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United States of America (Duluth) 16

THEORETICAL POPULATION BIOLOGY

OCTOBER 1992 - VOLUME 42, NUMBER 2

93.16.01 - English - Marcus W. FELDMAN, Department of Biological Sciences, Stanford University, Stanford, California 94305 (U.S.A.), and Kenichi AOKI, Department of Anthropology, Faculty of Science, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113 (Japan)

Assortative Mating and Grandparental Transmission Facilitate the Persistence of a Sign Language (p. 107-116)

Conditions for the persistence (i.e., protection from loss) of a sign language are investigated assuming monogenic recessive inheritance of deafness, assortative mating for deafness or for hearing, and cultural transmission of the sign language to deaf individuals from their deaf parents and deaf maternal grandparents. A new method is introduced to deal with the problem of grandparental transmission in which the basic variables are the frequencies of triplets comprising a mother, a father, and their daughter of permissible phenogenotypes. Usual stability analysis is then done on the system of linear recursions in the frequencies of these triplets, derived on the assumption that signers (users of the sign language) are rare. It is shown that assortative mating is the most important factor contributing to persistence, but that grandparental transmission can also have a significant effect when assortment is as strong as observed in England and the United States. (BIOLOGY, THEORY, MODELS)

93.16.02 - English - Jan EKMAN, Department of Zoology, University of Stockholm, S-10691 Stockholm (Sweden), and Björn ROSANDER, Centre of Applied Mathematics, University of Gothenburg, S-41296 Gothenburg (Sweden)

Survival Enhancement through Food Sharing: A Means for Parental Control of Natal Dispersal (p. 117-129)

The value of food sharing among relatives is analyzed for a situation where fitness equals survival. In seasonal environments the minimum food abundance may set a limit to group living. Delayed dispersal is predicted to be linked to relaxed winter competition and high parental survival. Enhanced survival for the offspring when parents share food could be a sufficient reason to delay dispersal, while early dispersal in advance of food shortage periods may be induced by a competitive relationship. At low resource abundance dominant parents do best by being competitive and retaining all resources. For food abundance higher than the expected requirements food sharing with independent offspring is possible, although it has a non-zero cost. Food sharing parents still retain most of resources to themselves, but the resource share given to subordinate offspring gradually gets larger when food abundance increases. Except for at very low food abundance, where subordinates may adopt a "suicidal" behaviour and cede their resources to the dominant, there is a conflict over how to share the resources. (BIOLOGY, THEORY, MODELS)

93.16.03 - English - David GREENHALGH, Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH (U.K.)

Some Threshold and Stability Results for Epidemic Models with a Density-Dependent Death Rate (p. 130-151)

Most classical models for infectious diseases assume that the birth and death rates of individuals and the meeting rates between susceptible and infected individuals do not depend on the total number of individuals in the population. While these assumptions are valid in some situations they are less valid in others. For example, for diseases in animal an insects populations competition for scarce resources might well mean that the death rate depends on the number of individuals. The present paper examines two epidemic models where the death rate is density dependent. For each model the possible equilibrium levels of disease incidence are determined and the stability of these equilibrium levels to small perturbations is discussed. The biological interpretation of these results is presented together with the results of some numerical simulations. (BIOLOGY, THEORY, MODELS)

93.16.04 - English - Wolfgang EBENHÖH, Fachbereich 6, Mathematik, Universität Oldenburg, Postfach 25 03, D-2900 Oldenburg (Germany)

Temporal Organization in a Multi-species Model (p. 152-171)

A multi-species model with a periodically changing environment and strict resource limitation is presented. The species differ in length and phase of their growth intervals. Although there is only one limiting resource, a very large number of species coexist. For almost any set of permanently coexisting species, invasion of additional species is possible, which either exclude others or not. The key to the high diversity in this model is temporal organization, which interferes with an approximate trade-off condition between the length of the growth interval and the growth rate for coexisting species. In some cases temporal organized species cooperate indirectly, and support each other. (BIOLOGY, THEORY, MODELS)

93.16.05 - English - James S. CLARK, Department of Botany, University of Georgia, Athens, Georgia 30602 (U.S.A.)

Density-independent Mortality, Density Compensation, Gap Formation, and Self-thinning in Plant Populations (p. 172-198)

