Spatial spread of sexually transmitted diseases within susceptible populations at demographic steady state
Carlos Castillo-Chavez Bingtuan Li
Mathematical Biosciences & Engineering 2008, 5(4): 713-727 doi: 10.3934/mbe.2008.5.713
In this study, we expand on the susceptible-infected-susceptible (SIS) heterosexual mixing setting by including the movement of individuals of both genders in a spatial domain in order to more comprehensively address the transmission dynamics of competing strains of sexually-transmitted pathogens. In prior models, these transmission dynamics have only been studied in the context of nonexplicitly mobile heterosexually active populations at the demographic steady state, or, explicitly in the simplest context of SIS frameworks whose limiting systems are order preserving. We introduce reaction-diffusion equations to study the dynamics of sexually-transmitted diseases (STDs) in spatially mobile heterosexually active populations. To accomplish this, we study a single-strain STD model, and discuss in what forms and at what speed the disease spreads to noninfected regions as it expands its spatial range. The dynamics of two competing distinct strains of the same pathogen on this population are then considered. The focus is on the investigation of the spatial transition dynamics between the two endemic equilibria supported by the nonspatial corresponding model. We establish conditions for the successful invasion of a population living in endemic conditions by introducing a strain with higher fitness. It is shown that there exists a unique spreading speed (where the spreading speed is characterized as the slowest speed of a class of traveling waves connecting two endemic equilibria) at which the infectious population carrying the invading stronger strain spreads into the space where an equilibrium distribution has been established by the population with the weaker strain. Finally, we give sufficient conditions under which an explicit formula for the spreading speed can be found.
keywords: sexually transmitted disease traveling wave spreading speed disease spread
Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream
Bingtuan Li William F. Fagan Garrett Otto Chunwei Wang
Discrete & Continuous Dynamical Systems - B 2014, 19(10): 3267-3281 doi: 10.3934/dcdsb.2014.19.3267
We propose a reaction-advection-diffusion model to study competition between two species in a stream. We divide each species into two compartments, individuals inhabiting the benthos and individuals drifting in the stream. We assume that the growth of and competitive interactions between the populations take place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-advection-diffusion equations and two ordinary differential equations. Here, we provide a thorough study for the corresponding single species model, which has been previously proposed. We next give formulas for the rightward spreading and leftward spreading speed for the model. We show that rightward spreading speed can be characterized as is the slowest speed of a class of traveling wave speeds. We provide sharp conditions for the spreading speeds to be positive. For the two species competition model, we investigate how a species spreads into its competitor's environment. Formulas for the spreading speeds are provided under linear determinacy conditions. We demonstrate that under certain conditions, the invading species can spread upstream. Lastly, we study the existence of traveling wave solutions for the two species competition model.
keywords: linear determinacy traveling wave solution. spreading speed Reaction-diffusion system
Traveling wave solutions in an integro-differential competition model
Liang Zhang Bingtuan Li
Discrete & Continuous Dynamical Systems - B 2012, 17(1): 417-428 doi: 10.3934/dcdsb.2012.17.417
We study the existence of traveling wave solutions for the two-species Lotka-Volterra competition model in the form of integro-differential equations. The model incorporates asymmetric dispersal kernels that describe long distance dispersal processes of competing species in space. Using lower and upper traveling wave solutions, we show that the model has traveling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.
keywords: competition dispersal kernel traveling wave solution. Integro-differential equation
Some remarks on traveling wave solutions in competition models
Bingtuan Li
Discrete & Continuous Dynamical Systems - B 2009, 12(2): 389-399 doi: 10.3934/dcdsb.2009.12.389
We study the existence of traveling wave solutions for competition models in the form of integro-difference equations. We show that for a two-species competition model it is possible that two species spread at different speeds, and there exists a traveling wave solution. For an $m$-species competition model, under the assumption that species have the same dispersal and growth properties but have different competition abilities, we establish the existence of traveling wave solutions.
