In a stochastic Lotka-Volterra model on a two-dimensional square lattice with periodic boundary conditions and subject to occupation restrictions, there exists an extinction threshold for the predator population that separates a stable active two-species coexistence phase from an inactive state wherein only prey survive. When investigating the non-equilibrium relaxation of the predator density in the vicinity of the phase transition point, we observe critical slowing-down and algebraic decay of the predator density at the extinction critical point. The numerically determined critical exponents are in accord with the established values of the directed percolation universality class. Following a sudden predation rate change to its critical value, one finds critical aging for the predator density autocorrelation function that is also governed by universal scaling exponents. This aging scaling signature of the active-to-absorbing state phase transition emerges at significantly earlier times than the stationary critical power laws, and could thus serve as an advanced indicator of the (predator) population's proximity to its extinction threshold. In order to study boundary effects, we split the system into two patches: Upon setting the predation rates at two distinct values, one half of the system resides in an absorbing state where only the prey survives, while the other half attains a stable coexistence state wherein both species remain active. At the domain boundary, we observe a marked enhancement of the predator population density, the minimum value of the correlation length, and the maximum attenuation rate. Boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern, with two different reaction rates assigned to neighboring patches. We furthermore add another predator species into the system with the purpose of studying possible origins of biodiversity. Predators are characterized with individual predation efficiencies and death rates, to which "Darwinian" evolutionary adaptation is introduced. We find that direct competition between predator species and character displacement together play an important role in yielding stable communities. We develop another variant of the lattice predator-prey model to help understand the killer- prey relationship of two different types of E. coli in a biological experiment, wherein the prey colonies disperse all over the plate while the killer cell population resides at the center, and a "kill zone" of prey forms immediately surrounding the killer, beyond which the prey population gradually increases outward. / Ph. D. / We utilize Monte-Carlo simulations to study population dynamics of Lotka–Volterra model and its variants. Our research topics include the non-equilibrium phase transition from a predator-prey coexistence state to an absorbing state wherein only prey survive, boundary effects in a spatially inhomogeneous system, the stabilization of a three species system with direct competition and “Darwinian” evolutionary adaption introduced, and the formation of spatial patterns in a biological experiment of two killer and prey E. coli species.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

We study diffusion-reaction-advection models describing population dynamics of aquatic organisms subject to a constant drift, with reflecting upstream and outflow downstream boundary conditions. We consider three different models: single logistically growing species, two and three competing species. In the case of a single population, we determine conditions for existence, uniqueness and stability of non-trivial steady-state solutions. We analyze the dependence of such solutions on advection speed, growth rate and length of the habitat. Such analysis offers a possible explanation of the "drift paradox" in our context. We also introduce a spatially implicit ODE (nonspatial approximation) model which captures the essential behavior of the original PDE model. In the case of two competing species, we use a diffusion-advection version of the Lotka-Volterra competition model. Combining numerical and analytical techniques, in both the spatial and nonspatial approximation settings, we describe the effect of advection on competitive outcomes. Finally, in the case of three species, we use the nonspatial approximation approach to analyze and classify the possible scenarios as we change the flow speed in the habitat.