The timing and relative contributions of mortality agents in plant populations depend on the ways in which density-dependent (DD) mortality compensates for density-independent (DI) mortality rate. This relationship, in turn, determines the degree of canopy closure, or its complement, gap area. A model is presented that integrates the effects of DD and DI mortality agents, and it permits exploration of the "density compensation" (DC) effect, or the degree to which mortality classed by DI factors is alleviated by reductions in DD mortality, and changing gap area through time. DI mortality factors can be additive, and they affect the entire population early and late in life. If DI mortality is low, crowding results in a DD mortality rate that is rather constant when plants are growing rapidly, and DD mortality descreases as plants approach maximum size. DD mortality rate compensates for DI mortality rate in populations of plants that are young and at high density: Although increased DI mortality contributes to increasing gap area as plants mature, such an increase is not necessary for gap area to increase. Reduced density compensation with age (declining plant growth rate) results in increased gap area even if per-capita mortality rates descrease. Complete canopy coverage shifts form a stable to a neutrally stable equilibrium with advancing stand age. Net ecosystem production (NEP) is maximized earliest during stand development at somewhat less than complete canopy coverage. This maximum occurs later in stands having lower canopy caverage, because of low DD mortality rate, and in stands having higher canopy coverage, because of high density. Low canopy coverage early in stand development will be associated with delayed maximum NEP if that low canopy coverage is associated with low density and with early maximum NEP if low canopy coverage results instead from high DI mortality rate. The self-thinning rule arises as a special case of a more general relationship that demonstrates the effects of canopy and DI mortality on the percent change in density that attends a percent change in standing crop. Because of high DC early in stand development, the tendency of this slope coefficient to become more negative with declining canopy coverage prevails over the tendency of DI mortality to make this coefficient less negative. With declining DC late in stand development the situation is reserved. (BIOLOGY, THEORY, MODELS)

93.16.06 - English - Thomas F. HANSEN, Division of Zoology, Department of Biology, University of Oslo, P.O. Box 1050, Blindern, N-0316 Oslo 3 (Norway)

Evolution of Stability Parameters in Single-species Population Models: Stability or Chaos? (p. 199-217)

Several difference equations commonly used to model density-dependent population growth of a single population in a constant environment show a period doubling bifurcation scenario from stability to chaos as one or more parameters are varied. Models of evolutionary change in such parameters are studied analytically. Whether the population will evolve towards stability or towards chaos is model dependent. This is caused by model-dependent differences in the coupling of stability evolution to selection for increased carrying capacity, and differences in whether the parameters act predominantly at high or at low densities. Generally, stability is favored by selection at high densities while unstability is favored at low densities. As the former selection pressure tends to be stronger, it is likely that a deterministically fluctuating population with constant carrying capacity will evolve towards stability. (BIOLOGY, THEORY, MODELS)

93.16.07 - English - Ronald H. KARLSON, Ecology Program, and Howard M. TAYLOR, Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716 (U.S.A.)

Mixed Dispersal Strategies and Clonal Spreading of Risk: Predictions from a Branching Process Model (p. 218-233)

We present a branching process model of dimorphic dispersal and growth of a clonal lineage in order to predict the optimal proportion of dispersed offspring. This stochastic model incorporates the notions of intraspecific variation in dispersal capabilities and the statistical independence of dispersed offspring, but not intraspecific competition for sites. We show how some dispersal may be advantageous, even when the survivorship of dispersed offspring is substantially less than that of nondispersed offspring. Using a procedure which minimizes the extinction probability for the lineage (hence maximizing survivorship of the genetic individual), we predict that a high proportion of dispersed offspring should be associated with high numbers of asexual offspring, a low risk of mortality to offspring during dispersal, and high local survivorship of offspring. We give supportive evidence for the First prediction using data from the freshwater bryozoan Plumatella repens and suggest that future theoretical work on dispersal dimorphisms might profitably use a probabilistic approach such as the one presented here. (BIOLOGY, THEORY, MODELS)

DECEMBER 1992 - VOLUME 42, NUMBER 3

93.16.08 - English - G.D. RUXTON, W.S.C. GURNEY, Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH (Scotland), and A.M. DE ROOS, Department of Pure and Applied Ecology, University of Amsterdam, Kruislaan 302, 1090 SM Amsterdam (Netherlands)

Interference and Generation Cycles (p. 235-253)

In this paper we re-examine the derivation of all interference limited functional response due to Beddington (1975 J. Anim. Ecol. 44, 331-340) and extend his treatment to more realistic models of the interference process. We study the dynamic effects of interference in the context of a structured population model and show that the stabilising effect of interference against paradox of enrichment cycles is unaffected by age-structure. We also demonstrate that single generation cycles are much more weakly affected by interference than prey-escape cycles. Thus the net effect of weak interference is to prevent single generation cycles from being masked by the prey-escape cycles which would otherwise dominate the population dynamics. (BIOLOGY, THEORY, MODELS)

93.16.09 - English - Lev A. ZHIVOTOVSKY and Sergey GAVRILETS, Institute of General Genetics, Russian Adademy of Sciences, 3 Gubkin Street, GSP-1, B-333, Moscow 117809 (U.S.S.R.)

