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Analysis of fractional-order two-dimensional models
Šustková, Apolena ; Opluštil, Zdeněk (referee) ; Nechvátal, Luděk (advisor)
This bachelor's thesis deals with the analysis of fractional-order two-dimensional models. The analysis itself is preceded by the introduction to the basic issues concerning the integer-order and fractional-order theory. Investigations are carried out for two specific models, Lotka-Volterra model and the Brusselator, the focus is put primarily on stability of the equilibrium points. The results are supported by appropriate phase portraits that were, for the non-integer case, created using the code for numerical solution of fractional differential equations.
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Lotka-Volterra competition model on graphs
Skácelová, Radka ; Šremr, Jiří (referee) ; Čermák, Jan (advisor)
This bachelor thesis analyzes several mathematical models describing the co-existence of two species, especially the classic Lotka-Volterra model and its extensions. These models are described by a system of non-linear differential equations. The goal of this thesis is to develop an extended predator-prey model using the graph theory, to find stationary states of this model and to analyze their stability. The thesis is furthermore focused on a comparison between the obtained results for this model with the existing results for the competition model on graphs.
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The Lotka-Volterra population model and its generalizations
Zubková, Kateřina ; Tomášek, Petr (referee) ; Čermák, Jan (advisor)
This bachelor's thesis is focused on several dynamical systems of nonlinear differential equations originating from the Lotka-Volterra predator-prey model. The aim of the thesis is to discuss stability and attractivity of the singular solutions of the classical model and its generalizations, investigate its periodicity and impact of the change of the initial data and entry parameters on the system's behaviour. The attention is also paid to involvement of time delay into the studied models, and its influence of stability on singular solutions. From the formal viewpoint, the thesis contains description and application of main stability technique applied to these nonlinear models and testing of results on some data.
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Lotka-Volterra competition model on graphs
Skácelová, Radka ; Šremr, Jiří (referee) ; Čermák, Jan (advisor)
This bachelor thesis analyzes several mathematical models describing the co-existence of two species, especially the classic Lotka-Volterra model and its extensions. These models are described by a system of non-linear differential equations. The goal of this thesis is to develop an extended predator-prey model using the graph theory, to find stationary states of this model and to analyze their stability. The thesis is furthermore focused on a comparison between the obtained results for this model with the existing results for the competition model on graphs.
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Analysis of fractional-order two-dimensional models
Šustková, Apolena ; Opluštil, Zdeněk (referee) ; Nechvátal, Luděk (advisor)
This bachelor's thesis deals with the analysis of fractional-order two-dimensional models. The analysis itself is preceded by the introduction to the basic issues concerning the integer-order and fractional-order theory. Investigations are carried out for two specific models, Lotka-Volterra model and the Brusselator, the focus is put primarily on stability of the equilibrium points. The results are supported by appropriate phase portraits that were, for the non-integer case, created using the code for numerical solution of fractional differential equations.
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The Lotka-Volterra population model and its generalizations
Zubková, Kateřina ; Tomášek, Petr (referee) ; Čermák, Jan (advisor)
This bachelor's thesis is focused on several dynamical systems of nonlinear differential equations originating from the Lotka-Volterra predator-prey model. The aim of the thesis is to discuss stability and attractivity of the singular solutions of the classical model and its generalizations, investigate its periodicity and impact of the change of the initial data and entry parameters on the system's behaviour. The attention is also paid to involvement of time delay into the studied models, and its influence of stability on singular solutions. From the formal viewpoint, the thesis contains description and application of main stability technique applied to these nonlinear models and testing of results on some data.
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Optimalizace těžby přírodních zdrojů
Chrobok, Viktor ; Dlouhý, Martin (advisor) ; Kodera, Jan (referee) ; Vošvrda, Miroslav (referee)
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.
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Těžba v Predator-Prey modelu
Chrobok, Viktor ; Lagová, Milada (advisor) ; Kalčevová, Jana (referee)
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.
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