a little in the last paragraph when comparing 10 and 20 feet tall people (as a very simple univariate example) - but if you want me to expand further then remind me tomorrow (bedtime here!) @AidanRocke $\endgroup$ – rg255 Apr 19 '16 at 19:57

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