Abstracts and pdf files of talks
Classical and resource-based competition: A unifying graphical approach
A graphical technique is given for determining the outcome of two species competition for two resources in the chemostat with fixed input resource concentrations. This method is unifying in the sense that the graphical criterion leading to the various outcomes of competition are consistent across most of the spectrum of resource types regardless of the classification method used, and the resulting graphs bear a striking resemblance to the well-known phase portraits for two species Lotka-Volterra competition. Our graphical method complements that of Tilman. Both include zero net growth isoclines. However, instead of using the consumption vectors at potential coexistence equilibria to determine input resource concentrations leading to specific competitive outcomes, we introduce curves bounding the feasible set (the set of all potential equilibria). Existence of coexistence equilibria depends upon whether or not intersections of the zero net growth isoclines lie inside the feasible set. The single species equilibria are found where the feasible set boundary intersects the appropriate zero net growth isoclines. The competitive outcomes are then determined from the position of the single species equilibria (rather than from information at potential coexistence equilibria as in Tilman's method).
Joint with Gail Wolkowicz.
Piecewise linear models approximate real continuous dynamics up to invariant sets
Piecewise linear systems are often used as models of biological (in particular, genetic) regulatory networks and are usually obtained by simplifying a more realistic, but mathematically more complicated, continuous model of the given system. Generally, piecewise linear models have the advantage of being amenable to several analytical and computational techniques, and still provide a good qualitative description of the biological system. The trajectories of a given continuous system can thus be approximated by the trajectories of a corresponding piecewise linear system. In thiscontext, it is important to know how far the solutions of the simplified piecewise linear systems may deviate from those of the true continuous system. One would like to characterize the possible differences as well as the similarities between the solutions of the piecewise linear and those of the continuous system. Focusing on some examples, the current work characterizes regions of phase space where solutions of the continuous and piecewise linear model are still close, and quantifies (up to a distance d) the regions where substantial deviations are possible. From a biological point of view, and if d is small, it is not unreasonable to allow the existence of a d-sized region where the solutions of a system are not well defined or not known. In such cases, a piecewise linear model may be very appropriate to sudy the system outside the d-sized region. An example to be studied is the n-dim negative feedback loop. For n=2, the corresponding continuous and piecewise linear systems may differ ``substantially'' only inside a region delimited by two limit cycles. These limit cycles are defined as the periodic solutions of two extreme (``worst'' and ``best'') cases of the piecewise linear approximation. The ``best'' case limit cycle corresponds to a single point A, while the ``worst'' case limit cycle encloses a compact set, with diameter of order d, around the point A. Trajectories of the continuous system starting outside this set are bounded by trajectories of the ``worst'' case piecewise linear system, and eventually converge towards the interior of the d-sized compact set. This result recalls that of Glass and Pasternack (1978) on the existence of oscillations on a piecewise linear model of the negative feedback loop. The current work also contributes to extend Glass and Pasternack's result to the case of distinct degradation rates for all variables.
Joint with Jean-Luc Gouze.
A Model of Persister Formation and Dynamics
Biofilms are well known for their extreme tolerance to antibiotics. Recent experimental evidence has indicated the existence of a small fraction of specialized persister cells may be responsible for this tolerance. Although persister cells seem to exist in planktonic bacterial populations, within a biofilm the additional protection offered by the polymeric matrix allows persister cells to evade elimination and serve as a source for re-population. Whether persisters cells develop through interactions with toxin/antitoxin modules or are senescent bacteria is an open question. In this investigation we contrast results of the analysis of a mathematical model of the toxin/antitoxin hypothesis for bacteria in a chemostat with results incorporating the senescence hypothesis. We find that the persister fraction of the population as a function of washout rate provides a viable distinction between the two hypothesis. We also give simulation results that indicate that an alternating dose/withdrawal disinfection can be effective in clearing the entire persister and susceptible populations of bacteria. This treatment is restricted to planktonic bacterial populations, so we also consider coupling the kinetics of persister formation in a spatially extended model of biofilm growth and motion in the presence of a flowing fluid.