Quantitative Variability and Multilocus Polymorphism under Epistatic Selection (p. 254-283)

We study multilocus polymorphism under selection, using a class of fitness function that account for additive, dominant, and pairwise additive-by-additive epistatic interactions. The dynamic aquations are derived in terms of allele frequencies and disequilibria, using the notions of marginal systems and marginal fitness, without any approximations. Stationary values of allele frequencies and pairwise disequilibria under weak selection and calculated by regular perturbation techniques. We derive conditions for existence and stability of the multilocus polymorphic states. Using these results, we then analyze a number of models describing stabilizing selection on additive characters, with some other factors, and determine the conditions under which genetic quantitative variability is maintained. (BIOLOGY, THEORY, MODELS)

93.16.10 - English - Sabin LESSARD, Département de Mathematiques et de Statistique, Université de Montréal, C.P. 6128, Suc. "A", Montréal, QC H3C 3J7 (Canada)

Relatedness and Inclusive Fitness with Inbreeding (p. 284-307)

Relatedness arising in kin selection theory is measured by a variable taking as values two pedigree indices in populations with inbreeding when selection is weak. This variable reduces to a single pedigree index when inbreeding is caused by partial selfing or partial sib-mating. General inclusive fitness formulations of kin selection models based on such a variable of relatedness are proposed. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.11 - English - Marc MANGEL, Section of Zoology and Center for Population Biology, University of California, Davis, California 95616 (U.S.A.), and Bernard D. ROITBERG, Department of Biological Sciences, Simon Fraser University, Burnaby, British Columbia V5A 1S6 (Canada)

Behavioral Stabilization of Host-parasite Population Dynamics (p. 308-320)

We demonstrate that individual behavior can stabilize classical (Nicholson-Bailey) host-parasite population dynamics. Our model assumes that hosts can be divided into at least two phenotypes and that parasites either do not attact one of the phenotypes or attack them facultatively. The former case corresponds to a behavioral refuge (Hassell, 1978) and it is known that other kinds of refuges lead to stability of population dynamics. Behavioral refuges can stabilize the population dynamics in the same way that spatial refuges do. When parasites attack hosts facultatively within the year, strange attractors may arise in the year-to-year population dynamics, in response to the nonlinear nature of the facultative response to the distribution of host densities. (BIOLOGY, THEORY, MODELS)

93.16.12 - English - Franco SPIRITO, Department of Genetics and Molecular Biology, Faculty of M.F.N. Sciences, "La Sapienza" University, 00185 Rome (Italy)

Conditions for the Persistence of Partial Zygotic Reproductive Isolation in an Island-continent Model (p. 321-332)

A theoretical study has been made of the contact between two populations (island-continent model) which are partially isolated insofar as they are monomorphic for different alleles at one or more strictly underdominant loci with multiplicative heterozygote fitnesses. This genetic model of an isolating mechanism is the simplest model of zygotic isolation and is also a very good representation of underdominant chromosomal rearrangements. The particular problem tackled is how the threshold value of the migration rate (mc), below which differentiation at the underdominant loci can persist, varies as a function of the number of underdominant loci involved (n), for a given level of selection against hybrids (Ó). It has been found that the stability of genetic differentiation causing a given level Ó of reproductive isolation decreases as the number of loci involved increases. This effect is particularly noticeable for low levels of isolation. These results have some relevance with regard to the question of whether the establishment of zygotic reproductive isolation between populations occurs more probably as a result of many mutations with small effects on fitness or of only at few mutations with significant effects. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.13 - English - W.J. EWENS, Department of Biology, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (U.S.A.)

An Optimizing Principle of Natural Selection in Evolutionary Population Genetics (p. 333-346)

This paper brings together two themes in evolutionary population genetics theory. The first concerns Fisher's Fundamental Theorem of Natural Selection: a recent interpretation of this theorem claims that it is an exact result, relating to the so-called "partial" increase in mean fitness. The second theme concerns the desire to find an optimality principle in genetic evolution. Such a principle is found here: of all gene frequency changes which lead to the same partial increase in mean fitness as the natural selection gene frequency changes, the natural selection values minimize a generalized distance measure between parent and daughter generation gene frequency values. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.14 - English - J.M. CUSHING, Department of Mathematics, Interdisciplinary Program in Applied Mathematics, Building 89, University of Arizona, Tucson, Arizona 85721 (U.S.A.)

A Size-structured Model for Cannibalism (p. 347-361)

A size-structured model for the dynamics of a cannibalistic population is derived under the assumption that cannibals (successfully) attack only smaller bodied victims, as is generally the case in the biological world. In addition to the resulting size-dependent death rate, the model incorporates the positive feedback mechanism resulting from the added resource energy obtained by the cannibal from the consumption of the victim. From the nonlinear partial integro-differential equation model, it is shown how to obtain a complete analysis of the global dynamics of the total population biomass. This analysis yields many dynamical features that have been attributed to cannibalism in the literature, including density self-regulation, a "life-boat strategy" phenomenon by which a population avoids extinction by practising cannibalism under circumstances when it would otherwise go extinct, and multiple stable positive equilibrium states and hysteresis. (BIOLOGY, THEORY, MODELS)

FEBRUARY 1993 - VOLUME 43, NUMBER 1

93.16.15 - English - Mirjam KRETZSCHMAR, National Institute of Public Health and Environmental Protection, P.O. Box 1, 3720 BA Bilthoven (Netherlands), and Frederick R. ADLER, Department of Zoology, University of California, Davis, CA 95616 (U.S.A.)

