Fisher’s Fundamental Theorem (Part 2)

Azimuth 2020-09-29

Here’s how Fisher stated his fundamental theorem:

The rate of increase of fitness of any species is equal to the genetic variance in fitness.

But clearly this is only going to be true under some conditions!

A lot of early criticism of Fisher’s fundamental theorem centered on the fact that the fitness of a species can vary due to changing external conditions. For example: suppose the Sun goes supernova. The fitness of all organisms on Earth will suddenly drop. So the conclusions of Fisher’s theorem can’t hold under these circumstances.

I find this obvious and thus uninteresting. So, let’s tackle situations where the fitness changes due to changing external conditions later. But first let’s see what happens if the fitness isn’t changing for these external reasons.

What’s ‘fitness’, anyway? To define this we need a mathematical model of how populations change with time. We’ll start with a very simple, very general model. While it’s often used in population biology, it will have very little to do with biology per se. Indeed, the reason I’m digging into Fisher’s fundamental theorem is that it has a mathematical aspect that doesn’t require much knowledge of biology to understand. Applying it to biology introduces lots of complications and caveats, but that won’t be my main focus here. I’m looking for the simple abstract core.

The Lotka–Volterra equation

The Lotka–Volterra equation is a simplified model of how populations change with time. Suppose we have $n$ different types of self-replicating entity. We will call these entities replicators. We will call the types of replicators species, but they do not need to be species in the biological sense!

For example, the replicators could be organisms of one single biological species, and the types could be different genotypes. Or the replicators could be genes, and the types could be alleles. Or the replicators could be restaurants, and the types could be restaurant chains. In what follows these details won’t matter: we’ll have just have different ‘species’ of ‘replicators’.

Let $P_i(t)$ or just $P_i$ for short, be the population of the $i$ th species at time $t.$ We will treat this population as a differentiable real-valued function of time, which is a reasonable approximation when the population is fairly large.

Let’s assume the population obeys the Lotka–Volterra equation:

$\displaystyle{ \frac{d P_i}{d t} = f_i(P_1, \dots, P_n) \, P_i }$

where each function $f_i$ depends in a differentiable way on all the populations. Thus each population $P_i$ changes at a rate proportional to $P_i,$ but the ‘constant of proportionality’ need not be constant: it depends on the populations of all the species.

We call $f_i$ the fitness function of the $i$ th species. Note: we are assuming this function does not depend on time.

To write the Lotka–Volterra equation more concisely, we can create a vector whose components are all the populations:

$P = (P_1, \dots , P_n).$

Let’s call this the population vector. In terms of the population vector, the Lotka–Volterra equation become

$\displaystyle{ \dot P_i = f_i(P) P_i}$

where the dot stands for a time derivative.

To define concepts like ‘mean fitness’ or ‘variance in fitness’ we need to introduce probability theory, and the replicator equation.

The replicator equation

Starting from the populations $P_i$ , we can work out the probability $p_i$ that a randomly chosen replicator belongs to the $i$ th species. More precisely, this is the fraction of replicators belonging to that species:

$\displaystyle{ p_i = \frac{P_i}{\sum_j P_j} }.$

As a mnemonic, remember that the big Population $P_i$ is being normalized to give a little probability $p_i.$ I once had someone scold me for two minutes during a talk I was giving on this subject, for using lower-case and upper-case P’s to mean different things. But it’s my blog and I’ll do what I want to.

How do these probabilities $p_i$ change with time? We can figure this out using the Lotka–Volterra equation. We pull out the trusty quotient rule and calculate:

$\displaystyle{ \dot{p}_i = \frac{\dot{P}_i \left(\sum_j P_j\right) - P_i \left(\sum_j \dot{P}_j \right)}{\left( \sum_j P_j \right)^2 } }$

Then the Lotka–Volterra equation gives

$\displaystyle{ \dot{p}_i = \frac{ f_i(P) P_i \; \left(\sum_j P_j\right) - P_i \left(\sum_j f_j(P) P_j \right)} {\left( \sum_j P_j \right)^2 } }$

Using the definition of $p_i$ this simplifies and we get

$\displaystyle{ \dot{p}_i = f_i(P) p_i - \left( \sum_j f_j(P) p_j \right) p_i }$

The expression in parentheses here has a nice meaning: it is the mean fitness. In other words, it is the average, or expected, fitness of a replicator chosen at random from the whole population. Let us write it thus:

$\displaystyle{ \langle f(P) \rangle = \sum_j f_j(P) p_j }$

This gives the replicator equation in its classic form:

$\displaystyle{ \dot{p}_i = \Big( f_i(P) - \langle f(P) \rangle \Big) \, p_i }$

where the dot stands for a time derivative. Thus, for the fraction of replicators of the $i$ th species to increase, their fitness must exceed the mean fitness.

The moral is clear:

To become numerous you have to be fit. To become predominant you have to be fitter than average.

This picture by David Wakeham illustrates the idea:

The fundamental theorem

What does the fundamental theorem of natural selection say, in this context? It says the rate of increase in mean fitness is equal to the variance of the fitness. As an equation, it says this:

$\displaystyle{ \frac{d}{d t} \langle f(P) \rangle = \sum_j \Big( f_j(P) - \langle f(P)\rangle \Big)^2 \, p_j }$

The left hand side is the rate of increase in mean fitness—or decrease, if it’s negative. The right hand side is the variance of the fitness: the thing whose square root is the standard deviation. This can never be negative!

A little calculation suggests that there’s no way in the world that this equation can be true without extra assumptions!

