Population turnover, behavioral conservatism, and rates of cultural evolution

Abstract Cultural evolution facilitates behavioral adaptation in many species. The pace of cultural evolution can be accelerated by population turnover, where newcomers (immigrants or juvenile recruits) introduce adaptive cultural traits into their new group. However, where newcomers are naïve to the challenges of their new group, population turnover could potentially slow the rate of cultural evolution. Here, we model cultural evolution with population turnover and show that even if turnover results in the replacement of experienced individuals with naïve ones, turnover can still accelerate cultural evolution if (1) the rate of social learning is more than twice as fast as the turnover rate and (b) newcomers are more likely to learn socially than behaviorally conservative existing group members. Although population turnover is a relatively simple factor, it is common to all animal societies, and variation in the turnover rate may potentially play an important role in explaining variation in the occurrence and rates of adaptive cultural evolution across species.

In our analysis, we assume that the population is initially composed only of inefficient individuals, that is (0) = 0 , (0) = 1 and (0) = 0 , because these conditions best describe the cultural evolution phenomenon we wish to model (see Methods section).However, for completeness, here we start by considering whether some initial conditions may be equilibria.If all individuals in the population are initially naïve, that is (0) = 0, (0) = 0 and (0) = 1, it can be easily shown that, independently of the values of the parameters,  ′ (0) = 0 and  ′ (0) = 0 .The system is at equilibrium and, from logical considerations, it is clear that this equilibrium is stable.The system does not change, because naïve individuals cannot learn the inefficient or efficient solutions independently (we assume no n-to-i or n-to-e innovation, see Methods section), but they can only learn them by copying inefficient or efficient individualsand there are no models available when (0) = 1.In all other cases, it can be easily shown that  ′ (0) and  ′ (0) do not simplify to zero for all parameter values, though they might for specific combinations of parameter values.We explore the possible equilibria when (0) ≠ 1 below.
The system is at equilibrium when  ′ () =  ′ () = 0. We indicate the values for the compartments at equilibrium by  * and  * (and  * = 1 −  * −  * ).Equations (S1-2) become: We solve this system of two equations to obtain  * and  * .Equation ( S3) is solved for  * = 0. We substitute this value in equation (S4) and solve it, obtaining two solutions  * = 1 and  * = /  .Considered that 0 ≤  * ,  * ,  * , ,   ≤ 1, the solutions to the system of equations are: To determine if and when these equilibria are stable, we identify the conditions under which both eigenvalues of the Jacobian matrix for the model evaluated at these points are negative.
The Jacobian matrix is given by: Therefore, the stable equilibria of the system are: This result is illustrated in figure 2e.Notice that these results hold for all initial conditions except (0) = 1.

SM 3 2 Effect of increasing turnover on the time when the efficient solution dominates
Our goal is to identify areas of the parameter space in which increasing population turnover, , results in the efficient solution dominating the population (( ℎ ) = 1/2) faster.Specifically, we want to identify the values of conservatism, , and turnover, , for which  ℎ does not change with .Since our system cannot be solved analytically, we did this numerically in Wolfram Mathematica (see the Mathematica notebook included as an electronic supplementary material).Using values of  ℎ obtained varying  and , we produced an interpolating function ℎ(, ).We then derived this with respect to  and set it to zero to obtain the set of values of  and  for which  ℎ does not vary with  (the black line in figure 2c). Figure S1 shows values of  ℎ as a function of  and , for nine different values of   .
See the Wolfram Mathematica notebook (.nb format), included as an electronic supplementary material, for the code used to produce the figures in the paper and in the Supplementary Materials.

Figure S1 -
Figure S1 -Time at e(th) = ½ (th, dark shades for low values, light shades for high values) as a function of population turnover (m) and behavioural conservatism (c), for different values of rate at which efficient individuals are copied (se).Above the black line, th decreases with increasing m, whereas below the black line th increases with increasing m.Parameter values μ = 0.01, si = 0.10, e0=0, i0=1, n0=0.