### Browsing by Author "Bica, Ion"

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Item A dynamical model of the coral-algae competition in a coral reef ecosystem(2020) Bica, Ion; Solomonovich, MarkA coral reef system is a biodiverse ecosystem in which coral is in mutual competitive partnership with algae. The survival of coral in this competition with algae is vital for the well-being of any coral reef ecosystem. In ideal conditions, the coral mass concentration and algae mass concentration are in a stable equilibrium. However, in practice, it is not always the case due to numerous factors of natural and anthropogenic origin. It is not easy to take into account all these factors when studying the question of survival of the reef ecosystem. We propose a dynamical system that describes the competition between coral and algae and contains terms that describe two major features inherent in this competition. The first one is the accelerated growth of algae when the amount of the turf algae exceeds a certain threshold, and it transforms into macroalgae, which grows much faster and has a detrimental effect on the corals. The second feature is associated with the grazing of the herbivory and other marine life on corals and algae. We apply both analytical and numerical techniques to study the system to find out what kind of equilibria such a system may exhibit. The results of our analysis show that although the boost in the growth of algae may devastate the corals, the latter may still survive if the algae are also subject to sufficiently intense grazing.Item Hydrostatic balance in meteorology(2020) Mucalica, Ana; Bica, IonMeteorology is a branch of geophysics that studies the properties of the atmosphere that “cushions” the Earth and all the phenomena that happen within it. The hydrostatic balance occurs when the pressure at any point in the fluid equals the weight of an air column of the unit section from above the point, and in these ideal conditions, we have the hydrostatic equation for the fluid in hydrostatic balance. The hydrostatic equation is viewed as the hydrostatic equilibrium condition, which provides an accurate approximation for the vertical dependence of the pressure field in the real atmosphere. In this talk, we manipulate a mathematical model describing the variation of pressure to altitude in the atmosphere using partial differential equations, and we will show how realistic charts based on real data are obtained, and compare them to charts obtained by using a climate change scenario when the temperature at sea-level is warmer by 2℃ than the one used in current charts. Work done in collaboration with Anneliese Ansorger, Cory Efird, Cassandra Lisitza, Ghristopher Macyk, Tarig Mergani, Adam Reeves, Rebecca Walton, Yaying Zhong, and Jett Ziehe, MacEwan University.Item Hypercylindrical art(2022) Bica, IonEvery form and shape that surrounds us has a purpose, and mathematics helps in understanding it. Mathematics provides a source of abstract forms, i.e., mathematical structures, which are deliberately emptied of any content and therefore adaptable to any content. Art History is a testimony of great artists who adapted abstract mathematical forms, giving them artistic content. The article was inspired by a piece of art that I purchased of a talented emerging Ecuadorian artist, Tanya Zevallos. Her “Cylindrical Abstract Construction #124” artwork inspired me to look at the mathematical perspective that she gave in her art and construct further abstract aesthetics using cylinders.Item Lorenz’s system - analysis of a sensitive system(2021) Lisitza, Cassandra; Bica, IonMeteorology is a branch of geophysics concerned with atmospheric processes and phenomena and atmospheric effects on our weather. Edward Lorenz was a devoted meteorologist who made several significant contributions to this field. We first describe the fascinating history of Lorenz’s discoveries and his revolutionary additions to the area of meteorology. In particular, he noted the extremely sensitive dependence on the initial conditions of a Chaotic system in the atmosphere, which is commonly referred to as the Butterfly Effect and pertains to Lorenz’s system of three Ordinary Differential Equations (ODEs) that models the atmospheric convection in the atmosphere. We conducted a novel, in-depth mathematical analysis of the Theorem of Existence and Uniqueness for a system of ODEs in general and addressed how it applies to Lorenz’s system. Further, we exhibited how Lorenz’s system is ill-posed using an application where we varied the initial parameters by minimal variations and noted relatively quick and drastic differences in the trajectories of the system.Item Modeling rogue waves with the Kadomtsev-Petviashvili equation(2018) Bica, Ion; Wanye, RandyIn this paper, we derive a new class of solutions for the Kadomtsev-Petviashvili (KP) equation, and we discuss their possible relevance to rogue waves. The nonlinear interaction of these solutions is considered.Item Modelling the Southwestern Alberta grizzly bear population using ordinary