We can start computing the left hand side:

$\begin{array}{ccl} \displaystyle{\frac{d}{d t} \langle f(P) \rangle} &=& \displaystyle{ \frac{d}{d t} \sum_j f_j(P) p_j } \\ \\ &=& \displaystyle{ \sum_j (\nabla f_j(P) \cdot \dot{P}) p_j + f_j(P) \dot{p}_j } \end{array}$

Before your eyes glaze over, let’s look at the two terms and think about what they mean. The first term says: as the population vector $P$ changes, the mean fitness will change since the fitnesses $f_j(P)$ depend on $P.$ The second term says: as $P$ changes, the mean fitness will change since the fraction $p_j$ of replicators of the $j$ th species will change.

We could continue the computation by using the Lotka–Volterra equation for $\dot{P}$ and the replicator equation for $\dot{p}.$ But it already looks like we’re doomed without invoking an extra assumption. The left hand side of Fisher’s fundamental theorem involves the gradients of the fitness functions, $\nabla f_j(P).$ The right hand side:

$\displaystyle{ \sum_j \Big( f_j(P) - \langle f(P)\rangle \Big)^2 \, p_j }$

does not!

This suggests an extra assumption we can make. Let’s assume those gradients $\nabla f_j$ vanish!

In other words, let’s assume that the fitness of each replicator is a constant, independent of the populations:

$f_j(P_1, \dots, P_n) = f_j$

where $f_j$ at right is just a number.

Then we can redo our computation of the rate of change of mean fitness. The gradient term doesn’t appear:

$\begin{array}{ccl} \displaystyle{\frac{d}{d t} \langle f(P) \rangle} &=& \displaystyle{ \frac{d}{d t} \sum_j f_j p_j } \\ \\ &=& \displaystyle{ \sum_j f_j \dot{p}_j } \end{array}$

We can use the replicator equation for $\dot{p}_j$ and get

$\begin{array}{ccl} \displaystyle{ \frac{d}{d t} \langle f \rangle} &=& \displaystyle{ \sum_j f_j \Big( f_j - \langle f \rangle \Big) p_j } \\ \\ &=& \displaystyle{ \sum_j f_j^2 p_j - f_j p_j \langle f \rangle } \\ \\ &=& \displaystyle{ \Big(\sum_j f_j^2 p_j\Big) - \langle f \rangle^2 } \end{array}$

This is the mean of the squares of the $f_j$ minus the square of their mean. And if you’ve done enough probability theory, you’ll recognize this as the variance! Remember, the variance is

$\begin{array}{ccl} \displaystyle{ \sum_j \Big( f_j - \langle f \rangle \Big)^2 \, p_j } &=& \displaystyle{ \sum_j f_j^2 \, p_j - 2 f_j \langle f \rangle \, p_j + \langle f \rangle^2 p_j } \\ \\ &=& \displaystyle{ \Big(\sum_j f_j^2 \, p_j\Big) - 2 \langle f \rangle^2 + \langle f \rangle^2 } \\ \\ &=& \displaystyle{ \Big(\sum_j f_j^2 p_j\Big) - \langle f \rangle^2 } \end{array}$

Same thing.

So, we’ve gotten a simple version of Fisher’s fundamental theorem. Given all the confusion swirling around this subject, let’s summarize it very clearly.

Theorem. Suppose the functions $P_i \colon \mathbb{R} \to (0,\infty)$ obey the equations

$\displaystyle{ \dot P_i = f_i P_i}$

for some constants $f_i.$ Define probabilities by

$\displaystyle{ p_i = \frac{P_i}{\sum_j P_j} }$

Define the mean fitness by

$\displaystyle{ \langle f \rangle = \sum_j f_j p_j }$

and the variance of the fitness by

$\displaystyle{ \mathrm{Var}(f) = \sum_j \Big( f_j - \langle f \rangle \Big)^2 \, p_j }$

Then the time derivative of the mean fitness is the variance of the fitness:

$\displaystyle{ \frac{d}{d t} \langle f \rangle = \mathrm{Var}(f) }$

This is nice—but as you can see, our extra assumption that the fitness functions are constants has trivialized the problem. The equations

$\displaystyle{ \dot P_i = f_i P_i}$

are easy to solve: all the populations change exponentially with time. We’re not seeing any of the interesting features of population biology, or even of dynamical systems in general. The theorem is just an observation about a collection of exponential functions growing or shrinking at different rates.

So, we should look for a more interesting theorem in this vicinity! And we will.

Acknowledgements

After writing this blog article I looked for a nice picture to grace it. I found one here:

• David Wakeham, Replicators and Fisher’s fundamental theorem, 30 November 2017.

I was mildly chagrined to discover that he said everything I just said more simply and cleanly… partially because he went straight to the case where the fitness functions are constants. I do want the more general case for later. But any mild chagrin I experienced was instantly offset by this remark:

Fisher likened the result to the second law of thermodynamics, but there is an amusing amount of disagreement about what Fisher meant and whether he was correct. Rather than look at Fisher’s tortuous proof (or the only slightly less tortuous results of latter-day interpreters) I’m going to look at a simpler setup due to John Baez, and (unlike Baez) use it to derive the original version of Fisher’s theorem.

So, I’m just catching up with Wakeham, but luckily an earlier blog article of mine helped him avoid “Fisher’s tortuous proof” and the “only slightly less tortuous results of latter-day interpreters”. We are making progress here!

(By the way, a quiz show I listen to recently asked about the difference between “tortuous” and “torturous”. They mean very different things, but this particular case either word would apply.)

My earlier blog article, in turn, was inspired by this paper:

• Marc Harper, Information geometry and evolutionary game theory.