Aggregated Distributions in Models for Patchy Populations (p. 1-30)

We investigate a model describing immigration, birth, and death of parasites on a dynamic host population. The model can also be interpreted as describing a herbivore population distributed on discrete patches of vegetation. We derive differential equations for the total number of hosts/patches and the mean number of parasites/herbivores per host/patch. The equations explicitly involve the variance-to-mean ratio of the distribution. It is shown that the positive equilibrium is stable if and only if the variance-to-mean ratio as a function of the mean increases with increasing mean. Thus aggregation of the parasites alone is not sufficient to stabilize the system; it is rather the density-dependent increase in parasite mortality due to a higher aggregation at higher mean parasite loads that causes stability. From this, it follows that introducing a distribution with a constant clumping parameter into the model artificially stabilizes the steady state. We derive a three-dimensional model based on an assumption about the form of the distribution of the parasites on the hosts, but without introducing additional parameters into the model. We compare stability results for this model for different types of aggregated distributions and show that the underlying distribution determines the qualitative results about the stability of the equilibrium. (BIOLOGY, THEORY, MODELS)

93.16.16 - English - G. DE LEO, Dipartimento di Biologia Evolutiva, Universita di Ferrara, Ferrara (Italy), L. DEL FURIA, Fondazione ENI Enrico Mattei, Milano (Italy), and M. GATTO, Dipartimento di Elettronica e Informazione, Politecnico di Milano, Milano (Italy)

The Interaction between Soil Acidity and Forest Dynamics: A Simple Model Exhibiting Catastrophic Behavior (p. 31-51)

Several hypotheses have been made to explain the forest decline due to acidic deposition. One of the most credited is the mobilization of toxic aluminium ions when soil pH falls below 4.2. A simple model is presented here that couples soil chemistry with tree biomass dynamics in order to investigate the influence of different proton loads on the existence, stability, and bifurcations of ecosystem equilibria. It is shown that, owing to the intrinsic nonlinear nature of the vegetation response to acid deposition, the equilibrium manifold can have, under certain conditions, the structure of a fold catastrophe. Increasing acidic load can thus drive the forest through a catastrophic transition from a viable equilibrium to extinction. Simulations using realistic ranges for model parameters and acidic input show that forests may indeed meet the conditions for a catastrophic collapse resulting from accumulation of acidic stress in the soil. (BIOLOGY, THEORY, MODELS)

93.16.17 - English - James W. ARCHIE, Department of Biology, California State University, Long Beach, California 90840 (U.S.A.), and Joseph FELSENSTEIN, Department of Genetics, University of Washington, Seattle, Washington 98195 (U.S.A.)

The Number of Evolutionary Steps on Random and Minimum Length Trees for Random Evolutionary Data (p. 52-79)

A model of evolutionarily uninformative data is derived and two separate character state distributions, one with two states (0,1) and one with missing-value data (0,1,{0,1}), are obtained. The expectation of number of steps on random trees is derived for both types of data and the variance in number of steps is derived for missing-value data. It is conjectured that the number of steps on random trees for these data should be asymptotically normal. Computer simulation is used to find approximations for the expected number and variance in number of steps of minimum length trees for both types of random evolutionary data. In both cases, minimum length trees are shorter than random trees and, although the lengths diverge in number, they converge in proportion. The length of minimum length trees is frequently used to evaluate the goodness of the tree as a representation of the data and the goodness of the data for constructing a plausible hypothesis of evolutionary relationships. In turn, as the length of a minimum length tree increases, confidence in the resulting tree decreases. The distributions of lengths of minimum length trees are of interest to determine if any information on phylogenetic branching is present in the data. The expected length of random trees can be used as a measure to compare to the lengths of minimum length trees for real data sets. (BIOLOGY, THEORY, MODELS)

93.16.18 - English - Steven N. EVANS, M.S. McPEEK and T.P. SPEED, Department of Statistics, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94720 (U.S.A.)

A Characterisation of Crossover Models That Possess Map Functions (p. 80-90)

This paper concerns genetic map functions and a particular probability model for crossovers. S, Karlin and U. Liberman (1979, Adv. Appl. Prob. 11, 479-501) and N. Risch and K. Lange (1979, Biometrics 39, 949-963) independently introduced what the former called the count-location (point) process model for crossovers, which leads to a probability model for multilocus recombination under the assumption of no chromatid interference. U. Liberman and S. Karlin (1984, Theor. Popul. Biol. 25, 331-346) later explored the constraints on genetic map functions resulting from the requirement that they be realisable in terms of a probability model for multilocus recombination. In this note we prove that under the assumption of no chromatid interference, the class of probability models for multilocus recombination that possess map functions is precisely the class of count-location processes. As a consequence, we give a complete analytic characterisation of the functions that can arise as map functions for some probability model of multilocus recombination under the assumption of no chromatid interference. (BIOLOGY, THEORY, MODELS)

93.16.19 - English - W.G. WILSON, Department of Chemistry, E. McCAULEY, Department of Biological Sciences, University of Calgary, Calgary, Alberta T2N 1N4 (Canada), and A.M. DE ROOS, Department of Pure and Applied Ecology, University of Amsterdam, Kruislaan 318, 1098 SM Amsterdam (Netherlands)

