Kinds of process and the levels of selection

Abstract

Most attempts to answer the question of whether populations of groups can undergo natural selection focus on properties of the groups themselves rather than the dynamics of the population of groups. Those approaches to group selection that do emphasize dynamics lack an account of the relevant notion of equivalent dynamics. I show that the theory of ‘dynamical kinds’ I proposed in Jantzen (Synthese 192(11):3617–3646, 2014) can be used as a framework for assessing dynamical equivalence. That theory is based upon the notion of a dynamical symmetry, a transformation of a system that commutes with its evolution through time. In the proposed framework, structured sets of dynamical symmetries are used to pick out equivalence classes of systems. These classes are large enough to encompass the range of phenomena we associate with natural selection, yet restrictive enough to guarantee a sort of causal homogeneity. By characterizing dynamical kinds via symmetry structures in this way, the question of levels of selection becomes a precise question about which populations respect the dynamical symmetries of Darwinian evolution. Standard population genetic models suggest that populations undergoing evolution by natural selection are partially characterized by a group of fitness-scaling symmetries. I demonstrate conditions under which these symmetries may be satisfied by populations of individuals, populations of groups of individuals, or both simultaneously.

Appendix: Violating the fitness-scaling symmetry

It is straightforward to show that fitness scaling is not a symmetry of Eqs. (6)–(8) unless the fitnesses $\alpha _1$ and $\beta _1$ are directly proportional under all possible transformations of the system that scale group fitnesses. Each of the dynamical equations in this case is a difference equation, expressing a type frequency at time $t+1$ in terms of type frequencies and fitnesses at time t. For ease of notation, let $\mathbf {s}(t)=\langle f_0(t),f_1(t),f_2(t),\omega _0(t), \omega _1(t), \omega _2(t) \rangle$ stand for state of the system at a time t. The variables $\omega _i$ are meant to represent the group type fitnesses. Group fitness $\omega _i$ in the discrete-time case represents the expected total number of offspring produced by a group of type i. Group fitnesses do not appear explicitly in Eqs. (6)–(8), but there is a simple connection between them and the particle fitnesses $\alpha _1, \alpha _2, \beta _0, \beta _1$. Since groups are assumed to be of fixed size, it must be the case that each $\omega _i$ is proportional to $\pi _i$ as defined by Kerr and Godfrey-Smith (2002). Really, each group fitness is the average of the particle fitnesses weighted by the proportion of each particle type appearing in the group. In the two-particle case, this means that $\omega _0 = \beta _0$, $\omega _2=\alpha _2$, and $\omega _1 = \frac{1}{2}(\alpha _1+\beta _1)$. It is this last fitness relation that is problematic.

To see why, we have to consider the relevant transformations. We are interested in fitness-scaling as a set of transformations of group-level variables. There is one fitness scaling transformation, $\mathbf {\sigma }_k$ for every positive real-valued k. These transformations have the effect of multiplying all group fitnesses by a common factor:

$$\begin{aligned} \mathbf {\sigma }_k(\mathbf {s}(t)) = \langle f_0(t),f_1(t),f_2(t),k\omega _0(t), k\omega _1(t), k\omega _2(t) \rangle \end{aligned}$$

According to the definition of dynamical symmetry, $\mathbf {\sigma }_k$ is a symmetry of the group dynamics just if the system state is the same whether we apply $\mathbf {\sigma }_k$ and then use Eqs. (6)–(8) to evolve the system, or first evolve the system and then apply $\mathbf {\sigma }_k$. Let $\varLambda _i$ be the function mapping the state of the system at time t to the value of $f_i$ at time $t+1$. If $\mathbf {\sigma }_k$ is a symmetry, then it must be that for $i=0,1,2$:

$$\begin{aligned} f_i(t+1)=\varLambda _i(f_0(t),f_1(t), f_2(t), k \omega _0(t), k \omega _1(t),k \omega _2(t)) \end{aligned}$$

(15)

Consider just $f_0(t+1)$. Dividing the numerator and denominator of the right-hand side of Eq. (6) by $\frac{1}{2}\beta _1$ gives:

$$\begin{aligned} f_0(t+1) = \frac{(2 \frac{\beta _0}{\beta _1} f_0(t)+f_1(t))^2}{\left( 2\frac{\alpha _2}{\beta _1} f_2(t)+ 2\frac{\beta _0}{\beta _1}f_0(t)+\frac{(\alpha _1+\beta _1)}{\beta _1}f_1(t)\right) ^2} \end{aligned}$$

(16)

A transformation mapping $\omega _0$ to $k \omega _0$ is identical with one which maps $\beta _0$ to $k \beta _0$ at the particle level. The same scaling transformation must also take $\alpha _2$ to $k \alpha _2$. However, the mapping from $\omega _1$ to $k \omega _1$ does not correspond to a unique transformation of particle-fitnesses. In the most general case, we have two functions, g and h such that $g(\alpha _1)+h(\beta _1)=k(\alpha _1+\beta _1)$. Any choice of functions satisfying this condition is equivalent to the single group fitness transformation—the transformation is infinitely degenerate from the particle perspective. However, if the transformation is to satisfy (15), then it must be the case that the coefficient of $f_1(t)$ in the denominator is constant under the scaling transformation. In other words, it must be the case that:

$$\begin{aligned} \frac{g(\alpha _1)+h(\beta _1)}{h(\beta _1)}=\frac{\alpha _1+\beta _1}{\beta _1} \end{aligned}$$

(17)

Since $g(\alpha _1)+h(\beta _1)=k(\alpha _1+\beta _1)$, we have that $\frac{k(\alpha _1+\beta _1)}{h(\beta _1)}=\frac{\alpha _1+\beta _1}{\beta _1}$, and thus $h(\beta _1)=k\beta _1$. Likewise, since, $\frac{g(\alpha _1)+k\beta _1}{k\beta _1}=\frac{\alpha _1+\beta _1}{\beta _1}$ it must be that $g(\alpha _1)=k\alpha _1$. Thus, in order for the group fitness scaling transformation to be a symmetry, it must be the case that:

$$\begin{aligned} \frac{g(\alpha _1)}{h(\beta _1)}=\frac{\alpha _1}{\beta _1} \end{aligned}$$

(18)

In plain language, the ratio of the particle fitnesses must remain fixed under all transformations of the group fitnesses. As a consequence, if the ratio of $\alpha _1$ to $\beta _1$ is not fixed under all group fitness scaling transformations, then fitness scaling is not a symmetry of the group-level dynamics, and the population of groups is not a Darwinian evolver.