Traveling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing (decreasing) traveling wave solutions are established. Some faults in previous studies are corrected.

Traveling wave solutions of integro-differential equations for modeling one-dimensional neuronal networks, are studied. Under moderate continuity assumptions, necessary and sufficient conditions for the existence and uniqueness of monotone increasing (decreasing) traveling wave solutions are established. Some faults in previous studies are corrected.

本篇論文討論的是SIR模型的反應擴散方程 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, 之行進波的存在性，其中模型描述的是在一個封閉區域裡流行疾病爆發的狀態。這裡的 β 是傳播係數，γ 是治癒或移除(即死亡)速率，s 是未被傳染個體數，i 是傳染源個體數，d_1、d_2、d_3分別為其擴散之係數。 　　我們將使用Schauder不動點定理(Schauder fixed point theorem)、Arzela-Ascoli定理和最大值原理(maximum principle)來證明：該系統存在最小速度為c=c*:=2√(d2( β - γ ))之行進波解。我們的結果回答了[11]裡所提出的開放式問題。 / In this thesis, we study the existence of traveling waves of a reaction-diffusion equation for a diffusive epidemic SIR model 　　　　　　　　　s_t = d_1 s_xx − βsi/(s + i), 　　　　　　　　　i_t = d_2 i_xx + βsi/(s + i) − γi, 　　　　　　　　　r_t = d_3 r_xx + γi, which describes an infectious disease outbreak in a closed population. Here β is the transmission coefficient, γ is the recovery or remove rate, and s, i, and r rep-resent numbers of susceptible individuals, infected individuals, and removed individuals, respectively, and d_1, d_2, and d_3 are their diffusion rates. We use the Schauder fixed point theorem, the Arzela-Ascoli theorem, and the maximum principle to show that this system has a traveling wave solution with minimum speed c=c*:=2√(d2( β - γ )). Our result answers an open problem proposed in [11].

The aim of this thesis is to investigate the dynamical behaviors of spatially extended systems with fluctuating time delays. In recent years, the study of spatially extended systems and systems with fluctuating delays has experienced a fast growth. In ubiquitous natural and laboratory situations, understanding the action of time-delayed signals is a crucial for understanding the dynamical behavior of these systems. Frequently, the length of the delay is found to change with time. Spatially extended systems are widely studied in many fields, such as chemistry, ecology, and biology. Self-organization, turbulence, and related nonlinear dynamic phenomena in spatially extended systems have developed into one of the most exciting topics in modern science. The first part of this thesis considers the discrete system. Diffusively coupled map lattices with a fluctuating delay are used in the study. The uncoupled local dynamics of the considered system are represented by the delayed logistic map. In particular, the influences of diffusive coupling and fluctuating delay are studied. To observe and understand the influences, the results for the considered system are compared with coupled map lattices without delay and with a constant delay as well as with the uncoupled logistic map with fluctuating delays. Identifying different patterns, determining the existence of traveling wave solutions, and specifying the fully synchronized stable state are the focus of this part of the study. The Lyapunov exponent, the master stability function, spectrum analysis, and the structure factor are used to characterize the different states and the transitions between them. The second part examines the continuous system. The delay is introduced into the reactionterm of the Fisher-KPP equation. The focus of this part of study is the time-delay-induced Turing instability in one-component reaction-diffusion systems. Turing instability has previously only been found in multiple-component reaction-diffusion systems. However, this work demonstrates with the help of the stability exponent that fluctuating delay can result in Turing instability in one-component reaction-diffusion systems as well. / Ziel der vorliegenden Arbeit ist die Untersuchung der Einflüsse der zeitlich fluktuierenden Verzögerungen in räumlich ausgedehnten diffusiven Systemen. Durch den Vergleich von Systemen mit konstanter Verzögerung bzw. Systemen ohne räumliche Kopplung erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise der Dynamik des räumlich ausgedehnten diffusiven Systems mit fluktuierenden Verzögerungen. Im ersten Teil werden diskrete Systeme in Form von diffusiven Coupled Map Lattices untersucht. Als die lokale iterierte Abbildung des betrachteten Systems wird die logistische Abbildung mit Verzögerung gewählt. In diesem Teil liegt der Fokus auf Musterbildung, Existenz von Multiattraktoren und laufenden Wellen sowie der Möglichkeit der vollen Synchronisation. Masterstabilitätsfunktion, Lyapunov Exponent und Spektrumsanalyse werden benutzt, um das dynamische Verhalten zu verstehen. Im zweiten Teil betrachten wir kontinuierliche Systeme. Hier wird die Fisher-KPP Gleichung mit Verzögerungen im Reaktionsteil untersucht. In diesem Teil liegt der Fokus auf der Existenz der Turing Instabilität. Mit Hilfe von analytischen und numerischen Berechnungen wird gezeigt, dass bei fluktuierenden Verzögerungen eine Turing Instabilität auch in 1-Komponenten-Reaktions-Diffusionsgleichungen gefunden werden kann

