1.6 Outline and main results of the thesis
1.6.1 Rare mutations in competitive Lotka-Volterra systems with mutation 30
mutation probability µ tends to zero. We derive an algorithmic description of the limiting deterministic jump process, which is simplified in the case of equal competition. Finally, we consider a cut-off model, where proliferation is only possible above a certain threshold population size. This chapter was published in the Journal of Mathematical Biology as joint work with Anton Bovier [112],
A. Kraut and A. Bovier, From adaptive dynamics to adaptive walks. Journal of Mathematical Biology, Volume 79 , Number 5, pp. 1699–1747, 2019.
Chapter 2 contains the published version, with only minor changes to correct some typing errors and to adapt the layout to the format of this thesis.
In the main result of Chapter 2 we derive a full description of the limiting jump process of (1.4), as µ → 0, see Theorem 2.12. As discussed above, we rescale the time as tlog 1/µ.
This is the time scale on which microscopic mutant populations can grow. The same limit is also considered in [26, 25], but in a much more restrictive setting. There, the trait graph is a subset of N with nearest neighbour mutations and very specific assumptions on the (invasion) fitness landscape. We consider the n-dimensional hypercube Hn = {0,1}n, due to its nice interpretation as sequences of (in)active genes and since the length of a shortest path between two traitsv, w∈Hnis exactly the 1-normkv−wk1. However, the results can easily be transferred to any finite, possibly directed graph.
The main difficulties of the proof, compared to previous results like [26], come from two sources. First, we allow for coexistence of multiple resident traits. This leads to more complicated calculations for the stability of the resident equilibrium state during the growth phase of the mutants. Instead of approximating the derivative of the population size of one resident trait from above and below, we prove a contraction close to the equilibrium state. To do so, we compare the Euclidean norm and a norm that is related to the positive definiteness of the competition kernel. The latter is an assumption that we make to also guarantee the existence of unique stable equilibria according to [35].
The second complication stems from the fact that we consider a trait graph with circles. As a consequence, two traits are connected by multiple different paths and mutant traits can arise due to different chains of consecutive single mutations. Since we consider a regime of very large mutation rates, this means that each subpopulation has a continuous influx of mutants from different sources. To approximate the growth rate of a population of a certain trait or better its population size, we carefully take into account all these different influences.
Here, a population can grow according to its own fitness or due to incoming mutants from a neighbouring trait. To derive the correct rates or sizes, we introduce a particular induction procedure. For all traits in parallel, we take into account the incoming mutants arising due to an increasing number of mutations, i.e. stemming from traits of increasing distance to the considered trait. This is a fundamentally different approach to the previous papers, where mutation was only going in one direction and one could therefore approximate the population sizes trait by trait, already knowing the size of the direct neighbour.
1.6 Outline and main results of the thesis
The first mutant trait to reach a macroscopic population size invades the resident population according to the Lotka-Volterra dynamics. It is determined by optimising the quotient of initial population size (as exponent of µ) and invasion fitness of the different traits. This is due to the fact that traits reaching a population size of order 1 cannot have gotten there purely due to incoming mutations (those result at most in a population size of order µ), but only due to their own growth with the rate of their invasion fitness. On the time scaletlog 1/µ, it then takes a time of order t=γ/fw,v to grow from µγ toµ0 at ratefw,v. Consequently, minimising this quotient not only gives the next trait to invade the population but also the time step between invasions. The exact bookkeeping of the population sizes of all other traits is necessary to determine the initial conditions for the next invasion step. This is important because the fitness landscape changes according to the new resident population and formerly unfit traits might become fit.
In Theorem 2.14, we present a simplified characterisation of the limiting process for the case of constant competition cv,w ≡ c. In this case, a trait that is once unfit stays unfit indefinitely. As a result, we do not need to keep track of the population sizes of all mi-croscopic populations, which simplifies the algorithmic description. Note that this is not a direct corollary from Theorem 2.12 since the assumption on the positive definiteness of the competition kernel can no longer be satisfied. However, we assume that the individual fitness r is different for each trait, which prevents coexistence and allows us to argue differently in the few places where the positive definiteness is used.
Finally, we propose a cut-off version of the deterministic system (1.4), where proliferation is only possible above a certain threshold for population size, depending on µ. This model is intended to mimic the simultaneous limit of large populations and rare mutations, where reproduction is only possible in populations with at least one individual. Depending on the choice of the threshold, populations of size µ` are able to reproduce, which corresponds to Kµ`K being of order at least one and hence µK ≥ K−1/`. In the case of ` = 1, we again characterise the limiting process in Theorem 2.18. Moreover, we give some results on the accessibility of traits in relation to fitness valleys.