Orientador: Prof. Dr. Roberto Venegeroles Nascimento / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2016. / Neste trabalho se estudaram aspectos básicos relacionados com a dinâmica de células cancerígenas do tipo B-Linfoma BCL1 e de gliomas fazendo ênfases neste último caso. O trabalho se iniciou revisando alguns modelos populacionais do câncer inspirados nos trabalhos de Lotka e Volterra o qual oferecem uma descrição muito simples da interação entre o câncer (presa) e o sistema imunológico (caçador). Posteriormente é revisado um modelo global espaço-temporal baseado nas equações de Fisher-Kolmogorov-Petrovsky- Piskounov (FKPP) [1] o qual permitiu aclarar a dicotomia entre proliferação e motilidade associada fortemente ao crescimento tumoral e à invasividade, respectivamente, das células cancerosas. A partir do modelo FKPP também se fez um estudo computacional mais detalhado aplicando diferentes protocolos de tratamentos para analisar seus efeitos sobre o crescimento e desenvolvimento de gliomas. O estudo sugere que um tratamento com maior tempo entre cada dose poderia ser mais ótimo do que um tratamento mais agressivo. Propõe-se também um modelo populacional local do câncer em que se tem em conta o caráter policlonal das células cancerígenas e as interações destas com o sistema imunológico natural e especifico. Neste último modelo se consegui apreciar fenômenos como dormancy state (estado de latência) e escape phase (fase de escape) para valores dos parâmetros correspondentes ao câncer de tipo B-Linfoma BCL1 [2] o qual explica os fenômenos de imunoedição e escape da imunovigilância [3] o qual poderia permitir propor novos protocolos de tratamentos mais apropriados.Depois se fez uma reparametrização do modelo baseado em algumas características mais próprias das células tumorais do tipo glioma e assumindo presença de imunodeficiência com o que se obtém coexistências oscilatórias periódicas tanto da população tumoral assim como das células do sistema imunológico o qual poderia explicar os casos clínicos de remissão e posterior reincidência tumoral. Finalmente se obtiveram baixo certas condições, uma dinâmica caótica na população tumoral o qual poderia explicar os casos clínicos em que se apresentam falta de controlabilidade da doença sobre tudo em pessoas idosas ou com algum quadro clinico que envolve alguma deficiência no funcionamento normal do sistema imunológico. / In this work we studied basic aspects of the dynamics of cancer cell type B-Lymphoma BCL1 and gliomas making strong emphasis in the latter case. We start reviewing some population models of cancer inspired in the work¿s of Lotka and Volterra, which offers a very simple description of the interaction between cancer (prey) and the immune system (Hunter). Subsequently revise a global model space-time based on the equations of Fisher-Kolmogorov-Petrovsky-Piskounov (FKPP) [1] which allowed elucidating the dichotomy between proliferation and strongly associated motility to tumor growth and invasiveness, respectively, of cancer cells. From the FKPP model also made a more detailed computer study applying different treatment protocols to analyze their effects on the growth and development of gliomas. The study suggests that treatment with longer time between each dose could be more optimal than a more aggressive treatment. Is studied also a local population cancer model that takes into account the polyclonal nature of cancer cells, and these interactions with the natural and specific immune system. In the latter model is able to appreciate phenomena as dormancy state and escape phase for values of parameters corresponding to lymphoma cancer BCL1 [2] which explains the phenomena of immunoediting and tumor escape immuno-surveillance [3] allowing elucidating treatments protocols more appropriate. A re-parameterization was made based on some features of tumor cells glioma type and assuming presence of immunodeficiency with that obtained coexistences periodic oscillatory both tumor populations as well as the immune system cells which could explain the clinical cases of remission and subsequent tumor recurrence. Finally obtained under certain conditions, a chaotic dynamics in tumor population which could explain the clinical cases that present lack of controllability of the disease on all in elderly or with some clinical picture involving some deficiency in the normal functioning of the immune system.

The thesis describes various modifications of the predator-prey model. The modifications are considering several harvesting methods. At the beginning a solution and a sensitivity analysis of the basic model are provided. The first modification is the percentage harvesting model, which could be easily converted to the basic model. Secondly a constant harvesting including a linearization is derived. A significant part is devoted to regulation models with special a focus on environmental applications and the stability of the system. Optimization algorithms for one and both species harvesting are derived and back-tested. One species harvesting is based on econometrical tools; the core of two species harvesting is the modified Newton's method. The economic applications of the model in macroeconomics and oligopoly theory are expanded using the methods derived in the thesis.

The paper is focused on the Predator-Prey model modified in the case of harvesting one or both populations. Firstly there is given a short description of the basic model and the sensitivity analysis. The first essential modification is percentage harvesting. This model could be easily converted to the basic one using a substitution. The next modification is constant harvesting. Solving this system requires linearization, which was properly done and brought valuable results applicable even for the basic or the percentage harvesting model. The next chapter describes regulation models, which could be used especially in applying environmental policies. All reasonable regulation models are shown after distinguishing between discrete and continuous harvesting. The last chapter contains an algorithm for maximizing the profit of a harvester using econometrical modelling tools.