The aim of this thesis is to investigate the dynamical behaviors of spatially extended systems with fluctuating time delays. In recent years, the study of spatially extended systems and systems with fluctuating delays has experienced a fast growth. In ubiquitous natural and laboratory situations, understanding the action of time-delayed signals is a crucial for understanding the dynamical behavior of these systems. Frequently, the length of the delay is found to change with time. Spatially extended systems are widely studied in many fields, such as chemistry, ecology, and biology. Self-organization, turbulence, and related nonlinear dynamic phenomena in spatially extended systems have developed into one of the most exciting topics in modern science. The first part of this thesis considers the discrete system. Diffusively coupled map lattices with a fluctuating delay are used in the study. The uncoupled local dynamics of the considered system are represented by the delayed logistic map. In particular, the influences of diffusive coupling and fluctuating delay are studied. To observe and understand the influences, the results for the considered system are compared with coupled map lattices without delay and with a constant delay as well as with the uncoupled logistic map with fluctuating delays. Identifying different patterns, determining the existence of traveling wave solutions, and specifying the fully synchronized stable state are the focus of this part of the study. The Lyapunov exponent, the master stability function, spectrum analysis, and the structure factor are used to characterize the different states and the transitions between them. The second part examines the continuous system. The delay is introduced into the reactionterm of the Fisher-KPP equation. The focus of this part of study is the time-delay-induced Turing instability in one-component reaction-diffusion systems. Turing instability has previously only been found in multiple-component reaction-diffusion systems. However, this work demonstrates with the help of the stability exponent that fluctuating delay can result in Turing instability in one-component reaction-diffusion systems as well. / Ziel der vorliegenden Arbeit ist die Untersuchung der Einflüsse der zeitlich fluktuierenden Verzögerungen in räumlich ausgedehnten diffusiven Systemen. Durch den Vergleich von Systemen mit konstanter Verzögerung bzw. Systemen ohne räumliche Kopplung erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise der Dynamik des räumlich ausgedehnten diffusiven Systems mit fluktuierenden Verzögerungen. Im ersten Teil werden diskrete Systeme in Form von diffusiven Coupled Map Lattices untersucht. Als die lokale iterierte Abbildung des betrachteten Systems wird die logistische Abbildung mit Verzögerung gewählt. In diesem Teil liegt der Fokus auf Musterbildung, Existenz von Multiattraktoren und laufenden Wellen sowie der Möglichkeit der vollen Synchronisation. Masterstabilitätsfunktion, Lyapunov Exponent und Spektrumsanalyse werden benutzt, um das dynamische Verhalten zu verstehen. Im zweiten Teil betrachten wir kontinuierliche Systeme. Hier wird die Fisher-KPP Gleichung mit Verzögerungen im Reaktionsteil untersucht. In diesem Teil liegt der Fokus auf der Existenz der Turing Instabilität. Mit Hilfe von analytischen und numerischen Berechnungen wird gezeigt, dass bei fluktuierenden Verzögerungen eine Turing Instabilität auch in 1-Komponenten-Reaktions-Diffusionsgleichungen gefunden werden kann