Frank C. Hoppensteadt

Conditions for sympatric speciation: A diploid model incorporating habitat fidelity and non-habitat assortative mating

SummaryThree types of genes have been proposed to promote sympatric speciation: habitat preference genes, assortative mating genes and habitat-based fitness genes. Previous computer models have analysed these genes separately or in pairs. In this paper we describe a multilocus model in which genes of all three types are considered simultaneously. Our computer simulations show that speciation occurs in complete sympatry under a broad range of conditions. The process includes an initial diversification phase during which a slight amount of divergence occurs, a quasi-equilibrium phase of stasis during which little or no detectable divergence occurs and a completion phase during which divergence is dramatic and gene flow between diverging habitat morphs is rapidly eliminated. Habitat preference genes and habitat-specific fitness genes become associated when assortative mating occurs due to habitat preference, but interbreeding between individuals adapted to different habitats occurs unless habitat preference is almost error free. However, ‘nonhabitat assortative mating’, when coupled with habitat preference can eliminate this interbreeding. Even when several loci contribute to the probability of expression of non-habitat assortative mating and the contributions of individual loci are small, gene flow between diverging portions of the population can terminate within less than 1000 generations.

THEORETICAL ANALYSIS OF DIVERGENCE IN MEAN FITNESS BETWEEN INITIALLY IDENTICAL POPULATIONS

Initially identical populations in identical environments may subsequently diverge from one another not only via the effects of genetic drift on neutral alleles, but also by selection on beneficial alleles that arise stochastically by mutation. In the simple case of one locus with two alleles in a haploid organism, a full range of combinations of population sizes, selection pressures, mutation rates and fixation probabilities reveals two qualitatively distinct dynamics of divergence among such initially identical populations. We define a non-dimensional parameter k that describes conditions for the occurrence of these different dynamics. One dynamic (k > 1) occurs when beneficial mutations are sufficiently common that substitutions within the populations are essentially simultaneous; the other dynamic (k < 1) occurs when beneficial mutations are so rare that substitutions are likely to occur as isolated events. If there are more than two alleles, or multiple loci, divergence among the populations can be sustained indefinitely if k < 1. The parameter k pertains to the nature of biological evolution and its tendency to be gradual or punctuated.

Analysis and computer simulation of accretion patterns in bacterial cultures

Patterned growth of bacteria created by interactions between the cells and moving gradients of nutrients and chemical buffers is observed frequently in laboratory experiments on agar pour plates. This has been investigated by several microbiologists and mathematicians usually focusing on some hysteretic mechanism, such as dependence of cell uptake kinetics on pH. We show here that a simpler mechanism, one based on cell torpor, can explain patterned growth. In particular, we suppose that the cell population comprises two subpopulations —one actively growing and the other inactive. Cells can switch between the two populations depending on the quality of their environment (nutrient availability, pH, etc.) We formulate here a model of this system, derive and analyze numerical schemes for solving it, and present several computer simulations of the system that illustrate various patterns formed. These compare favorably with observed experiments.

Oscillations of Linear Systems

A linear system of ordinary differential equations has the form n n

Control of Cell Volume and the Electrical Properties of Cell Membranes

dx/dt = Aleft( t right)x + fleft( t right)

Response of a solutivory chain to a nutrient pulse

n nGiven an N-dimensional vector f and an N × N-dimensional matrix A(t) of functions of t, we seek a solution vector x(t). We write x, f ∈ E N and A ∈ E N × N and sometimes x′ = dx/dt or ẋ = dx/dt.

Quasistatic-State Methods

Regular perturbation methods are based on Taylor’s formula and on implicit function theorems. However, there are many problems to which Taylor’s formula cannot be applied directly, in which case perturbation methods based on multiple time or space scales can often be used, sometimes even for chaotic systems.

Algebraic and Topological Aspects of Nonlinear Oscillations

Cell contain proteins, nucleic acids, and other macromolecules that often carry many negative charges per molecule. This electrical charge is balanced by positive ions (especially potassium, denoted by K +) that are dissolved in the intracellular water. These ions tend to draw water into the cells by osmosis, and the cells would swell and eventually burst if this osmotic effect were not offset by other factors.