| Time | Speaker and Title |
| 8:30am - 9:00am | Hal Smith (AARMS Distinguished Lecturer) |
| Bacterial Wall Attachment in a Flow Reactor | |
| 9:00am - 9:30am | Wenzhang Huang |
| Traveling Waves for Reaction-diffusion Equations | |
| from Application | |
| 9:30am - 10:00am | Julian López-Gómez |
| Spatial Heterogeneities in Mathematical Biology | |
| 10:00am - 10:30am | Coffee break |
| 10:30am - 11:00am | Meirong Zhang |
| Resonance Pockets of Hill's Equations with Two-step Potentials | |
| 11:00am - 11:30am | Steve Cantrell |
| Effects of Aggregative Movement on Population Dynamics in | |
| ``Island'' Habitats and Critical Patch Size | |
| 11:30am - 12:00pm | Xiao-Qiang Zhao |
| Global Attractivity in Monotone and Subhomogeneous | |
| Almost Periodic Systems | |
| 12:00pm - 1:00pm | Lunch |
| 2:30pm - 3:00pm | John Haddock |
| Instability in ``Delayed'' Dynamical Systems | |
| 3:00pm - 3:30pm | Yasuhiro Takeuchi |
| Global Stability of Time Delayed Chemostat Models for Bacteria | |
| and Virulent Phage | |
| 3:30pm - 4:00pm | Coffee break |
| 4:00pm - 4:30pm | Sue Ann Campbell |
| Neural Oscillators with Delayed Coupling | |
| 4:30pm - 5:00pm | Xingfu Zou |
| 3/2 Type Criteria for Global Asymptotic Stability for Continuous | |
| Lotka-Volterra Competition Systems of Pure-delay Type | |
| 5:00pm - 5:30pm | Mostafa Bacher |
| Integrated Semigroups Associated to a Linear Delay Differential | |
| Equation with Impulses | |
| 5:30pm - 6:30pm | Supper |
| Time | Speaker and Title |
| 8:30am - 9:00am | Pauline van der Drieche |
| Models for Transmission of Disease with Immigration of Infectives | |
| 9:00am - 9:30am | Chris Bauch |
| Deriving Better Deterministic Approximations to Dynamic Network | |
| Network Models of Sexually Transmitted Disease Spread | |
| 9:30am - 10:00am | Miriam Nuño |
| The Dynamics of Two-Strain Influenza with Quarantine | |
| and Cross-Immunity | |
| 10:00am - 10:30am | Coffee break |
| 10:30am - 11:00am | Fred Brauer |
| A Class of Characteristic Equations Arising in Epidemic Modeling | |
| 11:00am - 11:30am | Baojun Song |
| Tuberculosis Epidemics at Two Social Levels: The Role of | |
| Time Scales | |
| 11:30am - 12:00pm | James Watmough |
| Reproduction Numbers and Sub-threshold Endemic Equilibria for | |
| Compartmental Models of Disease Transmission | |
| 12:00pm - 1:00pm | Lunch |
| 2:30pm - 3:00pm | Stephen Gourley |
| Spatio-temporal Delays in Population Models on Finite | |
| Spatial Domains | |
| 3:00pm - 3:30pm | Weigu Li |
| Local First Integrals of Differential Systems and Diffeomorphisms | |
| 3:30pm - 4:00pm | Coffee break |
| 4:00pm - 4:30pm | Rebecca Culshaw |
| Control of HIV Infection by Maximising Immune Response | |
| 4:30pm - 5:00pm | Ying-Hen Hsieh |
| Modeling the Social Dynamics of Sex Industry: Its Implications | |
| for Spread of HIV/AIDS | |
| 5:00pm - 5:30pm | Connell McCluskey |
| An HIV/AIDS model with Staged Progression and Amelioration | |
| 5:30pm - 6:30pm | Supper |
| 7:30pm - 8:30pm | Panel discussion |
| Time | Speaker and Title |
| 8:30am - 9:00am | Jim Cushing |
| Lattice Effects and Complex Dynamics | |
| 9:00am - 9:30am | Jiong Ruan |
| Stability in Discrete-time Neural Networks | |
| 9:30am - 10:00am | Abdul-Aziz Yakubu |
| Dispersal and Intraspecific Competition in Discrete-time | |
| Patchy Environments | |
| 10:00am - 10:30am | Coffee break |
| 10:30am - 11:00am | Lin Wang |
| 3/2 Type Criteria for Global Attractivity of Lotka-Volterra | |
| Discrete System with Delays | |
| 11:00am - 11:30am | Jim Selgrade |
| Attractors for Discrete, Periodically Forced Systems with | |
| Applications to Population Models | |
| 11:30am - 12:00pm | Xinzhi Liu |
| Hybrid Control of a Three-species Population Growth Model | |
| 12:00pm - 1:00pm | Lunch |
| 2:30pm - 3:00pm | Yingfei Yi |
| Relaxation Oscillation in a Predator-prey System | |
| 3:00pm - 3:30pm | Josef Hofbauer |
| Robust Permanence for Ecological Differential Equations | |
| 3:30pm - 4:00pm | Coffee break |
| 4:00pm - 4:30pm | Maoan Han |
| On the Number and Distribution of Limit Cycles in a Cubic System | |
| 4:30pm - 5:00pm | Andy Foster |
| A Model for Predator Pit Behavior | |
| 5:00pm - 5:30pm | Huaiping Zhu |