differential equations(2019) Bica, Ion; Solomonovich, Mark; Burak, K.; Deutscher, K.; Garrett, A.; Peacock, H.The Alberta grizzly bear population was listed as “threatened” by the Alberta Wildlife Association in 2010. This particular species is important, as it is an umbrella species for a variety of other animals. Our goal in this project was to create a model using ordinary differential equations, based on the logistic growth model, to determine whether the Southern Alberta grizzly bear population is recoverable. We aimed to calculate the rate at which the population was growing and its carrying capacity.Item A modified susceptible-infected-recovered epidemiological model(2022) Bica, Ion; Zhai, Zhichun; Hu, RuiObjectives This paper proposes an infectious disease model incorporating two new model compartments, hospitalization, and intensive care unit. Methods The model dynamics are analyzed using the local and global stability theory of nonlinear systems of ordinary differential equations. For the numerical simulations, we used the Rosenbrock method for stiff initial value problems. We obtained numerical simulations using MAPLE software. The returned MAPLE procedure was called only for points inside the range on which the method evaluated the numerical solution of the system with specied initial conditions. Results We proposed a new model to describe the dynamics of microparasitic infections. Numerical simulations revealed that the proposed model fitted with the expected behaviour of microparasitic infections with "acute epidemicity." The numerical simulations showed consistency in the behaviour of the system. Conclusions The model proposed has "robust" dynamics, supported by the global stability of its endemic state and the consistency of the numerical simulations regarding the model's time evolution behaviour. The introduction of the hospitalization and intensive care unit compartments in the proposed model revealed that it is essential to consider such policies in the case of "acute epidemicity" of microparasitic infections.Item Periodic and solitary wave solutions for the one-dimensional cubic nonlinear Schrodinger model(2022) Bica, Ion; Mucalica, AnaUsing a similar approach as Korteweg and de Vries, [19], we obtain periodic solutions expressed in terms of the Jacobi elliptic function cn, [3], for the self-focusing and defocusing one-dimensional cubic nonlinear Schrodinger equations. We will show that solitary wave solutions are recovered through a limiting process after the elliptic modulus of the Jacobi elliptic function cn that describes the periodic solutions for the self-focusing nonlinear Schrodinger model.Item Robust optimal design when missing data happen at random(2023) Hu, Rui; Bica, Ion; Zhai, ZhichunIn this article, we investigate the robust optimal design problem for the prediction of response when the fitted regression models are only approximately specified, and observations might be missing completely at random. The intuitive idea is as follows: We assume that data are missing at random, and the complete case analysis is applied. To account for the occurrence of missing data, the design criterion we choose is the mean, for the missing indicator, of the averaged (over the design space) mean squared errors of the predictions. To describe the uncertainty in the specification of the real underlying model, we impose a neighborhood structure on the regression response and maximize, analytically, the Mean of the averaged Mean squared Prediction Errors (MMPE), over the entire neighborhood. The maximized MMPE is the “worst” loss in the neighborhood of the fitted regression model. Minimizing the maximum MMPE over the class of designs, we obtain robust “minimax” designs. The robust designs constructed afford protection from increases in prediction errors resulting from model misspecifications.Item Sound signature detection by probability density function of normalized amplitudes(2019) Bica, Ion; Zhai, Zhichun; Hu, Rui; Melnyk, Mickey H.In this paper, we propose to use the probability density function of normalized amplitudes (PDFNA) to detect distinctive sounds in classical music. Based on data sets generated by waveform audio files (WAV files), we use the kernel method to estimate the probability density function. The confidence interval of the kernel density estimator is also given. In order to illustrate our method, we used the audio data collected from recordings of three composers; Johann Sebastian Bach (1686-1750), Ludwig van Beethoven (1770-1827) and Franz Schubert (1797-1828).Item The two physics governing the one-dimensional cubic nonlinear Schrödinger equation(2021) Mucalica, Ana; Bica, IonIn 1926, in his quest to explain the quantum probabilistic nature of particles, Erwin Schrödinger proposed a nonrelativistic wave equation that required only one initial condition, i.e., the initial displacement of an electron. His equation describes the wave-particle duality discovered by Louis de Broglie in 1924. Furthermore, Schrödinger's wave equation is dimensionless, allowing the equation to be a