This thesis investigates a resource driven probabilistic cellular automata (PCA) model of plant competition in terms of local interactions, spatial distributions, and invasion. The model also incorporates herbivores and carnivores and examines their effect on plant populations and community structure. Comparisons are drawn between the model, field studies and other mathematical models. Chapter 1 provides a background of relevant concepts from plants and animal ecology, details a number of mathematical models used in this field and describes the model relevant models and results in the literature. It concludes with a comparison of the features of the most germane models and field studies. Chapter 2 primarily focuses on plants, argues for the model we have chosen, recaptures previous results which are similar to some natural phenomena, and makes a preliminary investigation of community behaviour and disturbance. It then describes the effect of introducing biomass for plants on species behaviour, and their spatial distributions. Chapter 3 deals with competition between different species, and aspects of invasion. Coexistence between functionally different plants can occur, join count statistics and measures for patch location on the torus are developed and applied. Chapter 4 derives a generalised probabilistic model for ruderal monocultures, finds numerical solutions for these and investigates models for vegetatively growing species of plants. Chapter 5 examines the population effects of herbivory (i.e. importance of spatial correlation of disturbance) and analogies to competitor-stress tolerator-ruderal (CSR) primary plant types, as well as plant successional rates and factors affecting community composition. Equilibrium species composition corresponded to CSR theory when plant immigration was introduced. Chapter 6 investigates the basic effects of carnivory, and discusses parallels between probabilistic cellular automata and field studies.

>Magister Scientiae - MSc / In this thesis, we consider some interesting mathematical models arising in ecology. Our focus is on the exploration of robust numerical solvers which are applicable to models arising in mathematical ecology. To begin with, we consider a simple but nonlinear second-order time-dependent partial differential equation, namely, the Allen-Cahn equation. We discuss the construction of a class of exponential time differencing methods to solve this particular problem. This is then followed by a discussion on the extension of this approach to solve a more difficult fourth-order time-dependent partial differential equation, namely, Kuramoto-Sivashinsky equation. This equation is nonlinear. Further applications include the extension of this approach to solve a complex predator-prey system which is a system of fourth-order time-dependent non-linear partial differential equations. For each of these differential equation models, we presented numerical simulation results. / 2025

Seasonal shifts in predation habits, from a generalist in the summer to a specialist in the winter, have been documented for the great horned owl (Bulbo virginialis) in the boreal forest. This shift occurs largely due to varying prey availability. There is little study of this switching behaviour in the current literature. Since season length is predicted to change under future climate scenarios, it is important to understand resulting effects on species dynamics. Previous work has been done on a two-species seasonal model for the great horned owl and its focal prey, the snowshoe hare (Lepus americanus). In this thesis, we extend the model by adding one of the hare's most important predators, the Canadian lynx (Lynx canadensis). We study the qualitative behaviour of this model as season length changes using tools and techniques from dynamical systems. Our main approach is to determine when the lynx and the owl may invade the system at low density and ask whether mutual invasion of the predators implies stable coexistence in the three-species model. We observe that, as summer length increases, mutual invasion is less likely, and we expect to see extinction of the lynx. However, in all cases where mutual invasion was satisfied, the three species stably coexist.

Temperature isoclines are shifting with global warming. To survive, species with thermal niches must shift their geographical ranges to stay within the bounds of their suitable habitat, or acclimate to a new environment. Mathematical models that study range shifts are called moving-habitat models. The literature is rich and includes modelling with reaction-diffusion equations. Much of this literature represents space by the real line, with a handful studying 2-dimensional domains that are unbounded in at least one direction. The suitable habitat is represented by the set over which the demographics (reaction term) has a positive net growth rate. In some cases, this is a bounded set, in others, it is not. The unsuitable habitat is represented by the set over which the net growth rate is negative. The environmental shift is captured by an imposed shift of the suitable habitat. Individuals respond to their environment via their movement behaviour and many display habitat-dependent dispersal rates and a habitat bias. Such behaviour corresponds to a jump in density across the interface of suitable and unsuitable habitat. The questions motivating moving-habitat models are: when can a species track its shifting habitat and what is the impact of an environmental shift on a persisting species. Without closed form solutions, researchers rely on numerical methods to study the latter, and depending on the movement of the interface, the former may require numerical tools as well. We construct and analyse two numerical methods, a finite difference (FD) scheme and a finite element (FE) method in 1- and 2-dimensional space, respectively. The FD scheme can capture arbitrary movement of the boundary, and the FE method rather general shapes for the suitable habitat. The difficulty arises in capturing the jump across a shifting interface. We construct a reference frame in which the interfaces are fixed in time. We capture the jump in density with a clever placing of the nodes in the FD scheme, and through a Lagrange multiplier in the FE method. With biological applications, we demonstrate the power of our solvers in advancing research for moving-habitat models.