| Blown-up Techniques and Applications to a Degenerate | |
| Predator-prey Model | |
| 5:30pm - 6:30pm | Supper |
| 7:30pm - 8:30pm | Panel discussion |
| Time | Speaker and Title |
| 8:30am - 9:00am | Herb Freedman |
| Mathematical Models of Cancer Treatment | |
| 9:00am - 9:30am | Yun Tang |
| Some Dynamic Problems in Blood Coagulation Systems | |
| 9:30am - 10:00am | Mary Lou Zeeman |
| Modeling the Human Menstrual Cycle | |
| 10:00am - 10:30am | Coffee break |
| 10:30am - 11:00am | Bill Langford |
| Cheyne-Stokes Respiration | |
| 11:00am - 11:30am | Ross Cressman |
| N-Species Evolutionary Stability Concepts | |
| 11:30am - 12:00pm | Lunch |
| Afternoon | No talks (discussion, etc.) |
In this paper, we discuss the fundamental linear theory for a class of delay differential equations with impulses. We show using the general theory of integrated semigroups, that we can associate with any delay differential equation with impulses a strongly continuous semigroups.
The structure of sexual partnership networks has a large impact on the spread of sexually transmitted diseases through a population. The importance of the contact structure for STDs is in contrast to the situation for most other types of epidemics which can be adequately modeled by a mass-action assumption. The need to model network structure has brought about the development of stochastic network models, which in turn has stimulated a search for deterministic approximations to network models. We employ the technique of moment closure approximation (also known as correlation equations or pair approximation) to create an ODE-based model which can capture aspects of the network structure. We also find that deriving moment closure approximations for dynamic networks is significantly different from deriving such approximations for static, regular lattices. (Joint work with David Earn and Gail Wolkowicz).
We consider a class of characteristic equations which involve the Laplace transform of an arbitrary infective period distribution function. Such characteristic equations appear in many epidemic models and variable maturation population models. Our goal is to obtain sufficient conditions for all roots to have negative real part, corresponding to local asymptotic stability of an equilibrium, and also conditions under which roots with non - negative real part, corresponding to sustained oscillation about an equilibrium, can appear for some infective period distributions. We are able to show that an infective period of fixed length is the "most likely" to produce oscillations.
We consider two oscillators linked with nonlinear, time delayed coupling. When the oscillators are identical we show that mode interaction can lead to the coexistence of stable in-phase and anti-phase oscillations, or of either oscillation and stable nontrivial equilibria. We show that this behaviour persists for nonindentical oscillators with appropriate parameter values.
We consider diffusive logistic models for the population dynamics of a species in an isolated or "island" habitat patch with a dissipative boundary. If random diffusive motion by members of the species is modified by the tendency to slow down at low to medium densities in response to the presence of conspecifics in a manner akin to area-restricted search for prey by predators, there may be qualitative changes in the predictions of the models at the population level. In this article, we employ the concept of a "critical patch size" of the habitat for the invasibility and/or persistence of the species to examine these effects, thereby illustrating how changes in the individual's utilization of space may lead to qualitative changes at the level of a population.
The theory of ESSs for a single species has developed over the past thirty years into a well-accepted means to predict individual behavior in single-species frequency-dependent evolutionary models. Less well-known are the generalizations of this theory to two or more species. This talk will summarize the analogous development of the two-species ESS concept and report some recent results for dynamic stability in both continuous and discrete-time two-species systems. Extensions to multi (i.e. more than two) species will also be discussed.
