Global dynamics of a selection model for the growth of a population with genotypic fertility differences

A population growth model is considered for a one locus two allele problem with selection based entirely on fertility differences. A local stability analysis is carried out for the critical points — which include possible polymorphic states — of the resulting nonlinear differential equations. The methods of dynamical systems theory are applied to obtain limiting genotypic proportions for every initial state. Thus the results are global and there are no periodic solutions.Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-8130Research for this paper was partially supported by the National Science and Engineering Research Council of Canada Grant NSERC A-4823Research supported by NSF Grant MCS 7901069. A portion of the work was carried out while the author was a Visiting Professor at the University of Utah, Salt Lake City, Utah     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models are relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local (using index arguments). This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the case of uninhibited specific growth rate.Research supported by the National Council of Science, Republic of ChinaResearch supported by National Science Foundation Grant, DMS-9204490Research supported by the Natural Science and Engineering Council of Canada. This author's contribution was made while on research leave visiting the Department of Ecology and Evolutionary Biology at Princeton University. She would especially like to thank Simon Levin for his guidance as well as for providing an exceptional working environment     

Competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

A model of competition in the chemostat with an inhibitor is combined with a model of competition in the chemostat between plasmid-bearing and plasmid-free organism to produce a model that more closely approximates the way chemostat-like devices are used in biotechnology. The asymptotic behavior of the solutions of the resulting system of nonlinear differential equations is analyzed as a function of the relevant parameters. The techniques are those of dynamical systems although perturbation techniques are used when the parameter reflecting plasmid-loss is small.Research Supported by National Council of Science, Republic of ChinaResearch Supported by National Science Foundation Grant MCS 9204490     

The propagation of the nerve impulse.

A partial differential equation for the propagated action potential is derived using symmetry, charge conservation, and Ohm's law. Charge conservation analysis explicitly includes the gating charge when applied in the laboratory frame. When applied in the system of reference in which capacitive currents are zero, it yields a relation between orthogonal components of the ionic current allowing us to express the nonlinear ionic current in terms of the voltage-dependent membrane capacitance C(V) and the axial current that satisfies Ohm's law. The ionic current is shown to behave as C(V)V[C(V)V2]' at the foot of the action potential while the gating current behaves as C(V)V[Cg(V)V]' where Cg(V) is the capacitance associated with gating. Improved knowledge of the nonlinear current makes it possible to describe the propagated action potential in an approximated way with quasilinear partial differential equations. These equations have analytical solutions that travel with constant velocity, retain their shape, and account for other properties of the action potential. Furthermore, the quasilinear approximation is shown to be equivalent to the FitzHugh-Nagumo equation without recovery making apparent its physical content.     

Accurate computation of the stable solitary wave for the FitzHugh-Nagumo equations

Comparisons are made between three different methods for computing the stable solitary wave solution for the FitzHugh-Nagumo equations which consist of a nonlinear diffusion equation coupled to an ordinary differential equation in time. They model the Hodgkin-Huxley equations which describe the propagation of the nerve impulse down the axon. Two of the methods involve the travelling wave equations. Previous accurate numerical computations of these equations as an initial-value problem using a shooting method lead to inaccurate values for the wave speed; however, nonlinear corrections to the initial values are shown to yield accurate values. A boundary-value method applies asymptotic boundary conditions and uses a spline-collocation code called COLSYS for numerical solution of boundary-value problems which leads to accurate wave profiles and speeds. The third method is to solve an initial-boundary-value problem with an adaptive outgoing wave condition for the partial differential equations where the solitary wave emerges as the stable long time solution. The concept of a wave integral is introduced and they are derived to determine the wave speed used in the adaptive boundary condition and to measure the closeness of the computed solutions to the exact solitary wave solution.This work was supported in part by the Natural Sciences and Engineering Research Council Canada under Grant A4559 and by the John Simon Guggenheim Memorial Foundation     

Helical waves in the vortex electron anisotropic hydrodynamic model

Solutions to the vortex electron anisotropic hydrodynamic equations are investigated that describe nonlinear helical waves in an anisotropic magnetized plasma. The possibility of constructing such solutions is provided by the symmetry properties of the equations. An optimum family of one-dimensional subgroups of a symmetry group consistent with the equations is constructed that makes it possible to derive other, essentially different solutions.     