A Model for the Evolution of Competitive Coexistence
I will describe a model for the evolution of two competing species based on an evolutionary game theory (EGT) modification of the Leslie-Gower model for two competing species. The state variables of this model are population numbers or densities together with mean phenotypic traits of each species. The goal is to determine those circumstances under which the model predicts evolution paths that result in a change of the competitive outcome. For example, we are interested in circumstances that result in a reversal of the roles of competitive winner and loser. We are also interested in evolutionary paths that pass from state of competitive exclusion to one of competitive coexistence. If maximum competition intensity occurs when species have identical traits, we show that the model can predict the former scenario (although the final state will not be an ESS), but cannot predict the latter scenario. We introduce the notion of a 'boxer effect' (when maximum competitive intensity does not occur at identical traits), which when present can lead to circumstances that result in either evolutionary path (and to an ESS). These questions are motivated by laboratory observations of T. Park and P. S. Dawson using two species of Tribolium.
Joint work with Rosalyn Real, Thomas L Vincent and Robert F Costantino.
Multiple equilibria and global attractors in biochemical reaction network dynamics
Modern biological research provides countless examples of biochemical reaction networks. In order to understand the role played by some of the reactions in these networks, one often faces great difficulties in trying to interpret the effect of positive and negative feedbacks, nonlinear interactions, and other complex signaling between the nodes of the network. We will describe connections between reaction network structure and the capacity for multiple equilibria and other interesting dynamic behavior, and will discuss how these connections may impact the interpretation of experimental data.
Treatment for within-host virus dynamics
We revisit a standard model describing the infection cycle of a virus in an individual. One example of this model is furnished by HIV, a retrovirus which infects a particular class of immune cells, the CD4+ T cells, giving rise to infected T cells that spawn off new virus. However, the dynamics of other infections such as hepatitis, influenza and even the malaria parasite P. falciparum can be described by the same model. We will discuss the effect of periodic treatment on the system, and give tight bounds for the drug efficiencies required to eradicate the infection. We will also consider the problem of finding optimal treatment schedules that minimize various measures of the burden of the treatment on the patient.
Global analysis of a multi-strain virus model with mutations
We consider a within-host multi-strain virus model and allow mutations between different strains. In the absence of mutations, the fittest strain drives the remaining strains to extinction. Treating the mutation rate as a perturbation parameter, we show that the corresponding steady state persists, perhaps with small concentrations of some or all other strains, depending on the connectivity of the graph describing all possible mutations. Using a global perturbation result of H. Smith and P. Waltman, we show that the perturbed steady state remains globally asymptotically stable.
Joint with Sergei S. Pilyugin.
Quorum Sensing and Biofilm Modeling
Many bacteria use the size and density of their colonies to regulate the production of a large variety of substances. This phenomenon is called quorum sensing. Pseudomonas aeruginosa use the size and density of their colonies to regulate the production of a large variety of substances, including toxins. P. Aeruginosa in planktonic (non-colonized) form are non-toxic, but as a biofilm, they are highly toxic and well protected by the polymer gel in which they reside. However, they do not become toxic or begin to form polymer gel until the colony is of sufficient size to overwhelm the immune system. Before this, they cannot be detected by the immune system. We give a overview of several different mathematical models of quorum sensing.
A Mathematical Model for the Effects of HER2 Overexpression on Cell Cycle Progression in Breast Cancer
We present a mathematical model to study the effects of HER2 over-expression on cell-cycle progression in breast cancer. The model addresses the following question: How do changes in the number of HER2 and EGFR receptors during the cell-cycle affect the cell proliferation rate? In order to characterize the effects of HER2 overexpression on the cell cycle progression, we use a three-compartment cell cycle model with non-constant transition rates. Our new hypothesis is that the transition rates depend on the number of the cell surface HER2 receptors and their signaling properties. The model relates the different phases of the cell cycle transition rates to the signaling properties of the EGFR-HER2 receptors (through their binding kinetics), and the population dynamics of cells in the corresponding cell-cycle phase.
Joint with D. Isaacson.
Cooperative Boolean networks: long periodic orbits and generalizations of the Smale and Hirsch theorems
Boolean networks are an increasingly popular tool to model large scale protein networks - they rely on the simple assumption that the expression level of a protein at any given time is either 1 (full expression) or 0 (no expression). In this talk we study to what extent well-known theorems by Smale and Hirsch, which form part of the theory of (continuous) monotone dynamical systems, generalize or fail to do so in this discrete case. We also report that a Boolean network may fall within the so-called 'chaotic regime', even under the simultaneous assumption of several conditions which in randomized studies have been separately shown to correlate with ordered behavior. These properties include the cooperativity of the network, and the use of at most two inputs and outputs for every variable.