Long-term anti-HIV medication has been shown not only to have some severe side effects, but also to disrupt and suppress the cytotoxic ("killer cell") immune response. New treatment strategies have been suggested to offset this effect, including allowing patients "drug holidays" wherein the immune response has a chance to rebuild prior to the next administration of drugs. D. Wodarz and M. Nowak (Proceedings of the National Academy of Science, December 1999) showed via numerical analysis of an ODE system with treatment a parameter, that periodic treatment would indeed be beneficial in terms of maximising immune response and retaining healthy CD4+ cells. We modify Wodarz and Nowak's model by reducing its dimension and incorporating treatment as a control. We characterise the optimal control using Pontryagin's Maximum Principle and analyse the optimality system to determine whether periodic solutions exist.
Animals and plants come in whole numbers. The collection of feasible system states for population numbers is a "lattice" of integer values, and the corresponding set of states for densities is a discrete lattice of fractions. Nevertheless, mathematical population models that track "mean" dynamic tendencies utilize a continuum of system states. In purely deterministic systems, habitat size, in concert with the discreteness of animal numbers, can fundamentally alter the array of dynamic possibilities by altering the granularity of the lattice. When restricted to a finite lattice within state space deterministic model dynamics become simplified. A mean tendency for complex dynamics might remain unobserved if the habitat is too small. For example, on a finite lattice all orbits are eventually periodic. Nonetheless, complex dynamics of the underlying continuous state space model (quasi-periodicity, chaos, etc.) can exert their influence. In general, as the lattice mesh size decreases the attractor on the lattice approaches that of the continuous model. One way to decreases the lattice mesh size is to increase habitat size. Another way the complex dynamics of the continuous model become apparent on the lattice is by the addition of a small amount of noise. In this context noise is beneficial in that it helps reveal properties of the underlying deterministic dynamics lost because of a coarse phase space lattice (e.g., small habitat size). Typical stochastic model simulations on the lattice will contain a mixture of temporal episodes that display the simplified lattice attractor and episodes that display the more complex continuous attractor. Such "lattice effects" have been observed in real data obtained from laboratory experiments involving cultures of flour beetles (T. Castaneum) whose population dynamics have been manipulated into chaos. (Joint work with S. M. Henson, R. F. Costantino, R. A. Desharnais, S. M. Henson, B. Dennis).
A number of articles in ecological literature use the term "predator pit" to describe a behavior wherein a prey species becomes "trapped" at a population level lower than its previously observed carrying capacity. This stabilization of prey at a depleted level occurs subsequent to a sudden decline in its population size, which is attributed to predation. We develop a simple canonical ODE system to model this phenomenon. This is an interesting example of inverse mathematical model construction, where the objective is to construct a simplest possible model to account for observed behavior.
Models of cancer treatment by chemotherapy and immunotherapy are given as systems of ordinary differential equations. Criteria for the persistence or extinction of the cancer cells and for periodic solutions at low lever of cancer are developed.
Gomez@Mat.UCM.Es
We analyze the effect of varying coefficients in a general class of semilinear elliptic boundary value problems of sublinear and superlinear type. As a consequence of our analysis it follows the necessity of introducing a new class of generalized solutions, which are not distributional, to describe the asymptotics of the positive solutions of a general class of sublinear problems with vanishing coefficients. Those solutions are referred to as METASOLUTIONS in the specialized literature. Then, we analyze a general class of indefinite superlinear problems characterizing whether or not the problem possesses a stable positive solution and showing the uniqueness of the stable solution when it exists. The uniqueness result is very striking as there are models exhibiting an arbitrarily large number of positive solutions, as an effect of the spatial inhomogeneities of the problem.
REFERENCES
[1] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns. 146 (1998), 336-374.
[2] R. Gómez-Reñasco and J. López-Gómez, The effect of varying coefficients on the dynamics of a class of superlinear indefinite Reaction-Diffusion equations, J. Diff. Eqns. 167, (2000), 36-72.
[3] J. López-Gómez Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, El. J. of Diff. Eqns. Conf. 05 (2000), 135-171.
