Replicator Kinetics (2013)

Consider a population of heritable types where an individual of type \(i\), \(X_i\), acquires a payoff \(P_{ij}\) against \(X_j\) in pairwise interactions. Further assume that an individual’s birthrate is determined by its payoff. Conversely, the deathrate depends only on the total population density, $n$: \begin{eqnarray} X_i + X_j & \xrightarrow{P_{ij}} & 2 X_i + X_j & \text{(birth)} \cr X_i & \xrightarrow{\delta(n)} & \emptyset. & \text{(death)} \end{eqnarray} The birth process performs evolutionary selection, while the death process is purely ecological (because it drives population density but not frequency of types, on average). The deathrate could be any non-negative function (even $\delta(n)=0$ leaving only the birth process and a geometrically increasing population).

2.1 Rate equations

These two processes can be applied to simulate a finite population, or analyzed in the limit of an infinite, well-mixed population. Then we find the dynamics are very similar to the replicator equation and reproduce some important properties thereof. The Law of mass action allows to compute the rate equation for the density \(x_i=[X_i]\) of type \(i\): \begin{equation} \frac{dx_i}{dt} = x_i \left( \sum_j P_{ij} x_j - \delta(n) \right) \end{equation} where the total population density is \(n\equiv \sum_k x_k\) and follows \begin{equation} \frac{dn}{dt} = \sum_k \frac{dx_k}{dt} = \sum_{jk} P_{kj} x_j x_k - n \delta(n). \end{equation}

2.2 Fitness

Let $y_i=x_i/n$ be the frequency of type $i$. We define its fitness

as its per-capita average birthrate and the mean fitness

is the overall population per-capita average birthrate.

Then the frequency of type $i$ obeys

For comparison, the replicator equation is $dy_i/dt = y_i (f_i - \bar{f})$, differing only by a factor $n$. So our “replicator kinetics” have exactly the same stability properties as the replicator equation, the only difference is that the rate of evolution varies with the population size. In particular, the evolutionarily stable state is the same in both frameworks.

2.3 Ecology

In terms of fitness $f$ the density rate equation becomes \begin{equation} \frac{dn}{dt} = n \left(n \bar{f} - \delta(n) \right) \end{equation} where $\bar{f}$ itself varies with the population structure.

In the special case where we neglect selection $P_{ij}\equiv r=\text{constant}$ for all $i,j$ so only ecology pertains, then $\bar{f}=r$ and the ecological dynamics become \begin{equation} \frac{dn}{dt} = r n^2 \left(1 - \frac{n}{K} \right) \end{equation} where $K \equiv r n^2 / \delta(n)$. If $\delta(n) \propto n^2$ then $K$ is a constant carrying capacity. The dynamics are similar to logistic except the quadratic factor depresses the growth rate when the population is small, a weak Allee effect.

2.4 Bounded population

or equivalently $\delta(n) \propto n^2$, where $X$ represents an individual of any type.

\(X_k\) is simply a bystander who may be replaced by a child of \(X_i\), depending on its payoff against \(X_j\). In this case the population size is fixed so we recover exactly the replicator equation, without a fluctuating rate depending on the population size as in Equation 2.3.

We could also design a system with an ecological (non-selective) birth process and a selective death process (that depends on payoffs). Let $\pi \geq \max_{i,j} P_{ij}$ be a constant. Then \begin{eqnarray} X_i & \xrightarrow{\beta(n)} & 2 X_i & \text{(birth)} \cr X_i + X_j & \xrightarrow{\pi - P_{ij}} & X_j & \text{(death)} \end{eqnarray} where $\beta(n)$ is a birthrate that may depend on the population density but not its composition. For a stable, finite population it requires that $\beta(n) \prec n$ grows slower than $n$, such as spontaneous birth, $\beta(n) =\text{constant}$.

In this system, a higher payoff reduces the death rate, increasing the survival time.

4.1 Ecology

Let’s consider again the dynamics without selection: $P_{ij} = \text{constant} < \pi$ for all $i,j$ so $\bar{f} = \text{constant}$. Then we can define $K=\beta(n)/(\pi-\bar{f})$ and — applying Equations 2.1, 2.2, and 2.3 to the rate equations — we find \begin{equation} \frac{dn}{dt} = \beta(n) n \left(1 - \frac{n}{K} \right). \end{equation} If we have a constant birthrate $\beta(n)\equiv r = \text{constant}$ then we exactly recover the logistic equation.

Is it possible to formulate a system with logistic ecological behaviour but with selection acting on births instead of deaths? It’s tricky because evolutionary game theory requires pairwise interactions but the birth process must be first-order (spontaneous). To achieve this we need to separate the “game” from the births: Let’s introduce pairwise interactions that determine the birthrates of individuals as follows: \begin{eqnarray} Z_{ik} + Z_{jl} & \xrightarrow{\gamma} & Z_{ij} + Z_{jl} & \text{(game)} \cr Z_{ij} & \xrightarrow{P_{ij}} & 2 Z_{ij} & \text{(birth)} \cr Z_{ij} & \xrightarrow{\delta(n)} & \emptyset. & \text{(death)} \end{eqnarray} We use $Z$ to denote these individuals because they carry two pieces of information: their own type and the type of their last co-player in the game. The density of type $i$ (regardless of last co-player) is now $x_i = \sum_j z_{ij}$ and $n = \sum_i x_i$.