Spatial Instabilities within the Diffusive Lotka-Volterra System: Individual-based Simulation Results (p. 91-127)

A predator-prey system is studied via an individual-based simulation technique involving discrete Lotka-Volterra-type predator and prey individuals occupying a two-dimensional lattice of up to 256 sites by 256 sites, encompassing up to 65,536 predators and 65,536 prey. Spatial instabilities are found that break the system into "asynchronous regions" that can stabilize the "global" populations. These spatial heterogeneities are determined to be the result of discretizing space, time, and the population. Agreement is found with analytic results for the non-spatial Lotka-Volterra model and the spatial Lotka-Volterra model with diffusion when the discretizations are sealed to zero. It is argued from an individual-based modelling perspective, however, that this limiting procedure is biologically untenable. The conclusion is that under an individual-based model formulation of the Lotka-Volterra system, spatially heterogeneous population distributions are allowed. The specific form of these spatial distributions are shown to be strongly dependent on the prey diffusion rate and the specifies of implementing individual stochasticity. (BIOLOGY, THEORY, MODELS)

APRIL 1993 - VOLUME 43, NUMBER 2

93.16.20 - English - R.B. CAMPBELL, Department of Mathematics, University of Northern Iowa, Cedar Fallsl, Iowa 50614-0441 (U.S.A.)

The Importance of Mating Structure versus Progeny Distribution for Genetic Identity under Mutation (p. 129-140)

The relative importance of mating structure versus the distribution of progeny number for the genetic identity in a population is investigated. Both homozygosity and identity between individuals in a popu]ation at equilibrium with mutation are used as measures. Several regular systems of inbreeding, as well as random mating, are contrasted subject to the constraint of exactly two progeny per individual. Conversely, different distributions of progeny number are contrasted under both random mating and half-sib mating. Both the mating system and the progeny distribution are significant determinants of genetic identity, but sometimes these factors are interrelated so that causality cannot be assigned to either factor. When the mating structure or progeny distribution is changed, identity within and between individuals may both increase (or both decrease), but they may also change in opposite directions. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.21 - English - M.A. LEWIS, Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 (U.S.A.), and P. KAREIVA, Department of Applied Mathematics, University of Washington, Seattle, Washington 98195 (U.S.A.)

Allee Dynamics and the Spread of Invading Organisms (p. 141-158)

We examine how an Allee effect in local population dynamics (reduced reproductive success at low densities) influences the spatio-temporal dynamics of ecological invasions. Our approach is to use a partial differential equation model of dispersal and population growth, and then ask whether we can identify "rates of spread" for an invading organism subject to an Allee effect. Results indicate that an Allee effect may substantially reduce the rate at which the invader moves into a new environment. Analysis of spread in two spatial dimensions entails application of a singular perturbation theory approach. Here the two-dimensional spread velocity is given in terms of the one-dimensional asymptotic spread rate and the curvature of a boundary between invaded and non-invaded regions. Using this result, we show that invasions cannot propagate unless they initially exceed a critical area. This prediction is verified by numerically solving the original model. Numerical solutions are used throughout in demonstrating the nature of the two-dimensional spread. (BIOLOGY, THEORY, MODELS)

93.16.22 - English - Franceso DOVERI, Laboratorio di Informatica Territoriale e Ambientale, Politecnico di Milano, Milan (Italy) et al.

Seasonality and Chaos in a Plankton-fish Model (p. 159-183)

The dynamics of a plankton-fish model comprising phosphorus, algae, zooplankton and young fish are analyzed for different values of average light intensity, phosphorus concentration in the inflow, and adult fish biomass. Light intensity and water temperature are periodically varied during the year, while the other parameters are fixed at realistic values. The analysis is carried out with a continuation method for the study of the bifurcations of periodically forced continuous-time nonlinear systems. The large number of bifurcations of different types indicates that the dynamics of the model can be very complex. In fact, multiplicity of attractors, catastrophic transitions, subharmonics of various periods, cascades of period doublings, and strange attractors arise for suitable values of the parameters. The results are in agreement with the most recent theories on food chain systems and periodically forced predator-prey systems. They also suggest that large year-to-year differences in food chain dynamics need not always be attributable to external factors like interannual weather variability. (BIOLOGY, THEORY, MODELS)

93.16.23 - English - Claudia PAHL-WOSTL, Swiss Federal Institute of Technology, Zürich (Switzerland)

The Influence of a Hierarchy in Time Scales on the Dynamics of, and the Coexistence within, Ensembles of Predator-prey Pairs (p. 184-216)

