Aspects of population dynamics

dc.contributor.authorSwailem, Mohameden
dc.contributor.committeechairTauber, Uwe C.en
dc.contributor.committeememberAshkar, Ranaen
dc.contributor.committeememberSharpe, Eric R.en
dc.contributor.committeememberPleimling, Michel Jeanen
dc.description.abstractNatural ecologies are prone to stochastic effects and changing environments that shape their dynamical behavior. Ecological systems can be modeled through relatively simple population dynamics models. There is a plethora of models describing deterministic models of ecological systems evolving in a constant environment. However, stochasticity can lead to extinction or fixation events, noise-stabilized patterns, and nontrivial correlations. Likewise, changing environments can greatly affect the behavior and ultimate fate of ecological systems. In fact, the dynamics of evolution are mostly driven by randomness and changing environments. Therefore, it is of utmost importance to develop population dynamics models that are able to capture the effects of noise and environmental drive. In this thesis, we use both theoretical tools and simulations to investigate population dynamics in the following contexts: We study the stochastic spatial Lotka-Volterra (LV) model for predator-prey interaction subject to a periodically varying carrying capacity. The LV model with on-site lattice occupation restrictions that represent finite food resources for the prey exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by spatio-temporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The stationary regime of our periodically varying LV system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity switching environments, the mean-field rate equations predict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast- and slow periodic switching. The analysis of the mean-field equations in the fast-switching regime enables a semi-quantitative description of the stationary state. The mean-field analysis of the Lotka-Volterra predator-prey model with seasonally varying carrying capacity is extended to the resonant regime. This is done by introducing a homotopy mapping from this model to another model that allows for the application of Floquet theory. The stability of the coexistence fixed point is studied and the period doubling is related to a bifurcation point in the homotopy mapping. However, we find that the predator-prey ecology's coexistence is stable for most of its parameter region. We apply a perturbative Doi–Peliti field-theoretical analysis to the stochastic spatially extended symmetric Rock-Paper-Scissors (RPS) and May–Leonard (ML) models, in which three species compete cyclically. Compared to the two-species Lotka–Volterra predator-prey (LV) model, according to numerical simulations, these cyclical models appear to be less affected by intrinsic stochastic fluctuations. Indeed, we demonstrate that the qualitative features of the ML model are insensitive to intrinsic reaction noise. In contrast, and although not yet observed in numerical simulations, we find that the RPS model acquires significant fluctuation-induced renormalizations in the perturbative regime, similar to the LV model. We also study the formation of spatio-temporal structures in the framework of stability analysis and provide a clearcut explanation for the absence of spatial patterns in the RPS system, whereas the spontaneous emergence of spatio-temporal structures features prominently in the LV and the ML models. Stochastic reaction-diffusion models are employed to represent many complex physical, biological, societal, and ecological systems. The macroscopic reaction rates describing the large-scale, long-time kinetics in such systems are effective, scale-dependent renormalized parameters that need to be either measured experimentally or computed by means of a microscopic model. In a Monte Carlo simulation of stochastic reactiondiffusion systems, microscopic probabilities for specific events to happen serve as the input control parameters. To match the results of any computer simulation to observations or experiments carried out on the macroscale, a mapping is required between the microscopic probabilities that define the Monte Carlo algorithm and the macroscopic reaction rates that are experimentally measured. Finding the functional dependence of emergent macroscopic rates on the microscopic probabilities (subject to specific rules of interaction) is a very difficult problem, and there is currently no systematic, accurate analytical way to achieve this goal. Therefore, we introduce a straightforward numerical method of using lattice Monte Carlo simulations to evaluate the macroscopic reaction rates by directly obtaining the count statistics of how many events occur per simulation time step. Our technique is first tested on well-understood fundamental examples, namely restricted birth processes, diffusion-limited two-particle coagulation, and two-species pair annihilation kinetics. Next we utilize the thus gained experience to investigate how the microscopic algorithmic probabilities become coarse-grained into effective macroscopic rates in more complex model systems such as the Lotka–Volterra model for predator-prey competition and coexistence, as well as the rock-paper-scissors or cyclic Lotka–Volterra model as well as its May–Leonard variant that capture population dynamics with cyclic dominance motifs. Thereby we achieve a more thorough and deeper understanding of coarse-graining in spatially extended stochastic reactiondiffusion systems and the nontrivial relationships between the associated microscopic and macroscopic model parameters, with a focus on ecological systems. The proposed technique should generally provide a useful means to better fit Monte Carlo simulation results to experimental or observational data.en
dc.description.abstractgeneralPopulation dynamics models describe how the number of individuals of interacting species changes over time. This is used to understand the ultimate fate of ecological systems. An ecological system can exhibit long-time multi-species coexistence, the fixation of just one species (all other species go extinct), or total extinction of the system. Understanding the dynamics of the system can help predict the final state of the system from early observations, also, it can inform possible ways to steer the system into a desirable outcome. However, it is very difficult to model such systems due to their complexity. While great progress has been made in understanding well-mixed populations in constant environments, there is still much to learn about ecological systems under spatial and environmental variability. A complete understanding of ecological dynamics and how they couple to evolutionary dynamics requires models of populations that are random, and that take into account how different species might be more or less dominant in different environments. We contribute to investigating these models in the following way: Seasonal variations in temperature leads to a change in the availability of different crops. This affects the resources available for animal species to consume in one season compared to another season (e.g. summer and winter). We study a predator-prey model wherein the resource abundance available to the prey vary between two seasons. We showcase how this affects the system's coexistence regime, and spatial patterns. Cyclic models of predation are models where the food chain is cyclic, meaning that there is no "food chain" but rather a "food circle". We utilize theoretical tools to gain a better understanding of the spontaneous formation of well-known spatial patterns in cyclic predation models. The aim of population dynamics is to write a simple set of equations or models that can accurately capture the behavior of natural ecologies. This is rarely an easy task, because even if the microscopic interactions between species is known, it is very difficult to simplify this microscopic model to a simple set of macroscopic equations. We develop a technique that uses computer simulations to map microscopic interactions into simple rate equations. This work can inform better modeling of observational data.en
dc.description.degreeDoctor of Philosophyen
dc.publisherVirginia Techen
dc.rightsIn Copyrighten
dc.subjectnon-equilibrium statistical mechanicsen
dc.subjectdynamical systemsen
dc.subjectstochastic modelingen
dc.subjectpopulation dynamicsen
dc.subjectpattern formationen
dc.titleAspects of population dynamicsen
dc.typeDissertationen Polytechnic Institute and State Universityen of Philosophyen


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