Dissertations / Theses on the topic 'Lotka-Volterra systems'

This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.

>Magister Scientiae - MSc
In principle, an enrichment of resources in predator-prey systems show prompts destabilisation of a framework, accordingly, falling trophic communication, a phenomenon known to as the \Paradox of Enrichment" [54]. After it was rst genius postured by Rosenzweig [48], various resulting examines, including recently those of Mougi-Nishimura [43] as well as that of Bohannan-Lenski [8], were completed on this problem over numerous decades. Nonetheless, there has been a universal none acceptance of the \paradox" word within an ecological eld due to diverse interpretations [51]. In this dissertation, some theoretical exploratory works are being surveyed in line with giving a concise outline proposed responses to the paradox. Consequently, a quantity of di usion-driven models in mathematical ecology are evaluated and analysed. Accordingly, piloting the way for the spatial structured pattern (we denote it by SSP) formation in nonlinear systems of partial di erential equations [36, 40]. The central point of attention is on enrichment consequences which results toward a paradoxical state. For this purpose, evaluating a number of compartmental models in ecology similar to those of [48] will be of great assistance. Such displays have greater in uence in pattern formations due to diversity in meta-population. Studying the outcomes of initiating an enrichment into [9] of Braverman's model, with a nutrient density (denoted by n) and bacteria compactness (denoted by b) respectively, suits the purpose. The main objective behind being able to transform [9]'s system (2.16) into a new model as a result of enrichment. Accordingly, n has a logistic- type growth with linear di usion, while b poses a Holling Type II and nonlinear di usion r2 nb2 [9, 40]. Five fundamental questions are imposed in order to address and guide the study in accordance with the following sequence: (a) What will be the outcomes of introducing enrichment into [9]'s model? (b) How will such a process in (i) be done in order to change the system (2.16)'s stability state [50]? (c) Whether the paradox does exist in a particular situation or not [51]? Lastly, (d) If an absurdity in (d) does exist, is it reversible [8, 16, 54]? Based on the problem statement above, the investigation will include various matlab simulations. Therefore, being able to give analysis on a local asymptotic stability state when a small perturbation has been introduced [40]. It is for this reason that a bifurcation relevance comes into e ect [58]. There are principal de nitions that are undertaken as the research evolves around them. A study of quantitative response is presented in predator-prey systems in order to establish its stability properties. Due to tradeo s, there is a great likelihood that the growth rate, attack abilities and defense capacities of species have to be examined in line with reviewing parameters which favor stability conditions. Accordingly, an investigation must also re ect chances that leads to extinction or coexistence [7]. Nature is much more complex than scienti c models and laboratories [39]. Therefore, di erent mechanisms have to be integrated in order to establish stability even when a system has been under enrichment [51]. As a result, SSP system is modeled by way of reaction-di usion di erential equations simulated both spatially and temporally. The outcomes of such a system will be best suitable for real-world life situations which control similar behaviors in the future. Comparable models are used in the main compilation phase of dissertation and truly re ected in the literature. The SSP model can be regarded as between (2018-2011), with a stability control study which is of an original.

