In this thesis I investigate various mathematical problems that are either connected to or loosely inspired by evolutionary biology. The main focus is on variants of existing models with a special emphasis on dynamic models or those in which the random quantities that define the model change over time. The first two chapters are concerned with accessibility percolation on tree-like graphs. I first look at a dynamic model on finite (but arbitrarily large) trees and then consider a static model on infinite (but locally finite) trees. For the dynamic model I consider two different scalings of the height of the tree, with the second scaling corresponding to a ‘critical window’ for the first scaling. For each of these scalings, I show that there exist times within an interval of polynomial size in the height of the tree, at which an accessible path exists (with probability converging to 1 as the height of the tree goes to infinity). For the infinite tree, a criterion is defined that quantifies the amount of branching in the tree, and this criterion is shown to determine a phase transition for the existence of an accessible path. In the second part of the thesis, I investigate variants of a model of adaptive dynamics, in which two mutants compete for fixation. The first variant involves an established type and an invading type, with the growth rate of the invading type subject to periodic switches between two values. I show that the length of time taken to invade is averaged provided the period length grows sufficiently slowly with respect to the carrying capacity
K. If the period length decreases to zero with increasing
K, the probability of invasion is itself averaged. The second variant involves two non-established types with one type initiated after a delay. The probability that either type reaches a size proportional to
K is calculated for different initial starting sizes and it is shown that for fixed starting sizes, there is a phase transition in the duration of the allowed delay on the log
K timescale.
- Accessibility Percolation
- Dynamical Random Graphs
- Adaptive Dynamics
- Periodic Environments
Accessibility Percolation and Evolutionary Dynamics in Varying Environments
Bartos, T. (Author). 24 May 2023
Student thesis: Doctoral Thesis › PhD