In this talk I will discuss how to model and analyze nonlocal spatial effects, induced by time delays, in a diffusion model for a single species confined to a finite domain. The nonlocal spatial effect arises when account is taken of the fact that individuals have been at different points in space at previous times. In contrast to the infinite domain situation, on a finite domain an additional difficulty arises in that, as well as having been at different points in space at previous times, individuals may also have been interacting with the domain's boundaries. I will show how to correctly derive the spatial averaging kernels for finite domain problems, generalising previous works which have concentrated on the simpler case of an infinite domain. I will describe how the resulting model can be analysed and results obtained on linear stability, boundedness, global convergence of solutions and bifurcations. (Joint work with Joseph W.-H. So).
Instability for dynamical systems generated by (mostly autonomous) delay differential equations has received considerable attention over the past several years. Researchers have been busy refining older ideas while developing new ones, and general properties of dynamical systems often play a fundamental role in the results. Along these lines, new instability theorems will be given along with applications to autonomous and nonautonomous delay differential equations, where both finite and infinite delay equations are considered. During the presentation, results will be motivated via a classical stability diagram for equations with finite delay.
This paper concerns with limit cycles in a cubic system. Four limit cycles are found and their distributions are studied by using the methods of bifurcation theory and qualitative analysis.
We present sufficient conditions for robust permanence in terms of average Ljapunov functions. Schreiber's (JDE 2000) recent conditions for robust permanence in terms of invariant measures follow via the minimax theorem. We also study permanence of discretizations of such systems. (Joint work with Barnabas M. Garay).
A theoretical model for HIV transmission is proposed in a population which has the structure of two classes (direct and indirect) of commercial sex workers (CSW) and two classes of sexually active male customers with different levels of sexual activity. Behavior change occurs between the two classes of CSW's and two classes of males under the setting of the proliferation of HIV/AIDS epidemic. The model is proposed to study the effect of behavior change, in particular the effects of decreased sexual contacts and transmission probabilities due to intervention measures, on the overall transmission dynamics of HIV on the population level. In the disease-free case, the analysis of the social dynamics of the commercial sex industry is given in detail. For the full model local analysis will be given in some special cases and the basic reproduction numbers computed along with numerical examples to illustrate the implications of the behavior change in the context of the spread of HIV epidemic. (Joint work with Chien Hsun Chen).
Many ODE or FDE models from population dynamics or
ecology have connecting orbits from an unstable equilibrium to
a stable equilibrium. In is talk we will show that if the
diffusion term is added into the model, then for each
sufficiently large real number 
, the reaction-diffusion
equation has traveling wave solutions of wave speed
near a connection orbit (for ODE or FDE). Moreover, the set
of those traveling waves has dimension 
, where is the
dimension of the unstable manifold associated to the unstable
equilibrium.
Cheyne-Stokes respiration is a periodic breathing pattern, characterized by short intervals of deep breathing, each followed by an interval of very little or no breathing (known as apnea). Compartmental models of the human cardio-respiratory system have been constructed, to simulate the concentration of carbon dioxide in the various parts of the cardiovascular system and the lungs. The parameter boundaries on which a Hopf bifurcation gives birth to an period oscillation in the models are determined. The models predict that an increase in ventilation-perfusion ratio, or an increase in feedback gain, can give rise to stable oscillations. Physiologically, it is known that Cheyne-Stokes respiration is most likely to occur in people with illnesses such as heart disease, neurological disorders and metabolic alkalosis, or in healthy humans during sleep, acclimatization to high altitudes or following hyperventilation. The model agrees qualitatively with all of these observations. An increase in the partial pressure of inspired carbon dioxide in the model shifts the bifurcation curve upward, giving a greater area of stability and potentially curing Cheyne-Stokes respiration. (Joined work with Julie Atamanyk).
By using the theory of linear operators and normal forms we generalize a result of Poincare about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta about local first integrals of semi-quasihomogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of the first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.
In this paper, we shall investigate the problem of hybrid control of a three-species population growth model. It is shown that by impulsively regulating one species, the population of all three species can be maintained at a positive level, which otherwise would drop to a level of extinction for one of the species.
I will discuss an HIV/AIDS model for which there are several infectious stages. Many earlier models assume a monotone progression through the infective stages. With advances in drug therapies, it is necessary to modify this assumption. This model allows for amelioration where infected individuals move from more advanced stages of infection to less advanced stages.
The local stability of the disease free equilibrium is related to a
threshold parameter 
which is expressed in terms of model
parameters.