Работа содержит исследования трех моделей живых систем с дискретным временем. В первой главе рассматривается одномерная модель нейронной активности, задаваемая кусочно-гладким отображением. Показывается, что в случае одномерного отображения наличие случайного возмущения приводит к появлению всплесков (спайкингу). Исследуются два механизма генерации спайков, вызванных добавлением случайного возмущения в один из параметров. Иллюстрируется, что сосуществование двух аттракторов является не единственной причиной возникновения спайкинга. Для прогнозирования уровня интенсивности шума, необходимого для генерации спайков, применяется метод доверительных областей, который основан на функции стохастической чувствительности. Также находятся основные характеристики межспайковых интервалов в зависимости от интенсивности шума. Вторая глава работы посвящена применению метода функции стохастической чувствительности к аттракторам кусочно-гладкого одномерного отображения, описывающего динамику численности популяции. Первым этапом исследования является параметрический анализ возможных режимов детерминированной модели: определение зон существования устойчивых равновесий и хаотических аттракторов. Для определения параметрических границ хаотического аттрактора применяется теория критических точек. В случае, когда на систему оказывает влияние случайное воздействие, на основе техники функции стохастической чувствительности дается описание разброса случайных состояний вокруг равновесия и хаотического аттрактора. Проводится сравнительный анализ влияния параметрического и аддитивного шума на аттракторы системы. С помощью техники доверительных интервалов изучаются вероятностные механизмы вымирания популяции под действием шума. Анализируются изменения параметрических границ существования популяции под действием случайного возмущения. В третьей главе проводится анализ возможных динамических режимов детерминированной и стохастической модели Лотки-Вольтерры. В зависимости от двух параметров системы строится карта режимов. Изучаются параметрические зоны существования устойчивых равновесий, циклов, замкнутых инвариантных кривых, а также хаотических аттракторов. Описываются бифуркации удвоения периода, Неймарка--Саккера и кризиса. Демонстрируется сложная форма бассейнов притяжения. Помимо детерминированной системы подробно изучается стохастическая, описывающая влияние внешнего случайного воздействия. В случае хаоса дан алгоритм нахождения критических линий, описывающих границу хаотического аттрактора. Опираясь на найденную чувствительность аттракторов, строятся доверительные полосы и эллипсы, позволяющие описать разброс случайных состояний вокруг детерминированного аттрактора. / The work contains study of three models of biological systems with discrete time. In the first chapter a one-dimensional model of neural activity defined by a piecewise-smooth map is considered. It is shown that in the case of a one-dimensional model, the presence of a random disturbance leads to a spike generation. Two mechanisms of spike generation caused by the presence of a random disturbance in one of the parameters are investigated. It is illustrated that the coexistence of two attractors is not the only reason of spiking. To predict the level of noise intensity needed to generate spikes, the confidence-domain method is used, which is based on the stochastic sensitivity function. The main characteristics of interspike intervals depending on the intensity of the noise are also described. The second chapter is devoted to the application of the method of the stochastic sensitivity function to attractors of a piecewise-smooth one-dimensional map, which describes the population dynamics. The first stage of the study is a parametric analysis of the possible regimes of the deterministic model: determining the zones of existence of stable equilibria and chaotic attractors. The theory of critical points is used to determine the parametric boundaries of a chaotic attractor. In the case where the system is affected by a random noise, based on the stochastic sensitivity function, a description of the spread of random states around equilibrium and a chaotic attractor is given. A comparative analysis of the influence of parametric and additive noise on the attractors is carried out. Using the technique of confidence intervals, the probabilistic mechanisms of extinction of a population under the influence of noise are studied. Changes in the parametric boundaries of the existence of population under the influence of random disturbance are analyzed. In the third chapter the possible dynamic modes of the Lotka-Volterra model in determi\-nistic and stochastic cases are analyzed. Depending on the two parameters of the system, bifurcation diagram is constructed. Parametric zones of the existence of stable equilibria, cycles, closed invariant curves, and also chaotic attractors are studied. The bifurcations of the period doubling, Neimark--Sacker and the crisis are described. The complex shape of the basins of attraction is demonstrated. In addition to the deterministic system, the stochastic system is studied in detail, which describes the influence of external random disturbance. In the case of chaos, an algorithm for finding critical lines describing the boundary of a chaotic attractor is given. Based on the stochastic sensitivity function, confidence bands and ellipses are constructed to describe the spread of random states around a deterministic attractor.