The limiting processes of the deterministic system, corresponding to the different scenarios, behave similar to adaptive walks or flights in the sense of [132, 102, 144, 147]. In the case of Theorem 2.12, the fitness landscape is changing after each invasion. The limiting jump process can take arbitrary large steps and reaches a final state only if there exists a set of coexisting resident traits, for which every invasion fitness is non-positive. In the cases of constant competition, the fitness landscape defined by the individual fitnessesrv corresponds to the setting of adaptive walks. For non-restricted reproduction, as in Theorem 2.14, the limiting process moves along traits of increasing fitness r and eventually reaches the global maximum, where it remains. In Theorem 2.18, we only allow for nearest neighbour jumps.
Thus, the process always jumps to the fittest neighbour of a resident trait, which corresponds to the greedy adaptive walk in [147].
Note that in Chapter 2 we employ a slightly different notation. We consider the trait space X = Hn with elements x, y, z. The rescaled stochastic process is denoted by νtµ,K instead of NK(t)/K and the solution of the Lotka-Volterra system with mutation is denoted by ξtµ instead ofnµ(t). Finally, the parameters for competition areα(x, y) instead of cv,w.
1 Introduction
1.6.2 Simultaneous large population-rare mutation limit for moderate power law mutation
In Chapter 3 we study the simultaneous limit of large populations and rare mutations in the individual-based model (1.3), i.e. K → ∞ and µK → 0. For finite trait spaces and on the time scale logK, we consider moderate mutation probabilities that decay like K−1/α, α >0. This corresponds to regime 3, or regime 1 in the special case of α <1. We derive a full description of the limiting jump process, thus extending the results from Chapter 2 to the simultaneous limit and the results from [25] to a more general trait space. Moreover, we present multiple specific examples of interesting evolutionary scenarios that occur for this choice of mutation rate. This chapter is available as a preprint as joint work with Loren Coquille and Charline Smadi [42],
L. Coquille, A. Kraut, and C. Smadi,Stochastic individual-based models with power law mutation rates on a general finite trait space. Preprint, arXiv:2003.03452, 2020.
Chapter 3 contains the preprint, with only minor changes to correct some typing errors and adapt the layout to the format of this thesis.
The content of Chapter 3 is divided into two parts. In the first part, in Theorem 3.3 and Proposition 3.6, we give a full characterisation of the limiting process, both in terms of the exponents βvK, as defined in (1.8), and the rescaled population size NK/K. This result applies to general finite, possibly directed graphs and provides an algorithmic construction of the limit, as long as there exist unique stable equilibria to the Lotka-Volterra dynamics for the resident and invading mutant traits. For α larger than the longest distance within the trait graph (in terms of shortest path length), the limiting process coincides with the one in Theorem 2.12. For smaller α, the description is more intricate since not all mutant traits are present in the beginning. Therefore, we have to introduce intermediate time steps (between invasions) when new subpopulations arise due to mutation from other growing subpopulations.
Both results are proved simultaneously and the proof relies on the induction approach from Chapter 2, as well as an approximation by (logistic) birth-and-death processes with and without migration. We make extensive use of the limit results for such processes from Champagnat, Méléard, and Tran’s paper [38]. However, we have to generalise some of their results, for example to deal with coexisting resident traits.
In the second part we consider several specific graphs and parameter choices to demonstrate interesting and partially counter-intuitive behaviours that arise in the limiting process under this particular scaling of mutation rates. To name some examples, mutational paths can be longer or shorter than expected, taking seemingly unnecessary detours or skipping traits on the graph. Adding an edge in the graph, i.e. a possibility for mutation, can prolong the time to reach a fitness maximum. Moreover, the process can take arbitrarily large steps, in particular farther than the radius α, in which a resident trait can produce mutants. In this regime, the limiting process can get stuck in a local maximum of the (invasion) fitness landscape, surrounded by unfit traits in a radius ofα.
1.6 Outline and main results of the thesis
Moreover, we consider two cases where metastable stochastic behaviour arises in the other-wise deterministic limit, when considering an even more accelerated time scale in the spirit of [25]. In those cases, we see an effective random walk that jumps between clusters of traits, on which a metastable equilibrium is reached according to the deterministic dynam-ics. These clusters are separated by fitness valleys wider thanα. In Chapter 3 we give these metastability results without proof but provide a heuristic derivation.
1.6.3 Modelling of genetic variation as an escape mechanisms from cancer