Joint work with Winfried Just
Multi-valued characteristics and Morse decompositions
A theory of monotone input-output systems is one of a few mathematical approaches that has been successfully applied to complex models of biological and chemical interactions. Replacing some dynamic interactions between variables by a set of static inputs and characterizing the resulting open loop system by an input-output characteristic, the theory establishes convergence results for the original closed loop system. We significantly extend the theory to the situation when the open loop system has multiple stable equilibria and hence a multi-valued characteristic. We show that the information embedded in the characteristic can be used to construct a Morse decomposition of the invariant set of the closed loop system. These results can be used to predict bistability as well as suggest existence of periodic orbits for the closed loop system. The previous theory on global convergence is shown to apply locally to individual Morse sets and is seamlessly incorporated into our global theory. We apply our tools to a previously studied model of cell cycle maintenance. Our results show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways: it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions.
Multi-scale phenomena in modeling the onset of ventricular fibrillation
Over 400,000 Americans suffer sudden cardiac death (SCD) each year, half unexpected, with SCD frequently caused by ventricular fibrillation. A simulation study of "spontaneous VF" needs to span time scales from milliseconds for the upstroke of an activation to ~ 1 second for a single heartbeat seconds to 300 years, the expected waiting time to SCD in Americans age 35 and above. This range of 10 to 13 orders of magnitude makes direct simulation difficult even in even a very simple cardiac model. The talk will discuss a the use of a Markov approach, and Kramers rule and other scaling laws to approach the challenging, stiff problem of studying "spontaneous VF."
Metapopulation extinction risk raised to critical level by spatially variable dispersal
Dynamics of acyclic 1-dimensional maps
A continuous map f in an interval I is acyclic if the only periodic points are fixed points. Sarkovski's theorem implies this is equivalent to every point of period 2 being fixed. By an interesting theorem--- perhaps folklore?--- acyclicity is also equivalent to every omega limit point being fixed. Old and new conditions for acyclicity will be discussed, some of which involve f''' and the Schwarzian derivative. An example in I=[0,\infty) is xe^{b-x}, b \ge 0: it is acyclic if and only if b \le 2. Typical application: ax^3 +bx^2 +cx, with a>0>c, is acyclic if and only if c \le -1.
Persistence and Senescence in Bacterial Populations
It has been known for many years that small fractions of persister cells resist killing in many bacterial colony-antimicrobial confrontations. These persisters are not believed to be mutants. Rather it has been hypothesized that they are phenotypic variants. Current models allow cells to switch in and out of the persister phenotype. In this talk a different explanation is suggested, namely senescence, for persister formation. Using a model including age structure, it is shown that senescence provides a natural explanation for persister-related phenomena including for example the observations that persister fraction depends on growth phase in batch culture and dilution rate in continuous culture.
Competitive exclusion and coexistence in a continuous-time lottery model with temporal heterogeneity Temporal heterogeneity is one of the important factors promoting species coexistence. The lottery model proposed by Chesson and Warner (1981) plays an important role in understanding the role of temporal heterogeneity. In this talk, we derive an ODE model from the original discrete-time lottery model by mean of averaging and analyze it since ODE models are often mathematically more tractable than discrete-time models. In fact, the mathematical framework of competitive exclusion constructed by McGehee and Armstring (1977) and the Liapunov function constructed by Kubo and Iwasa (1996) are applicable to our ODE model. These applications help to understand the global dynamics of the original lottery model and show how to count the number of resources generated by temporal heterogeneity.
Rich dynamics of a delay differential equation model of hepatitis B virus infection
Chronic HBV affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or an endemic steady state. Our model admits periodic solutions. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.
Global Dynamics of Microbial Competition for Two Resources with Internal Storage
We discuss a chemostat model that describes competition between two species for two essential resources based on storage. The model incorporates internal resource storage variables that serve the direct connection between species growth and external resource availability. We show that the limiting system of the model basically exhibits the familiar Lotka-Volterra alternatives: competitive exclusion, coexistence, and bi-stability, and most of these results can be carried over to the original model.
Joint work with Hal Smith.
Carrying Simplices in Totally Competitive Systems
We give a survey of carrying simplices in totally competitive systems of ordinary differential equations. In particular, we concentrate on the problem of their smoothness. Some of the results have been obtained jointly with Jifa Jiang and Yi Wang.
A Database for Global Dynamics: Is Less is More?
In many applications there are models for dynamics but specific parameters are unknown or not directly measurable. This suggests the need to be able to search through parameter space for specific types of dynamical behavior. Ideally, this would be done computationally in some automated manner, and then later the researcher would be able to query the results. In this talk, we discuss computational topological methods which can extract coarse, but rigorous, global descriptions of dynamics and changes with respect to parameters. Moreover, we discuss ongoing efforts to develop a method for building databases that contain useful, rapidly identifiable information about the types of dynamics computed.