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Instituto Presbiteriano Mackenzie
In digital game development, it is not uncommon to split the development process in three stages: pre-production, production and post-production. Game planning occurs in pre-production, in which game concept ideas are discussed and defined. In this stage, developers start working on the game design, describing what the game is about, its theme, number of players, game objectives, and others. Game design also includes game mechanics,which describes game rules, what players can and cannot do and how the game systems work. The production stage involves coding and asset creation that are used to build the game. Once the game is done, developers reach the post-production stage, shipping the game and entering the maintenance phase (bug fixing and updates). As for systems, it is possible to model a system using mathematical equations and verify its behaviour via temporal analysis. From this, this thesis aimed to evaluate the possibility of using dynamical systems as a tool to help defining game mechanics for digital games, including definition and analysis of agents and objects and their interaction via temporal analysis of the system. The time travel concept was included to offer players the ability to modify the initial parameters of the system modelled in a game, as a way to solve the challenges and problems presented in the game by changing the system behaviour over time. A digital game was developed as proof of concept, and its mechanics was based on the Lotka-Volterra model with logistic growth, applied to a three-species food chain. An agent-based three-species prey-predator model was also included in the game, and both models' behavior and outcome were compared. A pretest was taken by 11 users to evaluate the use of dynamical system as game mechanics as well as the time travel feature available in the game developed in this thesis. The proof of concept was evaluated and, together with the pretest results, it was confirmed that dynamical systems as game mechanics is possible, as it establishes the relationship between species and set the rules of temporal evolution for the game.
Na área de desenvolvimento de jogos digitais, costuma-se dividir o processo de desenvolvimento em três etapas: pré-produção, produção e pós-produção. A pré-produção envolve o planejamento do jogo, em que conceitos sobre este são discutidos e a ideia a ser desenvolvida é selecionada. Nessa etapa, começa o trabalho de game design (projeto de jogo), no qual se define sobre o que é o jogo, o tema, quantidade de jogadores, objetivos, entre outros. Um dos elementos de jogo definido no game design é a mecânica, que indica as regras e funcionamento do jogo. A produção é a etapa em que código e recursos áudiovisuais săo criados a fim de construir o jogo elaborado na pré-produção. Após o jogo ser desenvolvido, entra-se na etapa de pós-produção, com a distribuição do jogo e manutenção (correções posteriores e atualizações). Quanto ao funcionamento de um sistema real ou fictício, é possível modelar um sistema por meio de equações matemáticas e analisar seu comportamento a partir da evolução temporal. A partir disso, este trabalho teve como objetivo avaliar a possibilidade do uso de sistemas dinâmicos como ferramenta para elaboração da mecânica de jogos digitais, a fim de definir e analisar comportamentos de agentes e objetos e suas interações por meio da evolução temporal do sistema. Propôs-se a inclusão de viagem no tempo para permitir que o jogador modificasse parâmetros iniciais do sistema modelado, redefinindo o comportamento do sistema com o passar do tempo, com o objetivo de solucionar os desafios e problemas dispostos no âmbito do jogo. Para a realização da prova de conceito foi desenvolvido um jogo digital, sendo que aplicou-se como mecânica o modelo de Lotka-Volterra com crescimento logístico para uma cadeia alimentar de três espécies, assim como um sistema presa-predador baseado em agentes, a fim de comparar o funcionamento e comportamento de ambos os modelos. Realizou-se um teste preliminar com 11 usuários para avaliar o jogo desenvolvido na presente pesquisa quanto ao uso de sistemas dinâmicos como mecânica e ŕ funcionalidade de viagem no tempo. Com a análise da prova de conceito e resultados obtidos com o teste preliminar, confirmou-se a possibilidade de aplicação de sistemas dinâmicos como mecânica em jogos digitais, sendo possível estabelecer a relação entre espécies e definir as regras de evolução temporal no âmbito do jogo.

We analyse the preservation of physical properties of numerical approximations tosolutions of the Lotka-Volterra system: its positivity and the conservation of theHamiltonian. We focus on two numerical methods : the symplectic Euler method andan explicit variant of it. We first state under which conditions they are symplectic andwe prove they are both Poisson integrators for the Lotka-Volterra system. Then, westudy under which conditions they stay positive. For the symplectic Euler method,we derive a simple condition under which the numerical approximation always stayspositive. For the explicit variant, there is no such simple condition. Using propertiesof Poisson integrators and backward error analysis, we prove that for initial conditionsin a given set in the positive quadrant, there exists a bound on the step size, such thatnumerical approximations with step sizes smaller than the bound stay positive overexponentially long time intervals. We also show how this bound can be estimated.We illustrate all our results by numerical experiments.