One approcah to developing general model concepts for ecological systems is to base them on the properties associated with the body weight of the component organisms. The dynamic characteristics of species may be described according to allometric relationships. In the model presented here an ensemble of predator-prey pairs, which share a common pool of an abiotic nutrient, are distributed along the body weight axis, introducing thus a hierarchy of different time scales. Model versions comprising a single predator-prey pair are characterized by a time-invariant steady state. As soon as further pairs are added the system becomes instable, exhibiting first periodic and the chaotic oscillations. In spite of the chaotic dynamics, certain statistical trends are independent of the initial conditions chosen. The variability in global system variables (nutrient concentration, metabolic activity) decreases with increasing number of pairs, revealing thus an increase in the efficiency of nutrient utilization. In the non-equilibrium situation an unlimited number of differently sized predator-prey pairs may coexist, the time-averaged biomass being evenly distributed among all pairs. Such observations contrast with expectations derived from the steady state. (BIOLOGY, THEORY, MODELS)

93.16.24 - English - Thomas NAGYLAKI, Department of Ecology, The University of Chicago, 1101 East 57th Street, Chicago, Illinois 60637 (U.S.A.), Philip T. KEENANT, Department of Mathematical Sciences, Rice University, Houston, Texas 77251 (U.S.A.), and Todd F. DUPONT, Departments of Computer Science and Mathematics, The University of Chicago, 1100 East 58th Street, Chicago, Illinois 60637 (U.S.A.)

The Influence of Spatial Inhomogeneities on Neutral Models of Geographical Variation. III. Migration across a Geographical Barrier (p. 217-249)

The equilibrium structure of the infinite, one-dimensional stepping-stone model with a geographical barrier is investigated in the diffusion approximation. The monoecious, diploid population is subdivided into an infinite linear array of equally large, panmictic colonies that exchange gametes symmetrically. Migration is reduced across the grographical barrier, but is otherwise uniform. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. The two dimensionless parameters in the theory are ß=4po¹2uVo and K, where po, u, Vo, and K represent the population density, mutation rate, variance of gametic dispersion per generation, and penetrability of the barrier, respectively. The characteristics length is ¹Vo/(2u). Relative to a homogeneous infinite habitat, the barrier raises the probability of identity if the two points of observation are on the same side and lowers it if they are on opposite sides. The former effect is moderate or small, but the latter is large unless transmission is high (K > > 1); genetic differentiation across the barrier is very strong for low transmission (K < < 1). For points of observation on the same side, the influence of the barrier is significant only if the proximal point is within a few characteristic lengths. Upper and lower bounds on the probability of identity are established, and approximations are derived for four cases: (i) low expected homozygosity (ß > > 1), (ii) high transmission, (iii) low transmission, and (iv) at least one point distant. (BIOLOGY, THEORY, MODELS)

JUNE 1993 - VOLUME 43, NUMBER 3

93.16.25 - English - Shripad TULJAPURKAR, Department of Biological Sciences, Stanford University, Stanford, California 94305 (U.S.A.), and Conrad ISTOCK, Department of Ecology and Evolution, University of Arizona, Tucson, Arizona 85721 (U.S.A.)

Environmental Uncertainty and Variable Diapause (p. 251-280)

We analyze a stage-structured model of a population that displays variable diapause in a randomly varying environment. The ruggedness of the environment is measured by the extent of random variation in per-capita reproductive success. We show how variable diapause and environmental characteristics affect the population's stochastic growth rate. In rugged unpredictable environments, phenotypes that show some tendency to diapause are found to have a higher growth rate than nondiapausing phenotypes. In harsh rugged environments, some tendency to diapause may be all that permits population persistence. Positive serial autocorrelation causes the optimal diapause fraction to decrease, while negative autocorrelation causes that fraction to increase. The structured model behaves very differently from a scalar model for large diapause fractions even in uncorrelated environments, and in many cases predicts a broad optimum. The difference between models is due to the extreme variability of stage structure in populations subject to even small variability when diapause tendency is high. (BIOLOGY, THEORY, MODELS)

93.16.26 - English - Sabin LESSARD, Département de Mathématique et de Statistique, Université de Montréal, C.P. 6128, Suc. "A", Montréal, QC H3C 3J7 (Canada)

Adaptive Topography in Fertility-viability Selection Models: An Alternative to Inclusive Fitness in Kin Selection Models (p. 281-309)

In this paper, we propose an alternative to inclusive fitness in kin selection models. The key point is to show that kin selection in family-structured models can be put into a context of fertility selection. First, we extend T. Nagylaki's (1987, Genetics 115, 367-375) result on approximate adaptive topographies in two-sex populations under weak selection by considering sex-differentiated fertilities of matings. The result is that the change in the geometric average and in the arithmetic average of the mean fertilities in male and female offspring is approximately equal to the additive genetic variance in fertility. Then we show that fertility-viability selection models and kin selection models with sex differences are special cases if appropriate fitness parameters are introduced and if genotype frequencies are considered at an appropriate time of the life cycle. Kin selection models with partial adoption of offspring giving rise to frequency-dependent selection are also studied. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.27 - English - Marino GATTO, Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Milano (Italy)

The Evolutionary Optimality of Oscillaltory and Chaotic Dynamics in Simple Population Models (p. 310-336)

