Estuda-se a relação entre vários conceitos de estabilidade (permanência, estabilidade assintótica e V L-estabilidade) para a equação de Lotka-Volterra e certas propriedades algébricas da Matriz de Interação. Analisa-se os pontos de equilibrio saturados. / We show a relation of several different concepts of stability (permanence, asymptotic stability and VL-stability) for the Lotka-Yolterra equations and certain algebraic properties of the Interaction Matrix. We analize the saturated equilibrium points.

Estuda-se a relação entre vários conceitos de estabilidade (permanência, estabilidade assintótica e V L-estabilidade) para a equação de Lotka-Volterra e certas propriedades algébricas da Matriz de Interação. Analisa-se os pontos de equilibrio saturados. / We show a relation of several different concepts of stability (permanence, asymptotic stability and VL-stability) for the Lotka-Yolterra equations and certain algebraic properties of the Interaction Matrix. We analize the saturated equilibrium points.

Estuda-se a relação entre vários conceitos de estabilidade (permanência, estabilidade assintótica e V L-estabilidade) para a equação de Lotka-Volterra e certas propriedades algébricas da Matriz de Interação. Analisa-se os pontos de equilibrio saturados. / We show a relation of several different concepts of stability (permanence, asymptotic stability and VL-stability) for the Lotka-Yolterra equations and certain algebraic properties of the Interaction Matrix. We analize the saturated equilibrium points.

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.

In this thesis, we study a class of multi-stable reaction-diffusion systems used to model competing species. Systems in this class possess uniform stable steady states representing semi-trivial solutions. We start by considering a bistable, interaction, where the interactions are of classic “Lotka-Volterra” type and we consider a particular problem with relevance to applications in population dynamics: essentially, we study under what conditions the interplay of relative motility (diffusion) and competitive strength can cause waves of invasion to be halted and reversed. By establishing rigorous results concerning related degenerate and near-degenerate systems,we build a picture of the dependence of the wave speed on system parameters. Our results lead us to conjecture that this class of competition model has three “zones of response” in which the wave direction is left-moving, reversible and right-moving, respectively and indeed that in all three zones, the wave speed is an increasing function of the relative motility. Moreover, we study the effects of domain size on planar and non-planar interfaces and show that curvature plays an important role in determining competitive outcomes. Finally, we study a 3-species Lotka-Volterra model, where the third species is treated as a bio-control agent or a bio-buffer and investigate under what conditions the third species can alter the existing competition interaction.

We consider the boundary control problem to stabilize the power of a signal and a pump propagating down a Raman amplifier. This is essentially an initial-boundary value problem (IBVP) of a hyperbolic system with Lotka-Volterra type nonlinearities. We treat the system as a control problem with states in the function space and use Lyapunov-based analysis to demonstrate asymptotic stability in the C_0 and the L_2-sense. The stability conditions are derived for closed-loop systems with a proportional controller and a dynamic controller, and confirmed by simulations in MATLAB.

We consider the boundary control problem to stabilize the power of a signal and a pump propagating down a Raman amplifier. This is essentially an initial-boundary value problem (IBVP) of a hyperbolic system with Lotka-Volterra type nonlinearities. We treat the system as a control problem with states in the function space and use Lyapunov-based analysis to demonstrate asymptotic stability in the C_0 and the L_2-sense. The stability conditions are derived for closed-loop systems with a proportional controller and a dynamic controller, and confirmed by simulations in MATLAB.

El presente trabajo de título tiene por objetivo mostrar el efe to de la difusión ruzada no-homogénea en la rea ión de equilibrios de oexisten ia, en un modelo de ompeten ia tipo Lotka-Volterra de dos espe ies. La difusión ruzada orresponde a una forma de introdu ir en el modelo la idea de que el ujo de individuos de una espe ie no solo es afe tado por el gradiente de su on entra ión, si no que es afe tado por una fun ión de la on entra ión de ambas espe ies, donde la omponente espa ial apare e de manera explí ita. Se desarrolla el sistema no-esta ionario, demostrando existen ia y uni idad de la solu ión bajo ondi iones ade uadas en los parámetros y en las ondi iones ini iales de este. Para la existen ia, la té ni a utilizada orresponde a a otamientos a priori de las solu iones del sistema, es de ir, suponiendo que la solu ión existe se puede demostrar que ésta y sus derivadas hasta el segundo orden deben estar a otadas y que di ha ota es indepediente del tiempo. Estas otas se obtienen gra ias a apli a iones ade uadas del prin ipio del máximo y del Lema de Hopf para e ua iones parabóli as. Esto ombinado on un argumento de punto jo permite on luir existen ia. La uni idad se demuestra por ontradi ión, apli ando un fa tor integrante ade uado e integra ión por partes. En el aso esta ionario se demuestran ondi iones para la existen ia de equilibrios de oexisten ia y se ara teriza su estabilidad. La existen ia de equilibrios de oexisten ia se ara teriza en términos de fun iones es alares relativamente simples, dependientes del parámetro de difusividad. Para ello se utiliza la teoría de bifur a iones por medio de la té ni a de redu ión de Lyapunov-S hmidt. La estabilidad de los equilibrios en ontrados se determina por medio del estudio del primer valor propio del problema esta ionario linealizado. Esto es su iente gra ias a resultados en la literatura existente. Así, los resultados de esta memoria son dos teoremas, uno de existen ia y uni idad para el sistema no-esta ionario y el otro de ondi iones para la existen ia de equilibrios de oexisten ia para el sistema esta ionario. Se on luye que, para este tipo de sistemas, basta on difusión ruzada no-homogénea pequeña para produ ir equilibrios de o existenia.

In this thesis, a system of ordinary differential equations (ODES) is presented to model the population dynamics between poachers and rhino as a predator-prey system in both South Africa (SA) and the Kruger National Park (KNP). The data used in this thesis consists mainly of government and police reports, as well publications from several NGOs and the limitations caused by this lack of applicable data are explored. The system dynamics are based on Lotka-Volterra differential equations, which are extended to include both a carrying capacity and the Allee effect. This thesis parameterises a model of the dynamics of the interaction between rhino and poachers for some time t and makes predictions based on the interpolation of the available data. The unknown rates and parameters relating to the behaviour of populations R and P are optimised by initially using a combination of educated guesses made from the available data or trial and error until set values are obtained. The remaining unknowns are numerically optimised based on the fixed value parameters. This is considered a constrained system, and the results obtained can only be viewed as constrained predictions based on parameter values obtained by a combination of trial and error and numerical optimisation; namely root mean square (RMS) error considering the available data and model solution at time t. Those parameter values obtained through RMS are regarded as error-minimising parameters within the scope of this research, and make up the final models which are referred to as the models which have been fitted to data. This thesis is an introductory, exploratory work into future attempts at modeling population dynamics with very little or no available data. The models are solved for in a constrained system, limiting the resulting predictions to constrained estimates based on the assigned values to unknown parameters. These solutions predict rhino stabilisation for both models, with active poachers dying out in the KNP but general co-existence observed across SA, within the constrained system.

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