mathematical model describing different physical phenomena. Introducing nonlinearity into the Schrödinger equation, we worked with the so-called self-focusing nonlinear Schrödinger equation. We showed that when the nonlinearity is perfectly balanced with the dispersion, the self-focusing nonlinear Schrödinger model describes the propagation of a soliton. In 1968 Peter Lax introduced the "Lax Pair," a pair of time-dependent matrices/operators describing the nature of a nonlinear evolution partial differential equation, to discuss solitons in continuous media. This procedure is what we call the scattering method for describing mathematically nonlinear processes in physics. We used the scattering method to find the Lax Pair for the nonlinear Schrodinger model, and we showed that the equation is a compatibility condition for the AKNS system. In 1974, Ablowitz, Kaup, Newell, and Segur (AKNS) introduced the inverse scattering transform to solve evolution nonlinear partial differential equations arising from compatibility conditions for the AKNS system. Rather than using the inverse scattering transform, we showed an intuitive approach in revealing the formation and propagation of a soliton for the self-focusing nonlinear Schrodinger equation, using a novel approach via cnoidal waves. The work will also include a novel theorem describing the steepening of the wavefront due to nonlinearity.Item Ursus arctos horribilis: dynamic modeling of Canadian population(2019) Bica, Ion; Solomonovich, Mark; Deutscher, K.; Garrett, A.; Burak, K.; Peacock, H.The grizzly bears are K-strategists and their innate tendency is to reach homeostasis. In the First Nations folklore grizzly bears are viewed as “spirits” that bring balance in their untamed habitat where they roam, this being an indication that they do not overpopulate their habitat and their gene flow is “designed” to reach homeostasis without surpassing it. In the present article we study the dynamics of the grizzly bear population in the Southwest Alberta, Canada. Based on the dynamical model with three parameters, we obtain estimates for the carrying capacity and the minimum viable population of the grizzly bear population in their dynamical habitat. The article starts with the discussion of the rationale for choosing the Logistic Growth Model as the most appropriate for describing the dynamics of grizzly population. In addition to the usual for this kind of models parameters of the growth rate and the carrying capacity, in the current model we consider the parameters of Minimum Viable Population (MVP) and Safe Harbour (SH) – a measurement introduced by the Alberta Grizzly Bear Recovery Plan. The first of these parameters (MVP) is determined by the essential number of the individuals that would allow the survival of the species. The latter measurement (SH) is related to the so-called Grizzly Bear Priority Areas, where the risk of mortality is low. Then, based on Verhulst model and Statistical data, the carrying capacity and growth rate for the female grizzly bears in Alberta have been obtained. Mathematical analysis of the model has shown that the equilibria at K (carrying capacity) and MVP·SH are, respectively, stable and unstable. The time of possible extinction for the populations with the initial conditions below the threshold MVP·SH has been numerically estimated. The correlation between the system parameters and its influence on the survival of the population has been analyzed and the recommendations on ensuring the survival have been given.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” [2] The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce selfsustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator, confirming the findings from the intuitive approach.Item Van der Pol oscillator – analysis of a non-conservative system(2020) Reeves, Adam; Bica, IonThe Van der Pol oscillator was introduced by Balthasar van der Pol, who was ”a famous scholar, a famous scientist, a famous administrator at the international level, he was equally well known for the clarity of his lectures (in several languages), his knowledge of the classics, his warm personality and his talents for friendship, and his love for music.” The oscillator describes the nonlinear oscillations for systems like a triode circuit, which produce self-sustained oscillations known as relaxation oscillations. Extensive studies have been done on the oscillator, for understanding it and for using it as an applied model for the heartbeat, for example. In this thesis, we will explain the nature of the oscillator from an original point of view, in the low-friction regime. First, we will give an intuitive physical explanation of the first order averaging method, a perturbation theory method, applied onto the oscillator. We will follow with an analytical approach of the first order averaging method, and we will show the mathematical complexity of it. We will conclude with the application of the first order averaging method to the Van der Pol oscillator,confirming the findings from the intuitive approach.