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Integrodifference equations are discrete-time models that possess many of the attributes of continuous-time reaction-diffusion equations. They arise naturally in population biology as models for organisms with discrete nonoverlapping generations and well-defined growth and dispersal stages. I examined the varied travelling waves that arise in some simple ecologically-interesting integro-difference equations. For a scalar equation with compensatory growth, I observed only simple travelling waves. For carefully chosen redistribution kernels, one may derive the speed and approximate the shape of the observed waveforms. A model with overcompensation exhibited flip bifurcations and travelling cycles in addition to simple travelling waves. Finally, a simple predator-prey system possessed periodic wave trains and a variety of travelling waves.

The goal of this investigation was to evaluate the role of the Y chromosome in spermatogenesis by a quantitative and qualitative analysis of spermatogenesis as it occurs in the absence of a significant portion of the Y chromosome, i.e., in XO,Sxr male mice. Although these mice have the testis-determining portion of the Y chromosome on their single X chromosome, they lack most of the Y chromosome. Since it was found that all sperm-specific structures were assembled in a normal spatial and temporal pattern in spermatids of XO,Sxr mice, the genes controlling these structures cannot be located on the Y chromosome outside of the Sxr region, and are more likely to be on autosomes or on the X chromosome. In spite of the assembly of the correct sperm-specific structures, spermatogenesis was not quantitatively normal in XO,Sxr mice and significantly reduced numbers of spermatids were found in the seminiferous tubules of these mice. Furthermore, two size classes of spermatids were found in the testes of XO,Sxr mice, normal and twice-normal size. These findings are suggestive of abnormalities of meiosis in XO,Sxr spermatocytes, which lack one of the two sex chromosomes, and may not implicate function of specific genes on the Y chromosome. Morphological abnormalities of spermatids, which were not unique to XO,Sxr mice, were observed and these may be due to either a defective testicular environment because of reduced numbers of germ cells or to the lack of critical Y chromosome-encoded products. Since pachytene spermatocytes of XO,Sxr mice exhibited a sex vesicle, it can be concluded that the assembly of this structure does not depend on the presence of either a complete Y chromosome or the pairing partner for the X chromosome.

Discrete-time growth-dispersal models readily exhibit diffusive instability. In some instances, this diffusive instability parallels that found in continuous-time reaction-diffusion equations. However, if a sufficiently eruptive prey is held in check by a predator, predator overdispersal may also lead to one or a series of diffusion-driven period-doubling bifurcations. Quite common discrete-time predator-prey models exhibit this new brand of diffusive instability.

The incidence of measles in New York City and Baltimore was studied using recently developed techniques in nonlinear dynamics. The data, monthly case reports for the years 1928-1963, suggest almost two dimensional, chaotic flows whose essential attributes are captured by one dimensional, unimodal maps. The effects of noise, inevitable in ecological and epidemiological systems are discussed.

The ability of morphologically abnormal mouse sperm to bind to the zona pellucida of the egg was examined with techniques of fertilization in vitro. After incubation with sperm, cumulus-free ova were scored by either phase microscopy or scanning electron microscopy for the number and type of sperm bound. The percentages of abnormal sperm bound to zonae were compared to the percentages of abnormal sperm in the inseminating suspension. In general, all abnormal classes (except broadly spatulate sperm) bound to zonae at a frequency significantly lower than their representation in the inseminating suspension. However, when the percentage of abnormal sperm was quite high, no significant difference existed between frequencies of abnormal sperm bound and in the inseminating solution. The percentage of abnormal sperm bound did not increase significantly over time. Scanning electron microscopy studies demonstrated that the association of abnormal sperm with the zona pellucida varied according to sperm morphology. Normal and some abnormal sperm bound at an angle perpendicular to the zona, while more grossly abnormal sperm bound to ova tangentially.