Mestrado em Mathematical Finance
As equações de Lotka-Volterra, também conhecidas por equações de predador-presa, são um conjunto de equações diferencias não-lineares construídas para descrever a relação dinâmica entre espécies na natureza. No entanto, desde a sua publicação vários autores têm vindo a provar que estes sistemas dinâmicos têm diversas aplicações fora da área da biologia. Este trabalho tem como objetivo aprofundar as possíveis aplicações destas equações ao sistema bancário e à economia. Considerando o sistema bancário, estudamos três possíveis sistemas dinâmicos que podem descrever a relação entre o volume de depósitos e empréstimos num banco. Também apontamos as semelhanças entre um sistema bancário de três níveis e uma cadeia alimentar e estudamos a sua estabilidade. Olhando para as aplicações à economia, começamos por estudar o famoso modelo de Goodwin para ciclos de desemprego e crescimento dos ordenados. Para terminar, apresentamos um par de equações predador-presa que descrevem a relação entre bens capitais e bens de consumo, e concluímos que os ciclos económicos são endógenos, auto-sustentáveis e não-lineares.
The Lotka-Volterra equations, frequently referred to as predator-prey equations, are a set of non-linear differential equations constructed to describe the interaction dynamics between different species in nature. Yet, since their publication many authors have proved that the applications of these equations go way beyond mathematical biology. The present work focuses on their application to the banking system and to economics. Regarding the banking system, we study three dynamical systems that may describe the relationship between deposit and loan growth in a bank's balance sheet. In addition, we look at the resemblance between a three level ecological food chain and a three level banking system, and study its stability. As for the applications to economics, we study the famous Goodwin's model for the cyclic behavior of wages and employment. To finish our work we present a pair of predator-prey equations that model the dynamical relationship between consumption and capital goods, finding that economic cycles are endogenous, self-sustained and non-linear.
Mestrado em Mathematical Finance
info:eu-repo/semantics/publishedVersion

Interacting species systems have complex dynamics that are often subject to change due to internal and external factors. Quantitative modelling approaches to capture demographic fluctuations can be insufficient in the presence of stochastic variation and uncertainty. This thesis establishes new modelling techniques to account for such demographic variations and develops novel numerical simulation tools for solving these systems. These explorations are extended for solving invasive species management problems where robust management actions and efficient use of allocated budgets are necessary.

Traditionally, differential-equation models for population dynamics have considered organisms as "fixed" entities in terms of their behaviour and characteristics. However, there have been many observations of adaptivity in organisms, both at the level of behaviour and as an evolutionary change of traits, in response to the environmental conditions. Taking such adaptiveness into account alters the qualitative dynamics of traditional models and is an important factor to be included, for example, when developing reliable model predictions under changing environmental conditions. In this thesis, we consider piecewise-smooth and smooth dynamical systems to represent adaptive change in a 1 predator-2 prey system. First, we derive a novel piecewise-smooth dynamical system for a predator switching between its preferred and alternative prey type in response to prey abundance. We consider a linear ecological trade-off and discover a novel bifurcation as we change the slope of the trade-off. Second, we reformulate the piecewise-smooth system as two novel 1 predator-2 prey smooth dynamical systems. As opposed to the piecewise-smooth system that includes a discontinuity in the vector fields and assumes that a predator switches its feeding strategy instantaneously, we relax this assumption in these systems and consider continuous change in a predator trait. We use plankton as our reference organism because they serve as an important model system. We compare the model simulations with data from Lake Constance on the German-Swiss-Austrian border and suggest possible mechanistic explanations for cycles in plankton concentrations in spring.

