A generalized model

The front–quenched coalescent model we presented here represents a scenario in which the initial colonization of a deme is easier than taking over a already colonized deme. The front–quenched coalescent is, of course, not the only way one can model such a situation.

The concept of high–density blocking [100] motivates the following modified rules for a colonization: All colonized demes remain active for the entire colonization process. Colonization attempts are successful with probabilityp1 if the target deme is empty and succeed with probabilityp2 < p1 if the target deme is occupied.

As in the models presented so far, it is not necessary to simulate the full forward population dynamics. Instead, we redefine the coalescent according to the forward model: In the bulk of the population, the lineages move with probabilityp_bulk to one of the neighboring demes, at the front the lineages move with probability pfront

to the parental deme.

For simplicity, set p_front = 1. Then p_bulk = 0 is equivalent to the fully quenched coalescent and pbulk = 1/4 is equivalent to the front–quenched coalescent of sec-tion 10.2.2. Choosing p_front = p_bulk neglects the colonization paths and if the population front is flat, we end up with the unquenched coalescent of section 10.2.3.

For this version of the coalescent, we suggest the term p–quenched coalescent. By tuning the two probabilities, it would be possible to define a ‘quench–factor’ for the coalescent.

Outlook

In this thesis, we modeled populations under different scenarios of range expansions in order to increase the understanding of the impact of range expansions on the neutral genetic diversity.

In part I, we introduced the concept of boundary–limited range expansions and established the distinction towards phenotype–limited range expansions. We were able to demonstrate that these two types can have quite different consequences on neutral genetic diversity.

Populations that track a relatively slow moving boundary during, for instance, a scenario of climate change can maintain high population density up to the expansion front. In this case, the loss of genetic diversity typically associated with range expansions can be drastically reduced and relatively high levels of diversity are expected even close to the expansion front.

Our observations have direct implications for experimental applications: for in-stance, range expansions are commonly identified on the basis of clines in genetic diversity. These clines are much harder and sometimes impossible to detect in boundary–limited range expansions. Based on our results, it is possible to assess the appropriate sampling scheme for the detection of a boundary–limited range expansion or to assert that such a sampling scheme does not exist.

In part II, we generalized the concept of the boundary–limitation to two–dimensional habitats. Our linear habitat model is, to a certain extent, equipped to deal with two-dimensional habitats but it describes only the clines of genetic diversity along the expansion axis. Therefore, we addressed the two–dimensional patterns of diversity in the context of a new model.

In two spatial dimensions, new phenomena arise. The well–mixed population front of the linear habitat is replaced by a front line. This front line is not only spatially structured (topologically one–dimensional) but can also be rough: some regions of the front will expand faster than others, some patches remain temporally unpopulated albeit accessible.

The boundary limitation does not only influence the population density at the front but it can limit the roughness of the front, determine the front shape or introduce an external source of noise to the process.

12. Discussion

The roughness of the front can have a crucial influence on the lineage movement and on the coalescent as we demonstrated with thefully quenched and thefront–quenched coalescent: lineages move super–diffusive in phenotype–limited and diffusive in boundary–limited scenarios. So far, we were not able to fully understand the details of the transition between diffusion and super–diffusion and this problem requires further investigation.

Based on the comparison between thefully quenched, the front–quenched and the unquenched coalescent we tried to quantify the impact of the detailed colonization paths on the patterns of neutral genetic diversity. Drastic differences are apparent between the fully quenched coalescent as compared to the other versions. However, we were not yet able to establish the transition. Thep–quenched coalescent proposed in section 12.5.2 is a promising concept to close that gap.

With the methods and observables developed and presented in this thesis, we aim at providing tools that are applicable in experiments. So far, no other publications made direct use of the concept of boundary–limited range expansion published in 2013 but we are confident that this will change soon. With the analysis of the spatial distribution of alleles we have provided guidelines for a new perspective on genetic data. After all, it is often the representation of data that permits deeper understanding.

A. Appendix to part I

A.1. Supplementary Figures

0 5 10

0 10 20 30

⇠ /

h t

i /⌧

position of the contraction front

range shift range expansion

FIGURE A.1.: Comparison of mean coalescence times (within deme sam-pling) between the range expansion model and the range shift model.

The habitat of the range shift model is extended from0 to 30 while the habitat of the range expansion model is extended from0to infinity. All boundaries move at the same velocity and the parameters of the expansion are as in Figure 4.1.

The distance in which the contraction front has a significant influence on the coalescence process is on the order of . The Figure is from the SI of [78].

A. Appendix to part I

0 0.01 0.02 0.03

0 1 2

⇠/

h T

i /K ⌧

K = 10 K = 10² K = 10³

FIGURE A.2.: Range shifts. Mean coalescence times (scaled by the deme size K) are shown for a habitat of length l = 100 undergoing a range shift with velocity v = 10 ². Note that as we increase deme sizes, the mean coalescence time becomes independent on the sampling location. Other parameters were m= 0.33 and v = 10 ². The Figure is from the SI of [78].