In this dissertation, we investigate whether evolutionary paths for simple biological systems exhibiting punctuated or intermittent behavior through computational modeling of asexually reproducing unicellular populations. Punctuated equilibrium is a controversial theory proposed by Gould and Eldredge in 1972 [1] to explain observations of species experiencing long periods of stasis followed by bursts of change in the fossil record. The challenges in interpreting the fossil record and in the modeling of complex ecosystems has resulted in limited progress in settling the debate around this conjecture. In our work, we aim to build computational models of a simple biological system to study the behavior of fitness evolution at the population scale where fitness refers to an organism's reproductive success. We are motivated by the results from the ongoing Long-Term Evolution Experiment (LTEE) [2], which observes punctuated patterns in the fitness of E. coli populations grown in a laboratory setting. We construct two models of organismic evolution whose results are consistent with the fitness trends observed in the LTEE. First, we develop a Monte Carlo Wright-Fisher model that we adapt to model the LTEE populations. Our adapted Wright-Fisher model exhibits punctuated patterns at the bacteria's generational time scale. Second, we develop and introduce the smooth-colony dynamics model to study a population of colonies, which is more naturally realistic than the adapted Wright-Fisher model's assumptions and the LTEE's laboratory environment. Our main result shows that while this more biologically realistic model fits the LTEE data, it does not display punctuated patterns in absence of epistasis, or the interactions of genes. This indicates that punctuated behavior at the generation time scale may be limited to specific conditions that may not be naturally occurring. Further work aims to expand the complexity of the smooth-colony dynamics model to provide additional insights into the evolutionary trends of simple biological systems.

We explore, via analytical and numerical methods, the Kelvin-Helmholtz instability in relativistic magnetized plasmas, with applications to astrophysical jets. We solve the single-fluid relativistic magnetohydrodynamic (RMHD) equations in conservative form using a scheme which is fourth-order in space and time. To invert the RMHD equations, we use a highly accurate, rapidly-convergent algorithm which improves upon such schemes as the Newton-Raphson method. Though the exact RMHD equations are marginally stable, numerical discretization renders them unstable. Artificial viscosity restores numerical stability, but also models physical processes (momentum transfer, conductivity). We use a two-dimensional slab geometry with periodic boundary conditions in both directions. The background velocity peaks along the central axis and vanishes asymptotically at the transverse boundaries. Remaining background quantities are uniform, with a flow-aligned background magnetic field. In both the linear and nonlinear analysis, we have successfully unified the HD, MHD, RHD, and RMHD regimes.

The early nonlinear regime corresponds to the formation of counter-rotating vortices, connected by filaments, which persist in the absence of a magnetic field. A magnetic field inhibits the vortices through a series of stages, namely, field amplification, vortex disruption, turbulent breakdown, and an approach to a flow-aligned equilibrium configuration. Similar stages have been discussed in MHD literature. We examine how and to what extent these stages manifest in RMHD for a set of representative field strengths. To characterize field strength, we define a relativistic extension of the Alfvenic Mach number M_A. We observe close complementarity between flow and magnetic field behavior. Weaker fields exhibit more vortex rotation, magnetic reconnection, jet-broadening, and intermediate turbulence. Sufficiently strong fields (M_A < 6) completely suppress vortex formation. Maximum jet deceleration, and viscous dissipation, occurs for intermediate vortex-disruptive fields, while maximum electromagnetic energy extraction occurs for the strongest fields which allow vortex formation. Our results are qualitatively similar to observations of numerous jets, including M87, whose knots may exhibit vortex-like behavior.

Highly relativistic flows destabilize the system, supporting modes with near-maximum growth at smaller wavelengths than the shear width of the velocity. This helps to explain early numerical breakdown of highly relativistic artificial viscosity simulations, a long-standing problem.