This work is devoted to investigating the algebraic and analytic integrability of first order polynomial partial differential equations via an understanding of the well-developed area of local and global integrability of polynomial vector fields. In the view of characteristics method, the search of first integrals of the first order partial differential equations P(x,y,z)∂z(x,y) ∂x +Q(x,y,z)∂z(x,y) ∂y = R(x,y,z), (1) is equivalent to the search of first integrals of the system of the ordinary differential equations dx/dt= P(x,y,z), dy/dt= Q(x,y,z), dz/dt= R(x,y,z). (2) The trajectories of (2) will be found by representing these trajectories as the intersection of level surfaces of first integrals of (1). We would like to investigate the integrability of the partial differential equation (1) around a singularity. This is a case where understanding of ordinary differential equations will help understanding of partial differential equations. Clearly, first integrals of the partial differential equation (1), are first integrals of the ordinary differential equations (2). So, if (2) has two first integrals φ1(x,y,z) =C1and φ2(x,y,z) =C2, where C1and C2 are constants, then the general solution of (1) is F(φ1,φ2) = 0, where F is an arbitrary function of φ1and φ2. We choose for our investigation a system with quadratic nonlinearities and such that the axes planes are invariant for the characteristics: this gives three dimensional Lotka– Volterra systems x' =dx/dt= P = x(λ +ax+by+cz), y' =dy/dt= Q = y(µ +dx+ey+ fz), z' =dz/dt= R = z(ν +gx+hy+kz), where λ,µ,ν 6= 0. v Several problems have been investigated in this work such as the study of local integrability and linearizability of three dimensional Lotka–Volterra equations with (λ:µ:ν)–resonance. More precisely, we give a complete set of necessary and sufficient conditions for both integrability and linearizability for three dimensional Lotka-Volterra systems for (1:−1:1), (2:−1:1) and (1:−2:1)–resonance. To prove their sufficiency, we mainly use the method of Darboux with the existence of inverse Jacobi multipliers, and the linearizability of a node in two variables with power-series arguments in the third variable. Also, more general three dimensional system have been investigated and necessary and sufficient conditions are obtained. In another approach, we also consider the applicability of an entirely different method which based on the monodromy method to prove the sufficiency of integrability of these systems. These investigations, in fact, mean that we generalized the classical centre-focus problem in two dimensional vector fields to three dimensional vector fields. In three dimensions, the possible mechanisms underling integrability are more difficult and computationally much harder. We also give a generalization of Singer’s theorem about the existence of Liouvillian first integrals in codimension 1 foliations in Cnas well as to three dimensional vector fields. Finally, we characterize the centres of the quasi-homogeneous planar polynomial differential systems of degree three. We show that at most one limit cycle can bifurcate from the periodic orbits of a centre of a cubic homogeneous polynomial system using the averaging theory of first order.

Thesis (S.M.)--Massachusetts Institute of Technology, System Design and Management Program, 2008.
Includes bibliographical references (leaves 78-80).
Scholars have developed a range of qualitative and quantitative models for generalizing the dynamics of technological innovation and identifying patterns of competition between rivals. This thesis compares two predominant approaches in the quantified modeling of technological innovation and competition. Multi-mode framework, based on the Lotka-Volterra equation barrowed from biological ecology, provide a rich setting for assessing the interaction between two or more technologies. A more recent approach uses System Dynamics to model the dynamics of innovative industries. A System Dynamics approach enables the development of very comprehensive models, which can cover multiple dimensions of innovation, and provides very broad insights for innovative and competitive landscape of an industry. As well as comparing these theories in detail, a case study is also performed on both of them. The phenomenal competition between two technologies in the consumer photography market; the recent battle between digital and film camera technology, is used as a test case and simulated by both models. Real market data is used as inputs to the simulations. Outputs are compared and interpreted with the realities of the current market conditions and predictions of industry analysts. Conclusions are derived on the strengths and weaknesses of both approaches. Directions for future research on model extensions incorporating other forms of innovation are given, such as collaborative interaction in SME networks.
by Hakkı Özgür Ünver.
S.M.

Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015
In the 1970’s John M. Smith and George R. Price [22] applied the theory of strategic games developed by John von Neumann and Oskar Morgenstern [42] in the 1940’s to investigate the dynamical processes of biological populations, giving rise to the field of the Evolutionary Game Theory (EGT). Some classes of ordinary differential equations (o.d.e.s) which plays a central role in EGT are the Lotka-Volterra systems (LV), the replicator equation, the bimatrix replicator and the polymatrix replicator. Many properties of the LV systems can be geometrically expressed in terms of its associated graph, constructed from the system’s interaction matrix. For the class of stably dissipative LV systems we prove that the rank of its defining matrix, which is the dimension of the associated invariant foliation, is completely determined by the system’s graph. In this thesis we also study analytic flows defined on polytopes. We present a theory that allows us to analyze the asymptotic dynamics of the flow along the heteroclinic network composed by the flowing-edges and the vertices of the polytope where the flow is defined. In this context, given a flow defined on a polytope, we give sufficient conditions for the existence of normally hyperbolic stable and unstable manifolds for heteroclinic cycles. In polymatrix games population is divided in a finite number of groups, each one with a finite number of strategies. Interactions between individuals of any two groups are allowed, including the same group. The differential equation associated to a polymatrix game, that we designate as polymatrix replicator, is defined in a polytope given by a finite product of simplices. Karl Sigmund and Josef Hofbauer [16] and Wolfgang Jansen [18] give sufficient conditions for permanence in the usual replicators. We generalize these results for polymatrix replicators. Also for polymatrix replicators we extend the concept of stably dissipativeness developed by Ray Redheffer et al. [25–29]. In this context we generalize a theorem of Waldyr Oliva et al. [6] about the Hamiltonian nature of the limit dynamics in “stably dissipative” polymatrix replicators. We present also some examples to illustrate fundamental results and concepts developed along the thesis.
Na década de 1970 John M. Smith e George R. Price [22] começaram a usar a teoria de jogos estratégicos desenvolvida por John von Neumann e Oskar Morgenstern [42] nos anos 1940 para investigar os processos dinâmicos de populações, dando assim origem à Teoria de Jogos Evolutivos (TJE). Algumas classes de equações diferenciais ordinárias (e.d.o.s) que têm um papel central na TJE são os sistemas Lotka-Volterra (LV), a equação do replicador, o replicador bimatricial e o replicador polimatricial. Muitas propriedades dos sistemas LV podem ser expressas geometricamente em termos do seu grafo associado, construído a partir da matriz de interaçcão do sistema. Para a classe dos sistemas LV estavelmente dissipativos provamos que a característica da sua matriz de interaçcão, que é a dimensão da folheação invariante associada, é completamente determinada pelo grafo do sistema. Nesta tese estudamos também fluxos analíticos definidos em politopos. Apresentamos uma teoria que nos permite analisar a dinâmica assintótica do fluxo ao longo da rede heteroclítica formada pelas arestas e vértices do politopo onde os fluxos estão definidos. Neste contexto, dado um fluxo definido num politopo, damos condições suficientes para a existência de variedades normalmente hiperbólicas estáveis e instáveis para ciclos heteroclíticos. Nos jogos polimatriciais a população é dividida num número finito de grupos, cada um com um n´úmero finito de estratégias. As interaçcões entre indivíduos de quaisquer dois grupos podem ocorrer, inclusive do mesmo grupo. A equação diferencial associada a um jogo polimatricial, que designamos por replicador polimatricial, está definida num politopo dado por um produto finito de simplexos. Karl Sigmund e Josef Hofbauer [16] e Wolfgang Jansen [18] apresentam condições suficientes para a permanência nos replicadores usuais. Nesta tese generalizamos esses resultados para os jogos polimatriciais. Também para os replicadores polimatriciais estendemos o conceito de estabilidade dissipativa desenvolvido por Ray Redheffer et al. [25–29]. Neste contexto generalizamos um teorema de Waldyr Oliva et al. [6] sobre a natureza Hamiltoniana da dinâmica limite em replicadores polimatriciais “estavelmente dissipativos”. Apresentamos ainda alguns exemplos para ilustrar resultados e conceitos fundamentais desenvolvidos ao longo da tese.
Fundo Social Europeu (FSE, Programa Operacional Potencial Humano - POPH)

博士
國立臺灣大學
數學研究所
100
In the present work, we study diffusive Lotka-Volterra systems of two-species and three-species. For competitive systems of two species, the tanh method is applied to construct exact traveling wave solutions. Based on the Fujita-type results, the method of shifted coexistence is developed to find blow-up solutions of cooperative systems of two species (with Xian Liao). For competitive-cooperative and competitive systems of three species, we employ the method of super- and subsolutions to establish the existence of traveling wave solutions. By using the generalized tanh method, it is shown that exact (with M. Mimura et al.) and semi-exact (with Yu-Sheng Chiou) traveling wave solutions exist for competitive systems of three species. In addition, nonexistence of traveling wave solutions to competitive systems of three species is also established by the maximum principle. Finally, we show solutions to competitive systems of three species can be constructed from the solutions of the heat equation. Further investigations include how to study diffusion-enhanced long-term coexistence, which is an interesting new phenomenon discovered by means of the solutions constructed from the heat equation.