Should I stay or should I go? On the evolution of dispersal.
Plants and animals often live in landscapes where environmental conditions vary from patch to patch and moment to moment. Since the fecundity and survivorship of an individual depends on these factors, an organism may decrease or increase its fitness by dispersing across the environment. Consequently, a fundamental question in evolutionary ecology is ''how do dispersal patterns evolve in spatially and temporally heterogeneity environments?'' To address this question, I will present analytical results about periodically forced models of competing species that only differ in their dispersal strategies. The analysis combine standard techniques from monotone maps with new results about one-parameter families of non-negative matrices. Several challenging problems in dynamical systems and matrix analysis will be posed.
Prevalent Behavior of Strongly Order Preserving Semiflows
Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or towards the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence, developed in Christensen:1972 and Yorke et al.:1992. For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show the prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.
Joint work with G. Enciso & M.W. Hirsch
Passivity-based techniques for cyclic and other feedback systems
This talk will describe work, done in collaboration with Murat Arcak, dealing with the analysis of biological networks by means of passivity theory. We show in this talk how the theory of interconnections of passive systems (a class of systems well-studied in control theory) leads to natural and far-reaching generalizations of the classical ``secant condition'' for cyclic negative feedback systems, as well as to new results for other feedback structures.
An infectious disease model with distributed susceptibility: a challenge for persistence theory.
We consider an S-I type infectious disease model where the susceptibles differ by their susceptibility to infection. This model presents several challenges. Even existence and uniqueness of solutions is non-trivial. Further it is difficult to linearize about the disease-free equilibrium in a rigorous way. This makes disease persistence a necessary alternative to linearized instability in the superthreshold case. Application of standard persistence theory faces the difficulty of finding a compact attractor. One can work around this obstacle by using integral equations.
Asymptotic Symmetry in Monotone Skew-Product Semiflows with applications
This talk is mainly concerned with the asymptotic symmetry of the monotone skew-product semiflows under a group action. It is shown that any stable minimal set is residually symmetric and any uniformly stable bounded trajectory is asymptotically symmetric. These results are then applied to study the spatio-temporal asymptotics of stable solutions of reaction-diffusion equations on a symmetric domain in time recurrent structures including almost periodicity and almost automorphy. In particular, the 1-covering property of omega limit sets is established for uniformly stable bounded solutions of R-D equations on a ball.
Classical and resource-based competition: A unifying graphical approach
A graphical technique is given for determining the outcome of two species competition for two resources in the chemostat with fixed input resource concentrations. This method is unifying in the sense that the graphical criterion leading to the various outcomes of competition are consistent across most of the spectrum of resource types regardless of the classification method used, and the resulting graphs bear a striking resemblance to the well-known phase portraits for two species Lotka-Volterra competition. Our graphical method complements that of Tilman. Both include zero net growth isoclines. However, instead of using the consumption vectors at potential coexistence equilibria to determine input resource concentrations leading to specific competitive outcomes, we introduce curves bounding the feasible set (the set of all potential equilibria). Existence of coexistence equilibria depends upon whether or not intersections of the zero net growth isoclines lie inside the feasible set. The single species equilibria are found where the feasible set boundary intersects the appropriate zero net growth isoclines. The competitive outcomes are then determined from the position of the single species equilibria (rather than from information at potential coexistence equilibria as in Tilman's method).
Joint with Mary Ballyk.
Computational Study of Blood Flow Effects on Growth of Thrombi Using a Multiscale Model
A multiscale model developed in * is employed to study the formation of platelet thrombi in blood vessels. The model involves interaction among a viscous, incompressible fluid; activated platelets and non-activated platelets; activating chemicals; fibrinogen; and the vessel walls. The macroscale dynamics of the pulsatile blood flow is described by the Navier-Stokes equations and is described at the continuum level. The microscale activities of the platelets are described through a stochastic Cellular Potts model (CPM). Simulation results demonstrate qualitative agreement between the mathematical model and the experiments. Based on the preliminary results, we further explore the blood flow effects on growth of thrombi.
Joint work with Nan Chen, Malgorzata M. Kamocka, Elliot D. Rosen and Mark Alber.
* Journal of The Royal Society Interface doi: 10.1098/rsif.2007.1202, in press
Hal Smith's contributions
