The importance of life history and population regulation for the evolution of social learning

Social learning and life history interact in human adaptation, but nearly all models of the evolution of social learning omit age structure and population regulation. Further progress is hindered by a poor appreciation of how life history affects selection on learning. We discuss why life history and age structure are important for social learning and present an exemplary model of the evolution of social learning in which demographic properties of the population arise endogenously from assumptions about per capita vital rates and different forms of population regulation. We find that, counterintuitively, a stronger reliance on social learning is favoured in organisms characterized by ‘fast’ life histories with high mortality and fertility rates compared to ‘slower’ life histories typical of primates. Long lifespans make early investment in learning more profitable and increase the probability that the environment switches within generations. Both effects favour more individual learning. Additionally, under fertility regulation (as opposed to mortality regulation), more juveniles are born shortly after switches in the environment when many adults are not adapted, creating selection for more individual learning. To explain the empirical association between social learning and long life spans and to appreciate the implications for human evolution, we need further modelling frameworks allowing strategic learning and cumulative culture. This article is part of the theme issue ‘Life history and learning: how childhood, caregiving and old age shape cognition and culture in humans and other animals’.


Summary
Appendix (A) describes the derivation of equilibrium population sizes from the model recursions for both modes of population regulation; (B) shows how principled simulation parameter combinations were derived from analytical expressions; (C) explains how effective vital rate parameters in Fig. 3 were calculated; (D) summarizes all parameters and values used in the simulation models and (E) contains further results and illustrations.

Fertility Regulation
Recall that the number of adults and juveniles under fertility regulation are given by the following recursions, respectively: N 0,t+1 = N 1,t be −δNt (S1) N 1,t+1 = (N 1,t + N 0,t )s (S2) Assuming we are at equilibrium and the population has reached its stable age distribution, we can insert the numbers of juveniles at equilibrium given by equation S1 into equation S2 which yields: Solving this equation forN yields an expression for population equilibrium under fertility regulation.

Mortality Regulation
The number of adults and juveniles under mortality regulation are given by the following recursions: Following the same strategy as above, we first insert the numbers of juveniles at equilibrium given by equation S4 into equation S5 which yields: Solving forN , we have the expression for equilibrium population size under mortality regulation.

B. Derivation of parameter combinations
We started by finding vital rate parameter values that result in the chosenN and L under the specified strength of fertility regulation and, then, solved for the unique value of mortality regulation parameter γ that gives the sameN for the same s and b. If baseline vital rates are constant across different forms of population regulation, expected life spans will necessarily be shorter under mortality regulation, as percapita survival is reduced for any positive population size. Under fertility regulation, s is simply given by: while the corresponding fertility rate b can then be calculated as follows: This gives the unique combination of s and b that yield the specified equilibrium population size given the strength of fertility regulation. A relatively short expected life span ofL = 3 years, for instance, is realized through a per-time-step survival probability of s = 0.66. Under relatively weak fertility regulation (δ = 1 1500 ; Fig.  S2A), this would require a fertility rate of 0.61 to produce an equilibrium population size of 300. Under stronger regulation (δ = 1 550 ; Fig. S2B), b would have to be 0.86 to produce the same population size. A relatively long expected life span of 7.5 years, in contrast, is realized through a per-time-step survival probability of s = 0.86. As fewer individuals die, comparatively lower fertility rates will be required to result in the sameN . With weak fertility regulation, for example, b = 0.188 will suffice to produce the sameN of 300.
To find an equivalent value for the mortality regulation parameter γ that results in the same equilibrium population size under the same constellation of s and b, we equated expressions 2.6 and 2.7 and solved for γ, which gives: The fraction, which is multiplied by δ, evaluates to a positive number and simply determines the slope of a line that, for any δ, gives the unique value of γ that results in the sameN .

C. Effective vital rates
The actual number of births is the result of many interacting factors, such as population size, vital rates and the proportion of adapted adults. Teasing apart these factors, we calculated effective vital rates for different times after an environmental change. Effective vital rates represent the actual per-individual probability of surviving or reproducing at any point in time and depend on baseline vital rates, population size, proportion of individuals (and adaptive behaviors) in different age classes and mean propensities for individual learning.
The effective fertility rate at timestep i after an environmental change is calculated as follows: J i gives the proportion of the population that are juveniles, q A,i represents the proportion of adults that possess adaptive behavior, b is the baseline fertility, β is the fertility advantage for adapted individuals, δ is the parameter controlling fertility regulation and N i is the average population size. The effective survival rate, on the other hand, is given by: ξ J,i represents the average propensity for individual learning among juveniles, q J,i gives the proportion of juveniles that are adapted, c is the recruitment cost of individual learning, s indicates baseline survival probability, σ is the survival advantage for adapted individuals and γ is the mortality regulation parameter.  Figure S1. Population growth curves for adults (pink), juveniles (green) and both combined (yellow) according to analytical models; fertility regulation on the left and mortality regulation on the right; plots show trajectories for equilibrium population sizes of 200, 350 and 500, respectively (from top to bottom). δ = 1/1000,L = 7.5