碩士
國立交通大學
應用數學系所
100
In this thesis, we review the investigations of dynamics for Lotka Volterra models and patch models in mathematical ecology. We study two open questions posed by Gourley and Kuang in 2005, which are concerned with how dispersal rates affect the competition in two-species patch model with various spatial distribution of their growth rate. It was conjectured that, in a high dispersal environment, the winning strategy for species depends on the growth rate in a single patch. That is, the species which has the greatest growth rate will win. On the other hand, the system may have a globally asymptotically stable positive equilibrium for a small enough dispersal rate. We have not solved the conjectures, but have better understanding on these issues.

碩士
國立臺灣大學
數學研究所
104
The N-barrier maximum principle (NBMP) is a technique to estimate the total density of traveling wave solutions to one-dimensional diffusive competitive Lotka-Volterra systems. In this study, two-species cases, which are considered in [4], are generalized to multi-species cases. In addition, the constraints of the tangent line method proposed in [4] to obtain a refined estimate is released.

碩士
國立臺灣大學
數學研究所
106
In the present paper, we show that an analogous N-barrier maximum principle (see [3,5,7]) remains true for lattice systems. This extends the results in [3,5,7] from continuous equations to discrete and non-local equations. In order to overcome the difficulty induced by a discrete and non-local version of the classical diffusion in the lattice and non-local systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology, and show that the three species cannot coexist under certain conditions.

abstract: There has been important progress in understanding ecological dynamics through the development of the theory of ecological stoichiometry. This fast growing theory provides new constraints and mechanisms that can be formulated into mathematical models. Stoichiometric models incorporate the effects of both food quantity and food quality into a single framework that produce rich dynamics. While the effects of nutrient deficiency on consumer growth are well understood, recent discoveries in ecological stoichiometry suggest that consumer dynamics are not only affected by insufficient food nutrient content (low phosphorus (P): carbon (C) ratio) but also by excess food nutrient content (high P:C). This phenomenon, known as the stoichiometric knife edge, in which animal growth is reduced not only by food with low P content but also by food with high P content, needs to be incorporated into mathematical models. Here we present Lotka-Volterra type models to investigate the growth response of Daphnia to algae of varying P:C ratios. Using a nonsmooth system of two ordinary differential equations (ODEs), we formulate the first model to incorporate the phenomenon of the stoichiometric knife edge. We then extend this stoichiometric model by mechanistically deriving and tracking free P in the environment. This resulting full knife edge model is a nonsmooth system of three ODEs. Bifurcation analysis and numerical simulations of the full model, that explicitly tracks phosphorus, leads to quantitatively different predictions than previous models that neglect to track free nutrients. The full model shows that the grazer population is sensitive to excess nutrient concentrations as a dynamical free nutrient pool induces extreme grazer population density changes. These modeling efforts provide insight on the effects of excess nutrient content on grazer dynamics and deepen our understanding of the effects of stoichiometry on the mechanisms governing population dynamics and the interactions between trophic levels.
Dissertation/Thesis
Ph.D. Applied Mathematics 2014

碩士
東吳大學
數學系
89
We consider the dynamics of an epidemiological model of infection by two competing strains of virus which can be transformed into the Lotka-Volterra system. We prove that equilibrium of coexistence is globally asymptotically stable whenever it exists. If there is no interior equilibrium, there exists a unique locally asymptotically stable equilrbrium corresponding to the strain with greater basic reprodctive number Ro which is called the winning strain. The global stability of winning strain is also obtained in each combination of parameters.

碩士
國立臺灣大學
數學研究所
93
The existence of traveling wave front solution is established for diffusive and cooperative Lotka-Volterra system with time delays. The approaches used in this paper are the upper-lower solution technique, the monotone iteration by Wu and Zou in reference [6] for delayed reaction-diffusion systems. From the theorem 2.1 in reference [5] and theorem 4.5* in references [6], we know that if time delay is sufficiently small and all conditions in the above theorems hold. And then we have a traveling wave front solution. Now, we hope that we can change the upper and lower solutions in reference [5] and reduce the constraints on time delays. Keywords: Traveling wave solution; Upper solution; Lower solution;Time delay.