The structure of the equations prohibits solving directly for endemic equilibria, so a new method is used to show that there can be at most one endemic equilibrium. The global stability of the endemic equilibrium is resolved for the case when the drug therapy is sufficiently effective.
The evolution of influenza type A virus is linked to a non-fixed evolutionary landscape driven by tight co-evolutionary interactions between hosts and influenza strains. Cross-immunity, host isolation, and age-structure are three factors responsible for the coexistence of multiple strains of influenza. Here we show that cross-immunity and host isolation alone may support multi-strain epidemics. Futher, we show it is possible to produce sustained oscillations with realistic periods. We establish these predictions via Hopf-bifurcation theory, and illustrate our results with numerical simulations. Period lenghts agree with reported data.
In this talk, I am going to present a generalized sufficient condition that guarantees the stability in discrete-time neural networks. I derive this condition by using the Lyapunov function method and conclude that the equilibrium of the discrete-time neural networks is globally asymptotically stable under the given conditions Consequently, for symmetric and asymmetric connections, the unique attractor is a fixed point as several conditions are satisfied. This criterion is an extension of the results of Chen and Aihara.
This research studies the effects of periodic forcing on attractors
for an autonomous system of nonlinear difference equations. Results indicate
that an attractor for a periodically forced dynamical system may inherit
structure from an attractor of the autonomous system and also from the
periodicity of the forcing. In particular, a method is presented which shows
that if the amplitude of the 
-periodic forcing is small enough then the
attractor for the forced system is the union of homeomorphic subsets.
Examples from population biology and genetics indicate that each subset is
also homeomorphic to the attractor of the original autonomous dynamical
system. (Joint work with James H. Roberds).
This talk will describe recent joint work with Dung Le, University of Texas at San Antonio (a recent ASU Ph.D.), and Don Jones at Arizona State University in which mathematical methods as well as computer simulations are used to study a mathematical model (in this case, a system of differential equations) of microbial colonization of a tubular bio-reactor which accounts for the ability of the bacteria to attach to the reactor wall surface forming a biofilm. The model can be viewed as a description of the microbial community in the mammalian large intestine or of the fouling, via bacterial contamination, of a commercial bio-reactor. Two steady state regimes for the bio-reactor are identified in this research. These are: (1) the complete washout of the microbes from the reactor and (2) the successful colonization of both the wall and bulk fluid by the microbes. Sharp conditions, involving model parameters, are obtained which determine which regime is applicable. This work, soon to be submitted for publication, is the most recent work in an on-going research effort over the past four years centered on mathematical models of the gut microflora.
Epidemic models with two different but connected epidemiological levels, the generalized household and the individual, are introduced in the context of the transmission dynamics of tuberculosis (TB). This multi-level social structure implicitly assumes that individuals are at higher risk of infection from closed contacts in generalized households (clusters) than from casual (random) contacts in a general population. Slow and fast epidemiological time scales are used to reduce the dimensionality of the model. Singular perturbation methods corroborate the results of time-scale approximations. The concept and impact of optimal average cluster or generalized household size on TB dynamics is discussed. The potential impact of the results on the spread of TB are discussed. (Joint work with Carlos Castillo-Chavez and Juan P. Aparicio).
The dynamical properties of the time delayed chemostat model described by
 |
 |
 |
|
 |
|
 |
|
 |
|
 |
(1) |
 |
|
 |
|
 |
|
 |
, 
, 
, 
and 
are a concentration of
the resource, the densities of two bacteria, an infected bacteria and
bacteriophage, respectively.
Further, 
is the rate of flow through the chemostat, 
is the input
concentration of the resource, 
is the attack constant of phage to
the first bacteria, 
is the bacteria's consumption rate of the
resource, 
is the latent period (the time delay between the attack by a
phage on the first bacteria and the resulting reproduction of new phages)
and 
is the reproduction rate of the phage from the infected first
bacteria.
The 
is the bacteria's taking up rate of the resource and satisfies
and increasing in 
.
Note that the first bacteria is assumed to be sensitive to predation of the phage but the second is immune to predation. Some experimental data show that two bacteria (the first is resident and the second is a mutant) and phage can coexist. We consider the boundedness of the solutions of (1) and local (or global) asymptotic stability of nonnegative equilibria. We further show that the coexistence is possible for short latent period and for large reproduction rate of the phage. (Joint work with Edoardo Beretta).