keywords: traveling wave integro-difference equation. Competition
Spreading speeds and traveling wave solutions in cooperative integral-differential systems
Changbing Hu Yang Kuang Bingtuan Li Hao Liu
Discrete & Continuous Dynamical Systems - B 2015, 20(6): 1663-1684 doi: 10.3934/dcdsb.2015.20.1663
We study a cooperative system of integro-differential equations. It is shown that the system in general has multiple spreading speeds, and when the linear determinacy conditions are satisfied all the spreading speeds are the same and equal to the spreading speed of the linearized system. The existence of traveling wave solutions is established via integral systems. It is shown that when the linear determinacy conditions are satisfied, if the unique spreading speed is not zero then it may be characterized as the slowest speed of a class of traveling wave solutions. Some examples are presented to illustrate the theoretical results.
keywords: linear determinacy Integral-differential system spreading speed traveling wave solution. integral system
Yang Kuang Jiaxu Li Bingtuan Li Urszula Ledzewicz Ami Radunskaya
Discrete & Continuous Dynamical Systems - B 2009, 12(2): i-ii doi: 10.3934/dcdsb.2009.12.2i
This special issue of Discrete and Continuous Dynamical Systems, Series B (DCDS-B), is based on the timely special session on dynamical systems in biology and medicine in the 7th AIMS Conference on Dynamical Systems and Differential Equations, which took place at University of Texas at Arlington, Texas, USA, in the period of May 18 - 21, 2008. All papers are carefully refereed and selected based on the mathematical originality and biological relevance of the presented research work.
   The bi-annual AIMS international Conference on Dynamical Systems and Differential Equations has grown steadily in size, quality and scope. Indeed, in a short period of 12 years, it has become the largest, most popular and well organized international meeting of its kind, featuring many impressive keynote speakers, dynamic and engaging special sessions, and most importantly, effective and economical conference management. A total of 867 researchers participated in this well organized meeting at Arlington.

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Dynamics of a ratio-dependent predator-prey system with a strong Allee effect
Yujing Gao Bingtuan Li
Discrete & Continuous Dynamical Systems - B 2013, 18(9): 2283-2313 doi: 10.3934/dcdsb.2013.18.2283
A ratio-dependent predator-prey model with a strong Allee effect in prey is studied. We show that the model has a Bogdanov-Takens bifurcation that is associated with a catastrophic crash of the predator population. Our analysis indicates that an unstable limit cycle bifurcates from a Hopf bifurcation, and it disappears due to a homoclinic bifurcation which can lead to different patterns of global population dynamics in the model. We study the heteroclinic orbits and determine all possible phase portraits when the Bogdanov-Takens bifurcation occurs. We also provide the conditions for nonexistence of limit cycle under which the global dynamics of the model can be determined.
keywords: Hopf bifurcation Bogdanov-Takens bifurcation limit cycle. heteroclinic orbit Ratio-dependent predator-prey system Allee effect
Some recent developments on linear determinacy
Carlos Castillo-Chavez Bingtuan Li Haiyan Wang
Mathematical Biosciences & Engineering 2013, 10(5&6): 1419-1436 doi: 10.3934/mbe.2013.10.1419
The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called ``linear determinacy'' conjecture equates full nonlinear model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading edge of the invasion. A survey that accounts for recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions covered in this survey are also discussed.
keywords: ecology population biology Dispersal integer difference integral equations nonlinear reaction diffusion difference equations.
Analysis of IVGTT glucose-insulin interaction models with time delay
Jiaxu Li Yang Kuang Bingtuan Li
Discrete & Continuous Dynamical Systems - B 2001, 1(1): 103-124 doi: 10.3934/dcdsb.2001.1.103
In the last three decades, several models on the interaction of glucose and insulin have appeared in the literature, the mostly used one is generally known as the "minimal model" which was first published in 1979 and modified in 1986. Recently, this minimal model has been questioned by De Gaetano and Arino [4] from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called "dynamic model". Their model makes use of certain simple and specific functions and introduces time delay in a particular way. The outcome is that the model always admits a globally asymptotically stable steady state. The objective of this paper is to find out if and how this outcome depends on the specific choice of functions and the way delay is incorporated. To this end, we generalize the dynamical model to allow more general functions and an alternative way of incorporating time delay. Our findings show that in theory, such models can possess unstable positive steady states. However, for all conceivable realistic data, such unstable steady states do not exist. Hence, our work indicates that the dynamic model does provide qualitatively robust dynamics for the purpose of clinic application. We also perform simulations based on data from a clinic study and point out some plausible but important implications.
keywords: insulin qualitative analysis. Glucose minimum model delay differential equations

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