Pattern sensitivity to boundary and initial conditions in reaction-diffusion models

We consider Turing-type reaction-diffusion equations and study (via computer simulations) how the relationship between initial conditions and the asymptotic steady state solutions varies as a function of the boundary conditions. The results indicate that boundary conditions which are non-homogeneous with respect to the kinetic steady state give rise to spatial patterns which are much less sensitive to variations in the initial conditions than those obtained with homogeneous boundary conditions, such as zero flux conditions. We also compare linear pattern predictions with the numerical solutions of the full nonlinear problem.This work supported in part by U.S. Army Grant DAJA 37-81-C-0220 and the Science and Engineering Research Council of Great Britain Grant GR/c/63595     

Modelling and simulation of vertebrate retina

In the present work we investigate the neuronal activities in a vertebrate retina by modelling and simulations using the results of (Oguztöreli, 1979). The basic retinal network considered here consists of interconnected five neurons: a receptor cell (rod or cone), a horizontal cell, a bipolar cell, an amacrine cell, and a retinal ganglion cell. The mathematical model for the basic network is a system of nonlinear ordinary integral differential difference equations. A number of simulations describing the dynamics of the neural activities in the basic network under different conditions are presented, actual and steady-state solutions are discussed. An algorithm is proposed for the determination of the system parameters experimentally.This work was supported by the Natural Sciences and Engineering Research Council Canada under Grant NSERCA-4345 through the University of Alberta     

Mathematical aspects of peridic catatonic schizophrenia

The mathematical study of periodic catatonic schizophrenia initiated by Danziger and Elmergreen is generalized by considering the class of differential equations that could be used to describe the periodic behavioral symptoms and periodic variation of biochemical levels in periodic catatonic schizophrenia. The existence of asymptotically stable periodic solutions is studied mathematically and the physical significance of such periodic solutions is discussed. The occurrence of relaxation oscillations is briefly considered. The research in this paper was supported partly by the U.S. Army Research Office (Durham) (Grant No. DA-ARO-D-31-124-72-C69) and partly by a Rutgers Research Council Faculty Fellowship.     

Global analysis of a model of plasmid-bearing,plasmid-free competition in a chemostat with inhibitions

A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat was proposed in a paper of Stephanopoulis and Lapidus. The model was in the form of a system of nonlinear ordinary differential equations. Such models were relevant to commercial production by genetically altered organisms in continuous culture. The analysis there was local. The rigorous global analysis was done in a paper of Hsu, Waltman and Wolkowicz in the case of the uninhibited specific growth rates. This paper provides a mathematically rigorous analysis of the global asymptotic behavior of the governing equations in the cases of combinations of inhibited and uninhibited specific growth rates.Research Supported by the National Council of Science, Republic of China     

Effect of asymmetric modification on the conformation of ascidiacyclamide analogs.

Ascidiacyclamide (ASC), cyclo(-Ile1-Oxz2-d-Val3-Thz4-)2 (Oxz=oxazoline and Thz=thiazole) has a C2-symmetric sequence, and the relationships between its conformation and symmetry have been studied. In a previous study, we performed asymmetric modifications in which an Ile residue was replaced by Gly, Leu or Phe to disturb the symmetry [Doi et al. (1999) Biopolymers49, 459-469]. In this study, the modifications were extended. The Ile1 residue was replaced by Gly, Ala, aminoisobutyric acid (Aib), Val, Leu, Phe or d-Ile, and the d-Val3 residue was replaced by Val. The structures of these analogs were analyzed by X-ray diffraction, 1H NMR and CD techniques. X-Ray diffraction analyses revealed that the [Ala1], [Aib1] and [Phe1]ASC analogs are folded, whereas [Val1]ASC has a square form. These structures are the first examples of folded structures for ASC analogs in the crystal state and are similar to the previously reported structures of [Gly1] and [Phe1]ASC in solution. The resonances of amide NH and Thz CH protons linearly shift with temperature changes; in particular, those of [Aib1], [d-Ile1] and [Val3]ASCs exhibited a large temperature dependence. DMSO titration caused nonlinear shifts of proton resonances for all analogs and largely affected [d-Ile1] and [Val3]ASCs. A similar tendency was observed upon the addition of acetone to peptide solutions. Regarding peptide concentration changes, amide NH and Thz CH protons of [Gly1]ASC showed a relatively large dependence. CD spectra of these analogs indicated approximately two patterns in MeCN solution, which were related to the crystal structures. However, all spectra showed a similar positive Cotton effect in TFE solution, except that of [Val3]ASC. In the cytotoxicity test using P388 cells, [Val1]ASC exhibited the strongest activity, whereas the epimers of ASC ([d-Ile1] and [Val3]ASCs), showed fairly moderate activities.     

