Bacteria occupy a wide range of niches with many different types coexisting. They compete directly, with some capable of producing antibiotics that kill other members of the niche. Despite this, long term survival of these ecosystems is possible. Here, we consider a lattice-based three-component system with antibiotic producers, non-producers (or cheaters), and susceptible cells competing. In our system, there is a metabolic cost tied to production rate, resulting in a decrease in growth rate for the producers. Non-producers behave as cheaters that gain the benefit of an antibiotic without the cost of producing it themselves. The susceptible cells are a faster growing different species. The model behaves in a fashion similar to the game “rock-paper-scissors", because producers beat susceptible cells, non-producers beat producers, and susceptible cells beat non-producers. We consider two spatial lattice models, one in which there is a nearest neighbour interaction between cells, and one in which the long-range diffusion of the antibiotic is explicitly included. We consider the parameter space in which the three cell types can coexist (taking into account cost and production rate), and determine the regions in which production rate is too high or too low to allow coexistence. We determine that antibiotic producers will evolve to an optimal production rate and that low-rate producers can outcompete complete cheaters (non-producers). We finally illustrate that the introduction of a fourth “resistant” cell type allows the system to survive with four members for some parameters. In other cases, addition of the resistant cells causes the extinction of the producers, which eventually favours the susceptible cells. / Thesis / Master of Science (MSc) / We looked at computational models of bacterial interaction involving producers, non-producers, and susceptible cell types that interacted in a manner similar to the game “rock-paper-scissors”. We determined that the system is stable for the long term for a given set of parameters, otherwise susceptible cells win as not enough antibiotic is being produced, or too much is being produced, significantly inhibiting the growth of producers. Moreover, we found that these systems can evolve, tending towards one production rate, in order to better allow the system to survive. Non-producers also evolve, tending to low production rates instead. These results have implications in understanding bacteria that cannot be cultured and perhaps aiding in the discovery of novel antibiotics.

Evolutionary game theory originated from the application of mathematical game theory to biological studies. Well-known examples in evolutionary game theory are the prisoner's dilemma, predator-prey models, the rock-paper-scissors game, etc. Recently, such well-known models have attracted increased interest in population dynamics to understand the emergence of biodiversity and species coexistence. Meanwhile, it has been realized that techniques from statistical physics can aid us to gain novel insights into this interdisciplinary field. In our research, we mainly employ individual-based Monte Carlo simulations to study emerging spatio-temporal patterns, spatial correlations, and the influence of quenched spatial disorder in rock-paper-scissors systems either with or without conserved total population number. In balanced rock-paper-scissors systems far away from the ``corner'' of configuration space, it is shown that quenched spatial disorder in the reaction rates has only minor effects on the co-evolutionary dynamics. However, in model variants with strongly asymmetric rates (i.e., ``corner'' rock-paper-scissors systems), we find that spatial rate variability can greatly enhance the fitness of both minor species in``corner'' systems, a phenomenon already observed in two-species Lotka-Volterra predator-prey models. Moreover, we numerically study the influence of either pure hopping processes or exchange processes on the emergence of spiral patterns in spatial rock-paper-scissors systems without conservation law (i.e., May-Leonard model). We also observe distinct extinction features for small spatial May-Leonard systems when the mobility rate crosses the critical threshold which separates the active coexistence state from an inactive absorbing state. In addition, through Monte Carlo simulation on a heterogeneous interacting agents model, we investigate the universal scaling properties in financial markets such as the fat-tail distributions in return and trading volume, the volatility clustering, and the long-range correlation in volatility. It is demonstrated that the long-tail feature in trading volume distribution results in the fat-tail distribution of asset return, and furthermore it is shown that the long tail in trading volume distribution is caused by the heterogeneity in traders' sensitivities to market risk. / Ph. D.