The problem is considered of whether natural selection favors genotypes characterized by oscillatory or chaotic population dynamics. This is done with reference to two simple one-dimensional models, which display a variety of dynamical patterns according to the different values of their parameters: the semelparous and iteroparous Ricker models. To find the optimal genotype (or genotypes) within a given feasibility set, the concept of Continously Stable Strategy (CSS) and a haploid model of competition between genotypes are used. The parameters subject to evolution are the intrinsic finite rate of increase and respectively the juvenile mortality in the semelparous model and the adult survival in the iteroparous one. In the semelparous case a single feasible CSS exists, while in the other case more than one CSS might exist. The dynamical nature of the optimal genotype (stable equilibrium, stable sustained oscillations or chaos) is basically determined by the shape of the set of feasibility for the parameters defining each genotype. However, if the feasibility set is drawn at random, the probability that the corresponding optimal genotype (or genotypes) be oscillatory or chaotic is quite low. This result, however, might not hold with more complex models. (BIOLOGY, THEORY, MODELS)

93.16.28 - English - Marjan SJERPS, Patsy HACCOU, Evert MEELIS, Institute of Theoretical Biology, and Eddi VAN DER MEIJDEN, Department of Population Biology, University of Leiden, P.O. Box 9516, 2300 RA Leiden (Netherlands)

Egg Distribution within Patches: An Optimality Problem for Insects (p. 337-367)

A puzzling phenomenon in insect oviposition is the variety of egg distributions over plants. A clumped distribution can cause food competition among the offspring, and consequently risky migration to other plants. On the other hand, egg clumping can initially be advantageous. To examine under which conditions egg clumping is optimal, we studied the optimal distribution of eggs within a patch. A three-step model is proposed, representing hatching and initial growth of the larvae, migration and pupalation. Assuming that egg clumping is advantageous in the first step, but disadvantageous in the subsequent steps, we analyze the trade-off between these effects. Five factors are investigated to see if they can contribute towards egg clumping: (1) size of egg supply; (2) number of plants per patch; (3) number of patches; (4) competition between insects; (5) small stochastic fluctuations in patch quality. Our results show that (1), (2), and (3) can contribute towards egg clumping, but (5) cannot. As an example, the model is applied to the cinnabar moth. We show that only (2) and (3) contribute towards the clumped egg distribution of the moth. (BIOLOGY, THEORY, MODELS)

AUGUST 1993 - VOLUME 44, NUMBER 1

93.16.29 - English - Marc MANGEL, Center for Population Biology, University of California, Davis, California 95616 (U.S.A.), and Charles TIER, Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60680 (U.S.A.)

Dynamics of Metapopulations with Demographic Stochasticity and Environmental Catastrophes (p. 1-31)

In the first part of the paper, a method is developed for computing extinction properties of populations that are subject to demographic and environmental noise (catastrophes). The theory requires estimation of demographic birth and death rates, rates of catastrophes, and distribution of deaths when catastrophes occur. The colonization probability (chance of successful immigration), mean extinction time, and the long time conditional distribution of population size are predicted. The results can be put into algorithmic form so that workers can concentrate on developing parameters from empirical data. In the second part, the results are compared to the exact solution of a model (due to MacArthur and Wilson) without catastrophes and shown to be extremely accurate. The MacArthur-Wilson model is then extended to include environmental catastrophes. Finally, a metapopulation model with linear birth and death rates, immigration, catastrophes occurring at a rate independent of population size, and individuals dying independently is proposed. (BIOLOGY, THEORY, MODELS)

93.16.30 - English - M. KRETZSCHMAR, National Institute of Public Health & Environmental Protection, P.O. Box 1, 3720 BA Bilthoven (Netherlands), R.M. NISBET, Department of Biological Sciences, University of California, Santa Barbara, CA 93106 (U.S.A.), and E. McCAULEY, Ecology Division, Department of Biological Sciences, University of Calgary, Calgary, Alberta T2N 1N4 (Canada)

A Predator-prey Model for Zooplankton Grazing on Competing Algal Populations (p. 32-66)

The typical freshwater algal community can be roughly categorised into two groups according to the distribution of biomass with cell size. Algae in the small size group are known to be edible for zooplankton grazers; large algae are much less edible. To model population dynamics of a size selective zooplankton predator like Daphnia we thus consider a 3-species predator-prey model with an "edible" and an "inedible" prey. We show that a system which in the absence of inedible algae exhibits cyclic dynamics may exhibit a stable equilibrium if inedible algae are introduced. This stabilisation can be caused by one or both of the following mechanisms: (a) "indirect interaction" involving interspecific competition between the prey which reduces the effective carrying capacity of the edible species, and (b) "direct interaction" in which there is a reduction of the attack rate on the edible prey when the predator spends time handling inedibles. We study the behaviour of the system under enrichment. With only the indirect interaction, the equilibrium density of the edible prey is unaffected by enrichment, but the density of both inedible prey and predator increases. With sufficient enrichment the system can always be destabilised. If there is direct interaction, the equilibrium density of all three species increases with enrichment; the edible fraction decreases, but has a lower bound. If the direct interaction is sufficiently strong there are regions of parameter space for which the system can never be destabilised by enrichment. (BIOLOGY, THEORY, MODELS)

93.16.31 - English - Robert SCHOEN and Young J. KIM, Department of Population Dynamics, Johns Hopkins University, Baltimore, Maryland 21205 (U.S.A.)