Optimal HIV treatment by maximising immune response

We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy CD4+ T cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.Research supported by the Natural Science and Engineering Research Council (NSERC) and the Mathematics of Information Technology and Complex Systems (MITACS) of Canada     

A passive dynamic walking robot that has a deterministic nonlinear gait

There is a growing body of evidence that the step-to-step variations present in human walking are related to the biomechanics of the locomotive system. However, we still have limited understanding of what biomechanical variables influence the observed nonlinear gait variations. It is necessary to develop reliable models that closely resemble the nonlinear gait dynamics in order to advance our knowledge in this scientific field. Previously, Goswami et al. [1998. A study of the passive gait of a compass-like biped robot: symmetry and chaos. International Journal of Robotic Research 17(12)] and Garcia et al. [1998. The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering 120(2), 281-288] have demonstrated that passive dynamic walking computer models can exhibit a cascade of bifurcations in their gait pattern that lead to a deterministic nonlinear gait pattern. These computer models suggest that the intrinsic mechanical dynamics may be at least partially responsible for the deterministic nonlinear gait pattern; however, this has not been shown for a physical walking robot. Here we use the largest Laypunov exponent and a surrogation analysis method to confirm and extend Garcia et al.'s and Goswami et al.'s original results to a physical passive dynamic walking robot. Experimental outcomes from our walking robot further support the notion that the deterministic nonlinear step-to-step variations present in gait may be partly governed by the intrinsic mechanical dynamics of the locomotive system. Furthermore the nonlinear analysis techniques used in this investigation offer novel methods for quantifying the nature of the step-to-step variations found in human and robotic gait.     

Alternative solution of the simplest model of enzymatic synthesis of nucleic acids by homotopy perturbation method

Development of methods for obtaining approximate analytical solutions of nonlinear differential equations and their systems is a rapidly developing field of mathematical physics. Earlier, an approximate solution of the simplest system of kinetic enzymatic equations for calculating dynamics of complementary strands of nucleic acids was obtained. In this study, we consider an alternative approach to selecting the basic linear approximation of the used method, which makes it possible to obtain more accurate analytical solutions of the set problem.     

Differential equations for moments of an interacting particle process on a lattice

Differential equations are derived for a class of interacting lattice particle processes first introduced in [11]. When there is one type of particle, the equations are in terms of the moments of boundary counts and particle counts. When there are two types of particles, the equations are in terms of the moments of certain neighbourhood statistics as well.This work has been supported by grants from the Natural Sciences and Engineering Research Council of Canada. WJB also acknowledges support of an Ontario Graduate Scholarship     

Determination of sedimentation coefficients for small peptides.

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

Transient behavior of population density with competition for resources

In this paper we derive a perturbation expansion of some nonlinear diffusion equations proposed by J. G. Skellam and M. Kimura. The solution is derived by converting the differential equations into integral equations by means of the Green's function for the diffusion equation. This research was supported in part by the Office of Naval Research under Contract Nonr-595(17).     

An appetite suppressant from Hoodia species

Studies conducted at the Council for Scientific and Industrial Research (CSIR, South Africa) identified extracts from Hoodia species, in particular Hoodia pilifera and Hoodia gordonii, as possessing appetite suppressing properties. Two pregnane glycosides were isolated by fractionation of the dried stems of H. gordonii. Their structures were determined as 3beta-[beta-D-thevetopyranosyl-(1-->4)-beta-D- cymaropyranosyl-(1-->4)-beta-D-cymaropyranosyloxy]-12beta-tigloyloxy-14beta-hydroxypregn-5-en-20-one (1) and 3beta-[beta-D-cymaropyranosyl-(1-->4)-beta-D-6-thevetopyranosyl-(1-->4)-beta-D-cymaropyranosyl-(1-->4)-beta-D-cymaropyranosyloxy]-12beta-tigloyloxy-14beta-hydroxypregn-5-en-20-one (2) on the basis of spectroscopic studies and conversion to known compounds. Compounds 1 and 2 were also isolated from H. pilifera. Compound 1 was tested for its appetite suppressant properties in rats by oral gavage at 6.25-50 mg/kg and the results showed that all doses resulted in a decrease of food consumption over an eight day period and a body mass decrease when compared to the control sample receiving only the vehicle. In a comparative study against a fenfluramine control sample, compound 1 resulted in a reduction in food intake over the study period, with a concomitant overall decrease in body weight while fenfluramine resulted in a small decrease in food intake, but an increase in body weight (though less than control group) over the same period of time.     

Three-Dimensional Flow of an Oldroyd-B Nanofluid towards Stretching Surface with Heat Generation/Absorption

This article addresses the steady three-dimensional flow of an Oldroyd-B nanofluid over a bidirectional stretching surface with heat generation/absorption effects. Suitable similarity transformations are employed to reduce the governing partial differential equations into coupled nonlinear ordinary differential equations. These nonlinear ordinary differential equations are then solved analytically by using the homotpy analysis method (HAM). Graphically results are presented and discussed for various parameters, namely, Deborah numbers and , heat generation/absorption parameter Prandtl parameter , Brownian motion parameters, thermophoresis parameter and Lewis number . We have seen that the increasing values of the Brownian motion parameter and thermophoresis parameter leads to an increase in the temperature field and thermal boundary layer thickness while the opposite behavior is observed for concentration field and concentration boundary layer thickness. To see the validity of the present work, the numerical results are compared with the analytical solutions obtained by Homotopy analysis method and noted an excellent agreement for the limiting cases.