Natural populations may be managed by humans for a number of reasons, with mathematical modelling playing an increasing role in the planning of such management and control strategies. In an increasingly heterogeneous, or `patchy' landscape, the interactions between distinct groups of individuals must be taken into account to predict meaningful management strategies. Invasive control strategies, involving reduction of populations, such as harvesting or culling have been shown to cause a level of disturbance, or spatial perturbation, to these groups, a factor which is largely ignored in the modelling literature. In this thesis, we present a series of deterministic, differential equation models which are used to investigate the impact of this disturbance in response to control. We address this impact in two scenarios. Firstly, in terms of a harvested population, where extinction must be prevented whilst maximising the yield obtained. Secondly, we address the impact of disturbance in an epidemic model, where the aim of the control strategy is to eradicate an endemic pathogen, or to prevent the invasion of a pathogen into a susceptible population. The movement of individuals between patches is modelled as both a constant rate, and a function which is increasing with population density. Finally, we discuss the 'optimal' control strategy in this context. We find that, whilst a population harvested from a coupled system is able to produce an inflated yield, this coupling can also cause the population to be more resistant to higher harvesting efforts, increasing the effort required to drive the population to extinction. Spatial perturbation raises this extinction threshold further still, providing a survival mechanism not only for the individuals that avoid being killed, but for the population as a whole. With regards to the eradication of disease, we show that disturbance may either raise or lower the pathogen exclusion threshold depending on the particular characteristics of the pathogen. In certain cases, we have shown that spatial perturbation may force a population to be susceptible to an infectious invasion where its natural carrying capacity would prevent this.

In this thesis we discuss the synchronization of stochastic population oscillators in ecology and neuroscience. Traditionally, the synchronization of oscillators has been studied in deterministic systems with various modes of synchrony induced by coupling between the oscillators. However, recent developments have shown that an ensemble of uncoupled oscillators can be synchronized by a common noise source alone. By considering the effects of noise-induced synchronization on biological oscillators, we are able to explain various biological phenomena in ecological and neurobiological contexts - most importantly, the long-observed Moran effect. Our formulation of the systems as limit cycle oscillators arising from populations of individuals, each with a random element to its behaviour, also allows us to examine the interaction between an external noise source and this intrinsic stochasticity. This provides possible explanations as to why in ecological systems large-amplitude cycles may not be observed in the wild. In neural population oscillators, we were able to observe not just synchronization, but also clustering in some pa- rameter regimes. Finally, we are also able to extend our methods to include coupling in our models. In particular, we examine the competing effects of dispersal and extrinsic noise on the synchronization of a pair of Rosenzweig-Macarthur predator-prey systems. We discover that common environmental noise will ultimately synchronize the oscillators, but that the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable steady-state probability density. Similar analysis on a coupled system of neural oscillators would be an interesting project for future work, which, among other future directions of research, is discussed in the concluding section of this thesis.

Orientador: João Frederico da Costa Azevedo Meyer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T04:55:31Z (GMT). No. of bitstreams: 1 CarmonaTabares_PauloCesar_D.pdf: 24565019 bytes, checksum: 8ebe9aed1d258a0712f49e9711f8d107 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho, construímos um modelo matemático para avaliar as conjecturas existentes acerca do impacto que tem o material inorgânico particulado (sedimento) nas populações bentônicas predominantes na Enseada Potter. Na construção do modelo são utilizadas informações do fenômeno, proporcionadas pelas pesquisas permanentes na região de estudo. Como resultado, logramos comprovar mediante simulações numéricas, o efeito que produz o sedimento na distribuição e abundância das espécies do substrato marinho, constatando neste ecossistema particular as consequências do aquecimento global nessa parte da região antártica. A modelagem é feita com um sistema de equações diferenciais parciais não- lineares sobre um domínio bidimensional irregular (descritiva da região original), o qual é discretizado nas variáveis espaciais por elementos finitos de primeira ordem e na variável temporal pelo Método de Crank-Nicolson. A resolução do sistema não-linear resultante é aproximada através de um método preditor-corretor cuja solução aproximada é visualizada e valorada qualitativamente usando gráficos evolutivos obtidos por simulações em ambiente MATLAB / Abstract: In this work, we built a mathematical model to evaluate existing conjectures about the impact that inorganic particulate material (sediment) has upon predominating benthic populations in Potter Cove. For the mathematical model, phenomena information was that provided by permanent researches in the study area. As a result, by means of numerical simulations, we were able to confirm the effect of sediment over distribution and abundance for species of marine substrate, verifying in this particular ecosystem, the effects of global warming in this specific Antarctic region. Modeling is done with a system of nonlinear partial differential equations over an irregular two-dimensional domain (descriptive of the original region), which is discretized in the spatial variables by first order finite elements and in the time variable by Crank-Nicolson. The resolution of the resulting nonlinear system is approximated by a predictor-corrector method and the solution is displayed and qualitatively valorized using evolutive graphics, obtain in a MATLAB environment / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

The motile behaviour of bacteria underlies many important aspects of their actions, including pathogenicity, foraging efficiency, and ability to form biofilms. In this thesis, we apply mathematical modelling and analysis to various aspects of the planktonic motility of flagellated bacteria, guided by experimental observations. We use data obtained by tracking free-swimming Rhodobacter sphaeroides under a microscope, taking advantage of the availability of a large dataset acquired using a recently developed, high-throughput protocol. A novel analysis method using a hidden Markov model for the identification of reorientation phases in the tracks is described. This is assessed and compared with an established method using a computational simulation study, which shows that the new method has a reduced error rate and less systematic bias. We proceed to apply the novel analysis method to experimental tracks, demonstrating that we are able to successfully identify reorientations and record the angle changes of each reorientation phase. The analysis pipeline developed here is an important proof of concept, demonstrating a rapid and cost-effective protocol for the investigation of myriad aspects of the motility of microorganisms. In addition, we use mathematical modelling and computational simulations to investigate the effect that the microscope sampling rate has on the observed tracking data. This is an important, but often overlooked aspect of experimental design, which affects the observed data in a complex manner. Finally, we examine the role of rotational diffusion in bacterial motility, testing various models against the analysed data. This provides strong evidence that R. sphaeroides undergoes some form of active reorientation, in contrast to the mainstream belief that the process is passive.