The usual predator-prey model has been successfully applied recently to the spread of HIV/AIDS in both Canada and the United States. In the predator-prey ODE model, the rate of increase of the total population is taken to be equal to the rate at which HIV negative people would increase without HIV/AIDS minus the rate at which they die. We use this idea to develop our new model. If we take one year to be the unit of time, which is the usual interval at which the data on HIV positive people is published in most countries, then the differential equations become difference equations. In this paper, we present such a difference equations model for spread of HIV/ AIDS in any society and compare the numbers predicted by this model with the actual numbers in both Canada and the United States. The two sets of numbers are very close. For appropriate values of the parameters, the model seems to have a chaotic solution without a chaotic attractor.

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied in each iteration of the process. In this talk, we study random maps with position dependent probabilities on the interval. Sufficient conditions for the existence of absolutely continuous invariant measures for weakly convex and concave random maps with position dependent probabilities will be presented.

Linear stability conditions for a first order 3-dimensional discrete dynamic have been derived in terms of the trace, determinant and sum of principle minors of the Jacobian evaluated at the equilibrium. The resulting set of inequalities is an improvement over the Gershgorin Theorem in that the set of inequalities provides conditions that are both necessary and sufficient for linear stability in three dimensions whereas the Gershgorin Theorem provides only sufficient conditions. Can that same technique be expanded to generalize the linear stability conditions into higher order discrete dynamics? Can the same method also best the Gershgorin Theorem in 4-dimensional discrete dynamics?

The probabilities of various biological asymptotic dynamics are computed for a stable system that is invaded by another competitive species. The asymptotic behaviors studied include extinction, weak extinction, permanence, and mutual exclusion. The model used is a discrete Lotka-Volterra system that model species that compete for the some resources. Among the results found are that the chance of permanence occurring in the invaded system is significantly higher than the probability of permanence in a purely random system, and that multiple extinctions that include the invading species and one of the original species are impossible.

(joint work with M. E. Alexander, C. Bowman, S.M. Moghadas, R. Summers and B.M. Sahai)

Although some preventive vaccines against certain human diseases (such as measles, rubella, chickenpox etc.) are known to offer lasting immunity, influenza vaccines offer only a short-term protection which wanes over time. The vaccine-induced protection against influenza varies between 70%-90% for young healthy people to approximately 30%-40% in geriatrics or in individuals with a weakened immune system. The public health objective of this study is to determine, via mathematical modelling, whether or not such an imperfect vaccine can be an effective public health tool for controlling (or even eradicating) the spread of influenza infection within a given population. A robust deterministic model for influenza transmission, which incorporates an imperfect vaccine, will be presented and analyzed qualitatively. The global stability analysis of the model reveals that the model can exhibit the phenomenon of bi-stability, characterized by the presence of stable multiple endemic equilibria even when the basic reproductive number of infection (R₀) is less than unity. The epidemiological consequence of this phenomenon, vis-a-vis the community-wide control and/or eradication, will be discussed.

The spontaneous formation of order in the form of spatial concentration patterns in an unstirred chemical medium, supported by dissipation of chemical free energy, has been considered often since a pioneering suggestion by Turing and work by Prigogine's group on non-equilibrium thermodynamics. The best-known experimental example is the oscillatory Belousov-Zhabotinskii (BZ) reaction, in which target patterns of outward-moving concentric rings are readily observed. One widely-studied question is whether "microscopic" fluctuations can cause nucleate targets, or whether a catalytic, nucleating heterogeneous center is required. Vidal and Pagola observed spontaneous activity with no nucleating particles visible at 6-micron resolution; however Zhang, Forster and Ross argued theoretically that this is impossible in the steady cycling regime of the BZ reaction. We describe here an explicit mechanism in a supercritical regime by which microscopic fluctuations can nucleate activity and reconcile these results with Zhang et al. Joint work with S.G. Sobel (Hofstra) and R.J. Field (Univ. of Montana). Partially supported by the NSF.

We investigate the periodic character and boundedness nature of positive solutions of the difference equation

(i)  Each positive solution is either eventually periodic or is unbounded and nonpersistent.

(ii)  For fixed { A_n}_n=0^¥ and {B_n}_n=0^¥, either every positive solution is eventually periodic or every positive solution is unbounded and nonpersistent.

(iii)  For fixed { A_n}_n=0^¥ and {B_n}_n=0^¥, if every positive solution is eventually periodic, then there exists an integer N ³ 1 such that every positive solution is eventually periodic with (not necessarily prime) period 4Npq.

Fitting experimental data to a chosen mathematical model amounts to determining the best choice of coefficients or parameters in the model. In the literature, various methods are used to search the parameter space. Such problems can often be framed as finding a contractive map with a particular fixed point. In this talk, we discuss how the parameters that minimize the "collage distance" associated with such maps provide in general an excellent starting point for further optimization.

The most important problem is how to construct convergent difference scheme. Since the convergence is a consequence of consistency and stability thus it is necessary to choose those approximating schemes that are stable. It is naturally to have stability in the norms of the spaces for which the original problem is stable. For the well-posed problems of mathematical physics these are the energy spaces where the squares of the norms express the total energy of the systems. Because of this, we have to analyze the derivation of the energy estimations in the differential case and to construct the scheme for which we can satisfy this derivation in the corresponding Hilbert space in the discrete case. However, the criteria of consistency and stability become complicated when applied to the solution of non-linear partial differential equations. Therefore, the difference scheme has to be conservative, namely, its conservation laws to be satisfied identically. Then the non-linearity is not invincible task.

The conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. In present work we summarize some of the analytical requirements necessary to be satisfied of the difference scheme. For illustration the vectorial operator splitting numerical scheme is examined for its conservation properties and other requirements. It is proven that the difference approximation of the advection operator conserves square of velocity components and the kinetic energy like the differential operator does, while pressure term conserves only the kinetic energy. Therefore, strong stability in solving higher Reynolds number flows can be achieved, which is confirmed through various numerical results.

Consider the max-type difference equation

In the theory of analytic dynamical systems we make an extensive use of normal forms in the neighbourhood of fixed points of diffeomorphisms or singular points of vector fields. Generically the normalizing changes of coordinates diverge and the analytic classification of singularities under changes of coordinates involves functional moduli. An explanation is given by unfolding the situation and studying the dynamics of the unfolded system. We present here a complete modulus for a germ of generic family unfolding a diffeomorphism with a parabolic point and also for a family unfolding a generic saddle-node of a 2-dimensional vector field and discuss the geometry of the systems in the family.

In this talk, we will discuss the long-term behavior of solutions of a class of nonautonomous delay differential equations. By appealing to the theory of monotone discrete-time dynamical systems, we obtain a threshold-type result on the global dynamics for scalar periodic delay differential equations, which is then lifted to a class of asymptotically periodic delay differential equations. If time permits, I will give an example for the application of this result to m-species competitive systems with stage structure.

(joint work with Sue Ann Campbell and Gail Wolkowicz)

Consider a predator-prey system with nonmonotonic functional response. We show that in this case there exists a Bogdanov-Takens bifurcation point of codimension 3, which acts as an organizing center for the system. I shall talk about various sequences of bifurcations when the death rate of the predator is varied. The bifurcation sequences involving Hopf bifurcations, homoclinic bifurcations as well as the saddle-node bifurcations of limit cycles are determined using information from the complete study of the Bogdanov-Takens bifurcation point of codimension 3 and the geometry of the system.