Appendix for: ‘Contrasting patterns of density-dependent selection at different life stages can create more than one fast-slow axis of life-history variation’

Abstract There has been much recent research interest in the existence of a major axis of life‐history variation along a fast–slow continuum within almost all major taxonomic groups. Eco‐evolutionary models of density‐dependent selection provide a general explanation for such observations of interspecific variation in the "pace of life." One issue, however, is that some large‐bodied long‐lived “slow” species (e.g., trees and large fish) often show an explosive “fast” type of reproduction with many small offspring, and species with “fast” adult life stages can have comparatively “slow” offspring life stages (e.g., mayflies). We attempt to explain such life‐history evolution using the same eco‐evolutionary modeling approach but with two life stages, separating adult reproductive strategies from offspring survival strategies. When the population dynamics in the two life stages are closely linked and affect each other, density‐dependent selection occurs in parallel on both reproduction and survival, producing the usual one‐dimensional fast–slow continuum (e.g., houseflies to blue whales). However, strong density dependence at either the adult reproduction or offspring survival life stage creates quasi‐independent population dynamics, allowing fast‐type reproduction alongside slow‐type survival (e.g., trees and large fish), or the perhaps rarer slow‐type reproduction alongside fast‐type survival (e.g., mayflies—short‐lived adults producing few long‐lived offspring). Therefore, most types of species life histories in nature can potentially be explained via the eco‐evolutionary consequences of density‐dependent selection given the possible separation of demographic effects at different life stages.


S1: Specific model assumptions
We use the following fecundity function F (z, N t ) = e r F (z)−γ F (z)g (Nt) (S1) where r F (z) is the density independent deterministic growth rate of the fecundity, while γ F (z) is the strength of density regulation and g(·) describes the shape of the density regulation. While the range of F (z, N t ) is [0, ∞), the survival function has to be between zero and one, so we assume the following surival function where r S (z) is the growth rate and γ S (z) is the strength of density regulation of the survival. Other choices of survival function includes the logistic function or probit function, but the function above is similar in shape and we can compute the results exact, without relying on any numerical approximations of the integration.
The environmental factors in fecundity and survival are where ε F,t and ε S,t are standard normally distributed, N (0, 1), so the expectations of the factors are equal to one. For a given value of Λ F,t , we can write and similarly, for a given value of Λ S,t , the survival is In our analysis we have assumed that we only have two phenotypes, so z = [z 1 , z 2 ] T has a bivariate normal distribution: p(z;z, P) = 1 wherez = [z 1 ,z 2 ] T and P is a 2 × 2 matrix with rows equal to P 1· = [σ 2 1 , ρσ 1 σ 2 ] and P 2· = [ρσ 1 σ 2 , σ 2 2 ]. Furthermore, we assume that the growth rates and strengths of density dependence have the following shapes Fecundity is only a function of phenotype z 1 , while survival is a function of z 2 . The shape of density regulation is of the Gompertz type (linear on log scale). Since we have assumed that fecundity and survival are functions of a single phenotype, z 1 and z 2 respectively, we can integrate out the other and are left with the marginal distribution of z 1 ∼ N (z 1 , σ 2 1 ) or z 2 ∼ N (z 2 , σ 2 2 ). The following equation will be usefull: For a given environment, we can compute the mean fecundityF (z t , N t )Λ F,t using Equation (S13) where and we getF To find the selection differential ∆z F,t using Equation (3) in the main text, we see that the nominator is F (z 1 , N t ) times the mean of the normal distribution in Equation (S13), so we get We define G as a 2 × 2 matrix with rows equal to G 1· = [G 11 , G 12 ] and G 2· = [G 12 , G 22 ] (G is symmetric), so we havez Similar to the mean fecundity, we find the mean survival for a given evnironment Λ S,t using Equation (S13). For clarity, we suppress the time notation t, so thatz F,t = [z 1,F ,z 2,F ] T . We now have and getS The selection differential is

S2: Details regarding Figure 3
Each grid cell in both subfigures are constructed similarly: 1. Simulate 100 times the evolution of mean phenotypesz 1 andz 2 , using the algorithm in the previous section, for 1000 generations (t max ).
2. Compute the mean value of the mean phenotypes at the 1000th generation over the 100 iterations, i.e. z 1000 = 1 100 100 j=1z 1000,(j) 3. Compute the difference betweenz 1000 for the current grid cell and the grid cell where σ 2 fec = σ 2 sur = 0 (bottom left corner of the figures) and divide it by the difference between the grid cell where σ 2 fec = σ 2 sur = 1 (top right corner) and where σ 2 fec = σ 2 sur = 0 (bottom left). This will give us differences in eitherz 1 orz 2 with respect to increases in σ 2 fec or σ 2 sur that are relative to largest difference, which occurs at the maximum value of the enivronmental noise. We denote this difference relative to the largest difference by∆.
4. In addition, we create a new variable which is the smallest value of the differenes inz 1 orz 2 , thus for each combination of σ 2 fec and σ 2 sur , we have three values [∆z 1 ,∆z 2 , min(∆z 1 ,∆z 2 )]. 5. Color each grid cell by applying: • a gradient of red colour with respect to∆z 1 , so that the colour is transparent (white) if∆z 1 = 0 or full (red) if∆z 1 = 1.
• a gradient of blue colour with respect to∆z 2 , so that the colour is transparent (white) if∆z 2 = 0 or full (blue) if∆z 2 = 1.
This computation and coloring scheme will result in a grid where cells are white if there is no change in either z 1 orz 2 compared to a population with no environmental variance (∆z 1 =∆z 2 = min(∆z 1 ,∆z 2 ) ≈ 1). The cell will have a red color if the change is mostly in thez 1 direction (∆z 1 ≈ 1 and∆z 2 ≈ 0), a blue color if the change is mostly in thez 2 (∆z 1 ≈ 0 and∆z 2 ≈ 1) direction and a more purple color if bothz 1 andz 2 are changing rather similarly (min(∆z 1 ,∆z 2 ) ≈ 1).