DCDS-B
Interaction of diffusion and delay
Karl Peter Hadeler Shigui Ruan
For reaction-diffusion equations with delay, the joint effects of diffusion and delay are studied. In particular, for two-dimensional systems where only the interaction between species is delayed, the interdependence of stability against delay and against diffusion (Turing instability) can be clearly exhibited. Turing instabilities occur largely independent of delay. But periodic oscillations, constant in space or with low spatial frequency, can be achieved via increasing the delay or changing the diffusion rates.
keywords: matrix stability. time delay Hopf bifurcation Turing instability Reaction-diffusion equations
CPAA
Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting
Jicai Huang Sanhong Liu Shigui Ruan Xinan Zhang
Recently, we (J. Huang, Y. Gong and S. Ruan, Discrete Contin. Dynam. Syst. B 18 (2013), 2101-2121) showed that a Leslie-Gower type predator-prey model with constant-yield predator harvesting has a Bogdanov-Takens singularity (cusp) of codimension 3 for some parameter values. In this paper, we prove analytically that the model undergoes Bogdanov-Takens bifurcation (cusp case) of codimension 3. To confirm the theoretical analysis and results, we also perform numerical simulations for various bifurcation scenarios, including the existence of two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1.
keywords: homoclinic bifurcation. constant-yield harvesting Bogdanov-Takens bifurcation of codimension 3 Hopf bifurcaton Predator-prey model
DCDS-B
Preface on the special issue of Discrete and Continuous Dynamical Systems- Series B in honor of Chris Cosner on the occasion of his 60th birthday
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
Chris Cosner turned 60 on June 3, 2012 and now, at age 62, continues his stellar career at the interface of mathematics and biology. He received his Ph.D. in 1977 at the University of California, Berkeley under the direction of Murray Protter, winning the Bernard Friedman prize for the best dissertation in applied mathematics. From 1977 until 1982 he was on the faculty of Texas A&M University. In 1982 he left A&M to join the faculty of the Department of Mathematics of the University of Miami as Associate Professor, rising to the rank of Professor in 1988. The academic year 2013-2014 marked his 32nd year of distinguished service to the University of Miami and its research and pedagogical missions.

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DCDS-B
Bifurcations of an SIRS epidemic model with nonlinear incidence rate
Zhixing Hu Ping Bi Wanbiao Ma Shigui Ruan
The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence $\beta SI^p/(1+\alpha I^q)$. The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.
keywords: nonlinear incidence rate stability Hopf bifurcation Bogdanov-Takens bifurcation. SIRS epidemic model
DCDS-B
Preface
Robert Stephen Cantrell Suzanne Lenhart Yuan Lou Shigui Ruan
The movement and dispersal of organisms have long been recognized as key components of ecological interactions and as such, they have figured prominently in mathematical models in ecology. More recently, dispersal has been recognized as an equally important consideration in epidemiology and in environmental science. Recognizing the increasing utility of employing mathematics to understand the role of movement and dispersal in ecology, epidemiology and environmental science, The University of Miami in December 2012 held a workshop entitled ``Everything Disperses to Miami: The Role of Movement and Dispersal in Ecology, Epidemiology and Environmental Science" (EDM).

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DCDS-B
Preface
P. Magal Shigui Ruan
This special issue is the proceedings of the International Workshop on Differential Equations in Mathematical Biology held in Le Havre, France, July 11-13, 2005. The workshop brought together internationals researchers in Differential Equations and Mathematical Biology to communicate with each other about their current work. The topics of the workshop included various types of differential equations and their applications to biology and other related subjects, such as, ecology, epidemiology, medicine, etc. There were more than 60 participants came from Algeria, Canada, Cameroun, Finland, France, Germany, Hungary, Italy, Japan, Lithuania, Mexico, The Netherlands, Portugal, Romania, Spain, South Africa, UK, and USA. The ple- nary speakers were Pierre Auger (IRD Bondy, France), Josef Hofbauer (University College London, UK), Michel Langlais (Bordeaux 2, France), Hal Smith (Arizona State, USA), Horst Thieme (Arizona State, USA), Glenn Webb (Vanderbilt, USA) and Jianhong Wu (York, Canada). There were also more than 40 presentations by other participants.
    The 17 articles which appear in this special issue are from the participants of the Workshop and from other leading researchers in these subjects. Topics include malaria intra-host models, stem cell dynamics, tumor invasion, reaction-diffusion systems for competition and predation, traveling waves, optimal control in age structured models, host-parasitoid models, predator-prey models, HIV infection, immune system memory, bacteria infection, innate immune response, and antibiotic treatment.