碩士
東海大學
數學系
91
In this thesis, we are concerned with the dynamical behavior of a two-species Lotka- Volterra mutualistic system with time delay. First of all, we use three different methods to discuss the global stability of the unique positive equilibrium point of a two-species Lotka-Volterra mutualistic system without time delay. Secondly, we study the change of the global stability for a two-species Lotka-Volterra mutualistic system with time delay. Finally, we illustrative our results by some examples.

碩士
國立中央大學
數學系
107
In this thesis, we study the stability of traveling wave solutions for the three species competition cooperation system, which is the discrete version of the Lotka-Volterra system. Applying the weighted energy method and the comparison principle, we can derive the result that the traveling wavefronts with large speed are exponentially stable.

碩士
國立臺灣大學
數學研究所
106
In this thesis, we study the Lotka-Volterra 3 species competition model. We first introduce the conclusion of existence of 2-species traveling wave solution. Next, the main idea to prove the existence of 3-species traveling wave solution is the perturbation method and the iteration argument. We will prove that when the competition rate between w and both u & v is small enough, other parameters satisfy some suitable condition, then there exist a nontrivial 3-species traveling wave solution.

碩士
國立成功大學
經營管理碩士學位學程(AMBA)
101
In recent years, the network and technology products often brought people a refreshing surprise and made a great change in lifestyle, which became more convenient and abundant. The case of smart phone can explain the situation clearly. The Android and the iOS systems are the major operating systems of mobile phone, and the consideration of operation system is one of the reasons of purchasing smart phones. The market status of iOS and Android are closed-end and open-end respectively, which are shown in different competitive relationships. This study would like to discuss the key factors which may affect the competitive relationship in different stages. The Lotka-Volterra Model is based on the growth curve which is to discover the interaction between the two competitive species. And this article would like to utilize the Lotka-Volterra Model as a research framework and to discover the different competitive relationships. The ‘system dynamics’ is used as the research tool to build the competitive model of Lotka-Volterra. It is to analyze the impact of the two competitor’s sales and to simulate the sales. According to the simulation result which shows that the Lotka-Volterra competitive model holds a reasonable capability of simulation. Moreover, the parameter adjustment of Lotka-Volterra model provides the factors which affect the sales for both competitors. In addition, the growth rates of application of both sides are added in the Lotka-Volterra competitive model in order to simulate the sales for both sides. This article presents the results of simulation which may reveal the Volterra Model has a better simulation capability for smart phones market. Therefore, the Lotka-Volterra competitive model proposed in this study can be beneficial to the analysis of the competitive interaction relationship between two competitors.

碩士
國立中央大學
數學研究所
99
In this thesis, we first consider a Lotka-Volterra competition-diffusion-advection model for two competing species in a heterogeneous environment. The two species are identical except for their dispersal strategies: One is just random diffusion while the other is "smarter"- a combination of random diffusion and a directed movement up the environmental gradient. In [3], Chen and Lou conjectured that if the environment function $m$ has multiple local maxima, then the "smarter" species must concentrate at all local maximum of m. Nevertheless, in [6], Lam and Ni found that the "smarter" species will die out if the local maximum of m is smaller than the density of the other species. In this article, we consider a model of three species and expect that the related results will be similar to those in [6].

