Using mathematical modelling to investigate the adaptive divergence of whitefish in Fennoscandia

Modern speciation theory has greatly benefited from a variety of simple mathematical models focusing on the conditions and patterns of speciation and diversification in the presence of gene flow. Unfortunately the application of general theoretical concepts and tools to specific ecological systems remains a challenge. Here we apply modeling tools to better understand adaptive divergence of whitefish during the postglacial period in lakes of northern Fennoscandia. These lakes harbor up to three different morphs associated with the three major lake habitats: littoral, pelagic, and profundal. Using large-scale individual-based simulations, we aim to identify factors required for in situ emergence of the pelagic and profundal morphs in lakes initially colonized by the littoral morph. The importance of some of the factors we identify and study - sufficiently large levels of initial genetic variation, size- and habitat-specific mating, sufficiently large carrying capacity of the new niche - is already well recognized. In addition, our model also points to two other factors that have been largely disregarded in theoretical studies: fitness-dependent dispersal and strong predation in the ancestral niche coupled with the lack of it in the new niche(s). We use our theoretical results to speculate about the process of diversification of whitefish in Fennoscandia and to identify potentially profitable directions for future empirical research.


Appendices
Demographic equilibrium 24 To estimate the carrying capacity parameters K s, j we proceeded in several steps. First, we estimated the birth rate for whitefish of different size using data in 43,44 . Second, using personal 26 observations on the proportions of mature fish in each niche for trimorphic lakes we came up with realistic values of their equilibrium densities (N) at 1000, 6000 and 150 individuals, for littoral, 28 pelagic and profundal morphs, respectively. These values serve as default values in our simulations. 30 Figure S1: Demographic analysis of each population. N s , v s and b s are the population density, survival rate, and fertility at size s, respectively. Subscripts u and m specify immature and mature individuals.
Then, using the Leslie matrices approach (see Figure S1) and assuming perfect adaptation and no predation, we find that the equilibrium population densities N s of fish of different size in different niches must satisfied the following equalities:

Littoral
Profundal Pelagic where b s is the average number of offspring per female of size s, and subscripts u and m denote 34 immature and mature individuals, respectively. Note that the division by three in all equations for N 0 above is due to the 2:1 male:female sex ratio in whitefish (Author personal observations Assuming adapted individuals (ω s, j (x) = 1), we can now calculate the carrying capacity parameter Assuming that b 3 = 64, from Eq. S1 we get the number of stage zero individuals 44 in the littoral environment N 0 = 15125.
We now have to set the other three survival rate v 0 , v 1 and v 2 . We know their product because 46 it is equal to the ratio of stage 3 and 0 individuals: ν 0 ν 1 ν 2 = N 3u N 0 = 0.019. One possible simple combination of surviving rates that appears to be realistic is (ν 0 , ν 1 , ν 2 ) = (0.2, 0.25, 0.4). With these 48 values, we have equilibrium densities at (N 0 , N 1 , N 2 , N 3u , N 3m ) = (15125, 3025, 756, 291, 709). Using equation S2, the carrying capacity parameters for sizes 0, 1 and 2 for the littoral morph become 50 K 0 = 3781, K 1 = 1008, K 2 = 504, K 3 = 2448.  Profundal morph. In order to find parameters for the carrying capacity of the produndal popu-52 lation, we need to make further assumptions: First, we set its fertility at b 2 = 16 which fits the known relationship between fish size and the number of eggs produced 43,44 . In our unpublished data, of 54 the 243 profundal whitefish sampled, the oldest one was 30 years old. Using the same logic as above, we find its survival rate ν 2 = 0.82 at stage two. We set the equilibrium density of the largest 56 profundal fish to N 2 = 150. Using those assumptions we get a profundal niche with the following equilibrium densities (N 0 , N 1 , N 2u , N 2m ) = (656,66,123,27) and survival (ν 0 , ν 1 , ν 2 ) = (0.10, 0.40, 0.82).

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Using equation S2, the carrying capacity parameters for sizes 0, 1, and 2 for the profundal morph become K 0 = 73, K 1 = 44, K 3 = 683.  Table S1 summarizes the estimates of equilibrium survival rates v and K parameters.