James F. Selgrade

Global asymptotic behavior of a two-dimensional difference equation modelling competition

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = xn/ a + cyn, yn+1 = yn/ b + dxn, n =0,1,..., where the parameters a and b are in (0, 1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a = b we find an explicit equation for the stable manifold.

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.

Convergence to equilibrium in a genetic model with differential viability between the sexes.

A single locus, diallelic selection model with female and male viability differences is studied. If the variables are ratios of allele frequencies in each sex, a 2-dimensional difference equation describes the model. Because of the strong monotonicity of the resulting map, every initial genotypic structure converges to an equilibrium structure assuming that no equilibrium has eigenvalues on the unit circle.

On the structure of attractors for discrete, periodically forced systems with application to population models

Abstract This work discusses the effects of periodic forcing on attracting cycles and more complicated attractors for autonomous systems of nonlinear difference equations. Results indicate that an attractor for a periodically forced dynamical system may inherit structure from an attractor of the autonomous (unforced) system and also from the periodicity of the forcing. In particular, a method is presented which shows that if the amplitude of the k-periodic forcing is small enough then the attractor for the forced system is the union of k homeomorphic subsets. Examples from population biology and genetics indicate that each subset is also homeomorphic to the attractor of the original autonomous dynamical system.

A model for hormonal control of the menstrual cycle: Structural consistency but sensitivity with regard to data

This study presents a 13-dimensional system of delayed differential equations which predicts serum concentrations of five hormones important for regulation of the menstrual cycle. Parameters for the system are fit to two different data sets for normally cycling women. For these best fit parameter sets, model simulations agree well with the two different data sets but one model also has an abnormal stable periodic solution, which may represent polycystic ovarian syndrome. This abnormal cycle occurs for the model in which the normal cycle has estradiol levels at the high end of the normal range. Differences in model behavior are explained by studying hysteresis curves in bifurcation diagrams with respect to sensitive model parameters. For instance, one sensitive parameter is indicative of the estradiol concentration that promotes pituitary synthesis of a large amount of luteinizing hormone, which is required for ovulation. Also, it is observed that models with greater early follicular growth rates may have a greater risk of cycling abnormally.

Population Interactions with Growth Rates Dependent on Weighted Densities

Differential and difference equation models of interacting populations are presented and analyzed. The per capita growth rates are functions of linear combinations of individual population densities. The asymptotic behavior of these systems is discussed, including the occurrence of strange attractors. For the two dimensional differential equation, a general formula for the stability coefficient of a Hopf bifurcation is derived.

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.

A lifelong model for the female reproductive cycle with an antimüllerian hormone treatment to delay menopause.

A system of 16 non-linear, delay differential equations with 66 parameters is developed to model hormonal regulation of the menstrual cycle of a woman from age 20 to 51. This mechanistic model predicts changes in follicle numbers and reproductive hormones that naturally occur over that time span. In particular, the model illustrates the decline in the pool of primordial follicles from age 20 to menopause as reported in the biological literature. Also, model simulations exhibit a decrease in antimüllerian hormone (AMH) and inhibin B and an increase in FSH with age corresponding to the experimental data. Model simulations using the administration of exogenous AMH show that the transfer of non-growing primordial follicles to the active state can be slowed enough to provide more follicles for development later in life and to cause a delay in the onset of menopause as measured by the number of primordial follicles remaining in the ovaries. Other effects of AMH agonists and antagonists are investigated in the setting of this model.

Frequency-dependent selection in logistic growth models

This paper describes the dynamics of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. We modify the basic density-dependent logistic growth model by inserting three possible types of frequency dependence in the fitness functions. These models are analyzed and contrasted with the purely density-dependent situation. Examples are given of periodic fluctuations in allele frequency and population density, which would be impossible for purely density-dependent fitness functions.

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay.

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.