A Tale of Two Equations Ludi Vitae or Motus Vita?

Abstract

In this paper, we investigate the meaning and significance of a remarkable fact: the equivalence of the core equations in two apparently disconnected areas in biology. The core equation in evolutionary game theory expresses the dynamic law known as the replicator equation, and the core equation of population dynamics expresses the dynamic law known as the Lotka–Volterra equation. They are proven equivalent by a simple transformation of variables. What does it mean? Does it mean that evolution via natural selection is better modeled as a process of game playing—ludi vitae—or that of mechanical motion—motus vita? We analyze the complexity of this question in this paper.

Appendix: The Equivalence of the Replicator Equation and the LV Equation

Here is the proof in the completely spelled-out form such that one can see how the simple transformation of variables works to transform one equation to the other (Hofbauer 1981). To prove Theorem 1, we map all the trajectories in $\{ {\mathbf{y}} \in {\mathbb{R}}^n_+ \mid y_n = 1 \}$ to those in the simplex $\{ {\mathbf{x}} \in S_n \mid x_n > 0 \}$ with the following maps:

$$\begin{aligned} {\mathbf{y}} \mapsto {\mathbf{x}}: \;\; x_i= & {} \frac{y_i}{\sum _{j=1}^n y_j}, i = 1,\ldots , n, \\ {\mathbf{x}} \mapsto {\mathbf{y}}: \;\; y_i= & {} \frac{y_i}{y_n} = \frac{x_i}{x_n}. \end{aligned}$$

replicator equation$\Rightarrow$LV equation (Hofbauer and Sigmund 1998)

We begin with an easily proven lemma:

$$\begin{aligned} \dot{\left( \frac{x_i}{x_j}\right) } = \left( \frac{x_i}{x_j}\right) [(A{\mathbf{x}})_i - (A{\mathbf{x}})_j]. \end{aligned}$$

where the two average fitness terms, being the same for $x_i$ and $x_j$, cancel out. Then, using ${\mathbf{x}} \mapsto {\mathbf{y}}$ we have,

$$\begin{aligned} {\dot{y}}_i = \dot{\left( \frac{x_i}{x_n}\right) } = \left( \frac{x_i}{x_n}\right) [(A{\mathbf{x}})_i - (A{\mathbf{x}})_n]. \end{aligned}$$

We are going from a n-dimensional simplex to an n-dimensional open space with $y_n = 1$, and it is well known that A is invariant under the addition of a constant on each column in the matrix (that’s how A can always be equiv-transformed into a matrix whose diagonal elements are all 0). So, we imagine A in the replicator equation as having already been transformed to a matrix with only 0 for the nth row. Thus, we have $(A{\mathbf{x}})_n = 0$ and therefore,

$$\begin{aligned} {\dot{y}}_i = \left( \frac{x_i}{x_n} \right) [(A{\mathbf{x}})_i] = y_i \left( \sum _i a_{ij} y_j \right) x_n, \;\;\; i = 1,\ldots ,n \end{aligned}$$

which can be written more explicitly as (for $x_n = {\text{constant}}$),

$$\begin{aligned} {\dot{y}}_i = y_i \left( a_{in} + \sum _{j=1}^{n-1} a_{ij}y_j \right) , \;\;\; i = 1,\ldots ,n-1 \end{aligned}$$

And this is the LV equation in the theorem since $a_{nn} = a_{nj} = 0, \; \forall j = 1,\ldots ,n-1.$

LV equation$\Rightarrow$replicator equation (Hofbauer 1981)

We begins with the first mapping

$$\begin{aligned} {\dot{x}}_i = \frac{{\dot{y}}_i \sum y_j - y_i \sum {\dot{y}}_j}{(\sum y_j)^2.}. \end{aligned}$$

Going backwards with the above proof, we have the following:

$$\begin{aligned} {\dot{y}}_i = y_i [(A\mathbf{y })_i] = y_i \left( \sum _i a_{ij} y_j \right) , \;\;\; i = 1,\ldots ,n \end{aligned}$$

and

$$\begin{aligned} \sum {\dot{y}}_j = \sum _j y_j \sum _k a_{jk} y_k = \sum _{j,k} a_{jk}y_j y_k \end{aligned}$$

where $i, j, k = 1,\ldots ,n$, and the $a_{ni}$’s are all 0.

Substituting these two expressions back to the first, and noticing that $x_i = y_i / \sum y$, we have

$$\begin{aligned} {\dot{x}}_i = x_i \left( \sum _{j=1}^n a_{ij}x_j \right) - \frac{x_i}{(\sum y)} \left( \sum _{j,k = 1}^n x_j a_{jk} x_k \right) . \end{aligned}$$

where all the summations are from 1 to n, and $\sum y$ is a number.

By absorbing $\sum y$ into the coefficients, we have recovered the replicator equation as,

$$\begin{aligned} {\dot{x}}_i = x_i \left( \sum _{j=1}^n a_{ij}x_j - \sum _{j,k = 1}^n x_j a_{jk} x_k \right) = x_i[(A{\mathbf{x}})_i - {\mathbf{x}} \cdot A{\mathbf{x}})]. \end{aligned}$$

Because the solutions of a replicator equation live on a n-dimensional simplex, which only has $n-1$ degrees of freedom, $y_n = 1$ is defined arbitrarily to lock out one degree of freedom in the LV equation’s solution space.