Cooperation and multilevel selection

A few days ago I introduced how higher levels of selection could occur via a "toy" example. Obviously it wasn't realistic, and as RPM pointed out a real population is not open ended in its growth potential. I simply wanted to allude to the seeds of how Simpson's Paradox might occur, where population structure is needed to explain overall trends. Now I'm going to dive into a somewhat more complicated model, one which Martin Nowak published last year in PNAS, Evolution of cooperation by multilevel selection. The paper is free, so if this post piques your interest I recommend you dive straight into the paper. I've blogged Nowak's Evolutionary Dynamics, a recent book which gives accessible summaries of his body of work within mathematical biology, and am comfortable with his formalisms. So that explains my choice of this particular paper (though I plan to hit a few others and perhaps blog them soon).

First, the model assumes limits in population size, a concern brought up earlier. Roughly, Nowak presents a number of demes, defined by m, and within those demes there are individuals, bounded by a maximum value, n. Nowak's simulations assume bounds of population size defined by n = m (where each deme has only one individual) and nXm (where all demes have maximum population sizes, n). The model is stochastic, not deterministic, so each individual within the population has an expectation of reproduction proportional to w, its fitness. When the deme reaches the maximum alloted size (local carrying capacity) there are two outcomes:

1) A random individual within the deme is replaced so that n does not increase, or

2) The deme fissions into two, with all individuals randomly assigned to the daughter demes. Additionally, another deme goes extinct to maintain the constancy of the total deme number, m.

The probability that scenario 2 occurs is q, and that 1 occurs is (1 - q). In the treatment below Nowak assumes that fission is a rare process, so q << (1 - q).

One can conceptualize this model as one where the number of individuals within demes vary across a range from 1 to n, but the deme number itself remains constant. This is analogous to population genetic models which assume that population size remains constant across generations. In this case the demes themselves can fluctuate in size, though their total number remains the same. If you imagine the demes as spheres in a lattice only occasionally does a deme "go extinct," to be replaced by another, though the demes themselves exhibit some pulsing increase (remember that daughter demes will be smaller than the parent after fission). The variations in size and the rate of deme extinction at any given time is stochastic.

Up to this point I haven't described the nature of the individuals, but only the structure of the population. Predictably Nowak postulates two morphs: cooperators & defectors. Cooperators provide a benefit, b, to all members of their deme, at a cost, c, to themselves. Defectors receive a benefit, b, at no cost, because they never cooperate. Now, imagine that within the deme there are two probabilities of fixation for the two strategies when they are introduced as mutations:

ρ_C = probability of fixation of the cooperator
ρ_D = probability of fixation of the defector

Remember that this is a stochastic model. Nevertheless, obviously the probability of fixation of a cooperator introduced into a defector deme is lower than the reverse, because a defector can "free ride." In other words, while the cooperator has to depend on chance to increase its frequency against the hand of within group selection, the defector is aided in its chance of fixation by within group selection, by likely necessity as it were. In this conception the cooperators & defectors are somewhat like deleterious and advantageous alleles: both variants can be fixed by stochastic processes, but the latter is aided by selective processes. But, in the situations where cooperators do fix the homogeneous deme is now very fit as all the cooperators synergistically boost each other others' fitness without the drag of free riders (also, note that there is frequency dependence in this insofar as cooperators exhibit reduced cost as they increase in frequency because they interact with fewer free riders, while defectors are affected by an inverse dynamic as they encounter fewer and fewer "suckers" as they increase). Assuming that defecting mutants do not arise, this deme does not have to worry about invasion from other strategies. Because of the aggregate fitness, w, a group of cooperators will be more likely to enter into fission because within group increase is elevated (i.e., rate of within deme replacement, 1 - q, and fission, q, is greater, because of the increased reproduction of individuals, which has group level consequences simply because of the nature of the sum of the parts). Obviously "mixed" groups will be of lower total fitness, while homogeneous defector groups will have the lowest fitness of all. Nowak has then constructed a model where:

1) Cooperation is resisted within group because of its vulnerability to free riding by defectors, though there is frequency dependence of the fitness of cooperators so that the difference between it and the defecting strategy decreases as the latter are reduced (though the defector still has higher fitness because they never pay a cost).

2) Cooperation is favored between group because the higher the percentage of cooperators the greater the rate of fission. Remember that the deme number, m, is kept fixed. The process would be one where the lattice of spheres which demarcate the demes would have cooperative demes giving rise to offspring far more often.

So you have a process where within and between group selection are at odds, and the overall dynamic is determined by the balance between them and the various parameters. I will skip a bit of algebra to present the first major equation:

b/c > 1 + [n/(m - 2)]

What does this mean? b/c is the ratio of the benefit to the cost. n/(m - 2) is the ratio between the maximum population size of the demes and the number of demes subtracted from 2. When the number of demes is very large, that is, where m >> 2, you can simplify the equation to:

b/c > 1 + n/m

Please note that Nowak is assuming weak selection here (on the order of selection coefficients of 0.10). In any case, what does this formalism tell us? As m → ∞ we see that all that is needed for cooperation to be favored is that the benefit must exceed the cost, that is, b/c > 1. In contrast, when n is very large the ratio of the benefit to the cost must also be large. The implication seems clear: between group selection is important when you have a large number of small groups. In contrast, a small number of large groups favors within group selection. Intuitively this seems appealing as biologically we know that when groups become very large they develop their own internal structure and the ability to "punish" free riders also becomes very difficult. In contrast, a small group is much more easy to imagine as an analogy to an "organism." On the scale of the individual this might also predict naively that intragenomic conflict should be more common in complex organisms than simple ones.

Of course, in reality groups aren't isolated from each other, there is usually migration. So Nowak introduced the parameter z, defined by as λ/q, where λ is the small probability that a random individual moves from one group to another. The equation is then amended as below:

b/c > 1 + z + n/m

Migration between groups increases the chances that defectors can invade homogeneous cooperator groups, so it works against between group selection. This is a large problem with group selection in a genetic context, migration can very quickly exhaust the between group variation necessary for evolution to work on this level.

I invite you to read the original paper, which has some charts and figures which can clarify some of the formalism above graphically. Additionally, the supplementary materials available online are helpful. So now you see a population where demes are constrained in their maximum size, and the total census size is limited by nXm. But of course, the model is still very simple in relation to biological reality....