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MBE
Letter to the editors
Shigui Ruan
Dear Editors:
I request that Mathematical Biosciences and Engineering publish this Letter of Correction to an article published in the journal for which I was the corresponding author, "The Effect of Global Travel on the Spread of SARS'' (2006;3(1):205-218). The goal of this article was to study the effect of global travel on the geographic spread of SARS. A multiregional compartmental model was proposed, mathematically analyzed, and numerically simulated to study how SARS spread out from Guangdong, China. The article consists of six sections: (1) an introduction, (2) a background section on medical geography theory, (3) the mathematical model, (4) mathematical analysis and results, (5) numerical simulations, and (6) discussion. Sections 3, 4, and 5 are the main parts of the article which are all original work.

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DCDS-B
Intraspecific interference and consumer-resource dynamics
Robert Stephen Cantrell Chris Cosner Shigui Ruan
In this paper we first consider a two consumer-one resource model with one of the consumer species exhibits intraspecific feeding interference but there is no interspecific competition between the two consumer species. We assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington-DeAngelis functional response. Using dynamical systems theory, it is shown that the two consumer species can coexist upon the single limiting resource in the sense of uniform persistence. Moreover, by constructing a Liapunov function it is shown that the system has a globally stable positive equilibrium. Second, we consider a model with an arbitrary number of consumers and one single limiting resource. By employing practical persistence techniques, it is shown that multiple consumer species can coexist upon a single resource as long as all consumers exhibit sufficiently strong conspecific interference, that is, each of them exhibits Beddington-DeAngelis functional response.
keywords: coexistence competition stability. Predators interference persistence
DCDS-B
Oscillations in age-structured models of consumer-resource mutualisms
Zhihua Liu Pierre Magal Shigui Ruan
In consumer-resource interactions, a resource is regarded as a biotic population that helps to maintain the population growth of its consumer, whereas a consumer exploits a resource and then reduces its growth rate. Bi-directional consumer-resource interactions describe the cases where each species acts as both a consumer and a resource of the other, which is the basis of many mutualisms. In uni-directional consumer-resource interactions one species acts as a consumer and the other as a material and/or energy resource while neither acts as both. In this paper we consider an age-structured model for uni-directional consumer-resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. Examples include a predator-prey system in which the prey is able to kill or consume predator eggs or larvae and the insect pollinator and the host plant relationship in which the plants provide food, seeds, nectar and other resources for the pollinators while the pollinators have both positive and negative effects on the plants. By carrying out local analysis and bifurcation analysis of the model, we discuss the stability of the positive equilibrium and show that under some conditions a non-trivial periodic solution through Hopf bifurcation appears when the maturation parameter passes through some critical values.
keywords: Consumer-resource interaction age-structure stability Hopf bifurcation periodic solutions.
MBE
Preface
Peter Hinow Pierre Magal Shigui Ruan
This special issue is dedicated to the 70th birthday of Glenn F. Webb. The topics of the 12 articles appearing in this special issue include evolutionary dynamics of population growth, spatio-temporal dynamics in reaction-diffusion biological models, transmission dynamics of infectious diseases, modeling of antibiotic-resistant bacteria in hospitals, analysis of Prion models, age-structured models in ecology and epidemiology, modeling of immune response to infections, modeling of cancer growth, etc. These topics partially represent the broad areas of Glenn's research interest.

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