Research Doctorate - Doctor of Philosophy (PhD)
It is critical to investigate the development, diffusion and utilisation of emerging technologies in relation with each other, as well as with incumbent technologies. This is particularly imperative in sectors such as energy and transportation that struggle with both economic concerns (e.g., resource scarcity) and ‘grand challenges’, such as unsustainable consumption and production and greenhouse gas (GHG) emissions. This thesis takes a biological perspective and investigates whether technologies in an industry interact with one another in the same way that populations do in an ecology. The biological inter-population relationships analogy is applied to the three powertrain technologies of internal combustion engine, hybrid and battery electric vehicles (ICEV, HEV, and BEV) in the United States (US) automotive industry. Inter-powertrain relationships are studied in the early technological lifecycle (TLC) stage, known as an ‘era of ferment’, as it is characterised by an increase in technological variations, intense competition, high market uncertainty, and the frequent exits and entries of firms. This thesis conducts a qualitative and quantitative explanatory–exploratory study through a three-staged research design of narrative (conceptualisation), quantification and simulation. In the first stage, it conducts a qualitative explanatory–exploratory study to construct the conceptual framework. Adopting the technological innovation system (TIS) framework and the biological relationship modes, a dynamic approach to socio-technical interactions between technologies is proposed, called ‘dynapstic’. The various dimensions of the dynapstic framework are initially demonstrated by narrating some case studies, especially in the transportation and energy sectors. In the second stage, it conducts a quantitative explanatory–exploratory study to quantify the individual dimensions of the dynapstic framework. The biological Lotka-Volterra (L-V) equations are applied to quantify the individual socio-technical dimensions of powertrain technologies for the period 1985 to 2016. In the final stage, this thesis conducts a simulative explanatory–exploratory study to comprehensively simulate all the individual dimensions of the dynapstic framework. Feeding all the L-V quantifications and estimations from the second stage, all the individual socio-technical dimensions of powertrain technologies are integrated via an extensive system dynamics (SD) modelling for the time horizon of 1985 to 2050. This thesis illustrates that the internal dynamics of one powertrain technology become coupled with the internal dynamics of another powertrain technology through what is referred to as ‘co-dynamics’ in the dynapstic framework. Some of the proposed co-dynamics are entrepreneurial spawning, policy transfer, knowledge recombination and resource redeployment. Co-dynamics are illustrated to carry a mix of positive, negative and neutral influences between powertrain technologies that shape the various biological relationship modes between them, such as competition symbiosis, commensalism, parasitism and amensalism. These co-dynamics eventually lead to the build-up of shared structural elements or ‘couplings’ between powertrain technologies, such as overlap actors, knowledge overlap, institutional overlap and resource overlap. The findings throughout the three stages reveal that while inter-technology relationships can be multimodal and multidimensional, their nature and extent may undergo temporal transitions and suspensions over time. This thesis extends the TLC and strategic management literatures by challenging the assumptions for pure competition and for explicit dimensions, as technologies are illustrated to interact with each other in other forms (e.g., symbiosis, parasitism and commensalism) and for implicit dimensions (e.g., knowledge, policies, expectations and collaborations). It additionally contributes to the path dependency and sustainability transition literatures by revealing that transition processes are not only a result of path dependence, path creation and path destruction, but also a result of cross-path socio-technical interactions via positive and negative internalities and externalities. In particular, the TIS framework is made more outward oriented—first, by accommodating co-dynamics as a complementary dynamical unit of analysis to the conventional structural unit of analysis ‘couplings’, and second by proposing two new TIS motors, ‘motor of creative destruction’ and ‘motor of creative accumulation’. Finally, it contributes to the sustainability transition literature by challenging the transitionary, parasitic definition of hybrid technologies. This thesis informs transition managers and policymakers that their policy mixes may possess a triple nature of ‘creation’, ‘destruction’ and ‘accumulation’. Because their policy mixes may not only generate positive or negative internalities for the intended technology, but may also bring about positive or negative externalities in the field of other technologies. Taking a biological perspective, six types of strategies are proposed: competition, symbiosis, parasitism, commensalism, amensalism and neutral strategies. Considering the temporal transitions and suspensions findings, public policy makers are recommended to create and alternate their strategies in accordance with the changing multimodal and multidimensional relationships, but maintain a balance between them by strategically and proactively reconfiguring, modifying, facilitating and coordinating them over time. While public policy makers should avoid devising policies that may eventually lead to the demise of both incumbent and emerging technologies, their pro-entrepreneurship public policies should be preceded by pro-incumbent public policies, for instance, through exit options or transition supports such as knowledge recombination, knowledge continuity mobilisation, resource redeployment, and entrepreneurial recycling. Such an understanding informs policy decisions of when, and to what extent, one should invest in emerging disruptive technologies, divest from the incumbent technology, or pursue an intermediate solution between the new and incumbent technologies, while avoiding any dead ends.