A nonlinear mathematical model was set up for the biochemical reaction in blood coagulation system. The stability of equilibria, the existence of periodic solutions and bifurcation behaviour were studied. The theory on relatively irreducible cooperative systems was established.
Models of disease transmission that include immigration of infective individuals and variable population size are constructed and analyzed. Threshold-like results are given, but in the presence of immigration of infectives there is no disease-free equlibrium. A model with general contact rate for a disease that confers no immunity admits a unique endemic equilibrium that is globally stable. For some other models, the global stability problem remains open. In a model for HIV transmission in a prison system, a considerable reduction of infectives is suggested by the incorporation of screening and quarantining of infectives.
In this talk, we consider a two-species competition system of discrete Lotka-Volterra type with delays proposed by May [1974]. Motivated by the 3/2 global attractivity result for the scalar discrete logistic model with delay in So and Yu [1995], we establish a new 3/2 type criterion for global attractivity of the positive equilibrium of the system.
Classical disease transmission models typically have only a single stable
equilibrium. There is a threshold level of the reproduction number, Ro,
such that if 
then the disease dies out and if 
then the
disease approaches an endemic level. In this simple case, disease control
is a `simple' matter of reducing the reproduction number. Many recent
models show bistability over a range of reproduction numbers, where both
the disease free equilibrium and an endemic equilibrium are stable. These
results have important consequences for disease control. We present a
general compartmental disease transmission model based on a system of
ordinary differential equations. An analysis of the local centre manifold
yields a simple criterion for the existence and stability of super- and
sub-threshold endemic equilibria for Ro near one. This criterion,
together with the definition of 
is illustrated by several models,
including multiple group, multiple strain, staged progression and
vector-host models.
A simple framework for the study of the effects of dispersal on the dynamics of a population experiencing density dependent discrete-time intra-specific competition in several patches is provided. The population dynamics in each patch are modeled by nonlinear functions of the densities and, therefore, capable of generating simple and complex (chaotic) dynamics. Conditions under which the full multi-patch system with dispersal behaves like a single patch system are discussed. The role of dispersal rates in generating multiple attractors where local populations are on simple cyclic non-chaotic attractors are studied. The results are applied to the bobwhite quail population model of Milton and Belair.
We consider the coexistence problem of two predators competing exploitatively for the same prey in a constant environment. It is shown that the coexistence occurs for a wide range of parameter values - as the result of the existence of stable relaxation cycles for a class of three dimensional, singularly perturbed predator-prey system.
The mathematics of the human menstrual cycle is remarkably understudied. Although several hundred biological papers are written on the topic every year, there is no established mathematical model for the cycle, and there have been only about a dozen mathematical papers addressing the subject in the past 30 years. It is a wide open topic, with a wealth of medical and epidemiological applications.
Viewed simply, the menstrual cycle is an attracting cycle of the feedback dynamics among several hormones: two produced at the pituitary (follicle stimulating hormone:FSH, and luteinizing hormone: LH), and three produced in the ovaries (estradiol: E2, progesterone: P4, and inhibin). Another attractor of the system is the post-menopausal steady state. Some of the subtleties that make the system particularly rich from the mathematical viewpoint are:
We will describe the cycle and some of the experimental data, and propose a mechanism for modeling the LH surge as a resonance phenomenon.
In this talk, we first briefly introduce the rotation number approach to eigenvalues. Then we study in detail the characteristic values of the Hill's equations with two-step periodic potentials. As a result, the global structure of resonance pockets is described completely. The results show that resonance pockets behave in a sensible and fairly rich way even in this simplest case.
In this talk, we will discuss the global attractivity in a class of skew-product semiflows and its applications to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations in population dynamics.
For the classic predator-prey model, when the natural death rate of the predator is zero, the model is an ODE with a line of equilibria. The blown-up technique is employed to study the bifurcation of the degenerate predator-prey system.
It is well-known that when
, then the positive equilibrium 
of the delayed logistic
equation (DLE)
![$x'(t)=rx(t)[1-x(t-\tau)]$](img36.gif){kind=small}
is globally asymptotically
stable. On the other hand, a Lotka-Volterra competition system of
pure-delay type is the result of coupling of such basic DLEs. Thus, one
expects that the corresponding 3/2 type criteria exist for the resulting
system. In this talk, we will explore such criteria for continuous
Lotka-Volterra competition systems of pure-delay type.