In terms of the frequencies $w_{ij} \equiv z_{ij}/n$ and $y_i \equiv x_i/n = \sum_j w_{ij}$, after some work (see Appendix) we find \begin{equation} \frac{dy_i}{dt} = \sum_j P_{ij} w_{ij} - y_i \sum_{kl} P_{kl} w_{kl}. \end{equation}

If we define the fitness of type $i$

as its per-capita average birthrate and the mean fitness \begin{equation} \bar{f} \equiv \sum_i f_i y_i = \sum_{ij} P_{ij} w_{ij} \end{equation} then we find

5.1 Ecology

Without selection, so $P_{ij}\equiv r=\text{constant}$ for all $i,j$, we require $\delta(n)=r n/K$ to get the logistic equation

To achieve that each individual should compete with all others through a reaction like, \begin{eqnarray} Z_{ik} + Z_{jl} & \xrightarrow{\delta} & Z_{jl}. & \text{(competitive death)} \end{eqnarray}

5.1 Discussion

In this section we went back to select births and ecological deaths but separated the “game” (assignment of birthrates) from the birth process. With a slightly modified fitness function (Equation 5.1) we were able to achieve all of our goals: (1) replicator equation-like frequency dynamics (Equation 5.2), and (2) logistic ecological behaviour (Equation 5.3), with (3) only elementary reactions.

The logistic equation derives from two simple reaction kinetic approaches: $N \rightleftharpoons 2 N$ or with explicit ecological “holes”, $H$: $N+H\rightarrow 2 N, N\rightarrow H$. Here the “holes” serve to limit the population without explicit competition; a successful birth won’t occur without a hole for the offspring to occupy. Since the total population of holes and individuals is conserved, the population is controlled. The former approach is similar to Select births, ecological deaths, suggesting that we should also consider the latter.

\begin{eqnarray} X_i + X_j + H & \xrightarrow{P_{ij}} & 2 X_i + X_j & \text{(birth)} \cr X_i & \xrightarrow{\delta(n)} & H. & \text{(death)} \end{eqnarray}

If the total population, including “holes” is conserved, $K\equiv \sum_l x_l + h$, then the rate equations for this reaction set can be written, substituting $h=K-\sum_l x_l$, as: \begin{equation} \frac{dx_i}{dt} = x_i \left( \left(K - \sum_l x_l\right) \sum_j P_{ij} x_j - \delta(n) \right) \end{equation} where the total population density \(n\) follows \begin{equation} \frac{dn}{dt} = \sum_k \frac{dx_k}{dt} = \left(K - \sum_l x_l\right) \sum_{jk} P_{kj} x_j x_k - n \delta(n). \end{equation}

The rate equation for the frequency comes out more complicated than before but it’s still “replicator-like” in the sense that the equilibria and stabilities are the same as the replicator equation: \begin{equation} \frac{dy_i}{dt} = n y_i \left(f_i - \bar{f}\right) \left(K - n \sum_l y_l\right). \end{equation}

We have looked at four different systems of elementary reactions that produce replicator equation-like dynamics (having the same stability properties, only different in the rate of approach). Two of theses cases also have the pleasing property that they “fall back” to logistic population dynamics in the absence of selection. A third is logistic with a weak Allee effect. The last, Replacement, is less realistic in that the population size is fixed.

Here are some ideas for future consideration:

• Can this approach be extended to \(N\)-player games (eg. the public goods game)?
• Is there a general “solution” for a finite population, eg. Fokker-Planck?

In terms of the frequencies $w_{ij} \equiv z_{ij}/n$ and $y_i \equiv x_i/n = \sum_j w_{ij}$

\begin{eqnarray} \frac{dw_{ij}}{dt} & = & \frac{1}{n} \left[ \frac{dz_{ij}}{dt} - w_{ij} \frac{dn}{dt} \right] \cr \frac{dx_i}{dt} & = & \sum_j \frac{dz_{ij}}{dt} \cr & = & \gamma (x_i n - n x_i) + \sum_j P_{ij} z_{ij} - \delta(n) x_i \cr & = & \sum_j P_{ij} z_{ij} - \delta(n) x_i. \cr \frac{dn}{dt} & = & \sum_i \frac{dx_i}{dt} \cr & = & n \left( \sum_{ij} P_{ij} w_{ij} - \delta(n) \right). \end{eqnarray} From Equation 8.1 \begin{eqnarray} \frac{dz_{ij}}{dt} & = & \gamma n^2 ( y_i y_j - w_{ij} ) + n (P_{ij} - \delta(n)) w_{ij} \cr \frac{dw_{ij}}{dt} & = & \frac{1}{n} \left[ \frac{dz_{ij}}{dt} - w_{ij} \frac{dn}{dt} \right] \cr & = & \gamma n (y_i y_j - w_{ij}) + (P_{ij} - \delta(n)) w_{ij} - w_{ij} \sum_{kl} P_{kl} w_{kl} + \delta(n) w_{ij} \cr \frac{dy_i}{dt} & = & \sum_j P_{ij} w_{ij} - y_i \sum_{kl} P_{kl} w_{kl}. \end{eqnarray}