Two-state Spatial Dynamics in the Absence of Age (p. 67-79)

This paper investigates the simplest multistate population model, a one-age-group, two-living-state model with constant rates of birth, death, and interstate movement. A general solution for the model is presented, and special attention is given to the process of convergence to stability and its relationship to spatial population momentum. The constant ultimate level of the speed of convergence, or the intrinsic force of convergence, is found to be twice the difference between the two eigenvalues of the model's transition matrix. Spatial population momentum is shown to be directly proportional to the initial growth rate of the population under the stationary rates and inversely proportional to the intrinsic force of convergence. A population increases in size during the transition to zero growth only when the faster growing state is a larger fraction of the initial population than of the stationary population. Hypothetical calculations with an urban/rural population model illustrate the importance of the speed of convergence, and show the great growth potential inherent in spatial momentum. (BIOLOGY, THEORY, MODELS)

93.16.32 - English - Franco SPIRITO, Marco RIZZONI and Carla ROSSI, Department of Genetics and Molecular Biology, Faculty of M.F.N. Sciences, "La Sapienza" University, 00185 Rome (Italy)

The Establishment of Underdominant Chromosomal Rearrangements in Multi-deme Systems with Local Extinction and Colonization (p. 80-94)

The fate of an underdominant chromosomal mutant was investigated in multideme models with high rates of local extinction and colonization. Four models with different patterns of colonization (number of colonists and place of origin of colonists) were studied by performing a large number of computer simulations with the Monte Carlo method for several sets of values of the following parameters: coefficient of selection against the heterozygote, extinction rate of each deme, deme size, and number of demes. The probability of the newly arisen rearrangement being established in the multi-deme system depends strongly on the pattern of colonization, other things being equal. In the three models in which there is absent or scarce mixing of gene pools of different demes when a new deme is founded, the fixation probability of the new chromosomal rearrangement is rather close to that calculated by R. Lande (1979, Evolution 33, 234-251; 1995, Heredity 54, 323-332), which is equal to the corresponding probability in a single deme divided by the number of demes. In the model with extensive mixing of gene pools of demes, the corresponding probability is lower (considerably in some cases). Furthermore, in the models where the fixation probability is higher, the analysis of the time of the process leads to the conclusion that in systems consisting of a large number of demes, overlapping of several different processes of fixation of chromosomal rearrangements occurs. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.33 - English - Norio YAMAMURA, Department of Natural Sciences, Saga Medical School, Nabeshima 5-1-1, Saga 849 (Japan)

Vertical Transmission and Evolution of Mutualism from Parasitism (p. 95-109)

Using an evolutionarily stable strategy (ESS) model, it is shown that vertical transmission, defined as the direct transfer of infection from a parent host to its progeny, is an important factor which can urge reduction of parasite virulence. Evolution of the vertical transmission rate from both points of view, the parasite and the host, is analyzed. There is a critical level on the rate, below which an evolutionary conflict arises (the parasite would want to increase the rate while the host would not), and above which both sides would correspond to increase the rate. Therefore, once the parasite dominates the evolutionary race so as to overcome the critical level, the one-way evolution begins toward a highly mutualistic relationship with a high vertical transmission rate. The changes in other parameters may decrease the critical level, initiating the one-way evolution. However, changes in traits, probably developed through long interrelationship in parasitism, do not necessarily induce the evolution of mutualism. Establishment of the ability to make use of metabolic and digestive wastes from the partner certainly facilitates the evolution of mutualism, while improvements in reproductive efficiency of the parasites and reduction of negative effects from exploitation in hosts contrarily disturb mutualism. Finally, available data are discussed in the light of predictions of the model. (BIOLOGY, THEORY, MODELS)

93.16.34 - English - Neil O'CONNEL and Montgomery SLATKIN, Department of Integrative Biology, University of California, Berkeley, CA 94720 (U.S.A.)

High Mutation Rate Loci in a Subdivided Population (p. 110-127)

Analytic and simulation studies were carried out in order to predict the average geographic area occupied by alleles in a continuously distributed population. The properties of three statistics were investigated: the sum of the squared distance between members of allelic classes, the sum of the root mean squared distances, and the sum of the squares of the numbers of alleles. The expectations of these quantities can be obtained analytically from both stepping-stone and branching diffusion models. The predictions of these two models are similar for wide ranges of parameter values and are consistent with the simulation results from a stepping-stone model. These results suggest that measures of the geographic distribution of alleles can be useful for estimating average dispersal distances at loci, such as minisatellite and microsatellite loci, at which mutation rates are high enough that they can be estimated with confidence. (BIOLOGY, THEORY, MODELS)

OCTOBER 1993 - VOLUME 44, NUMBER 2

93.16.35 - English - Kenneth LANGE, Department of Biomathematics, School of Medicine, University of California, Los Angeles, CA 90024 (U.S.A.)

A Stochastic Model for Genetic Linkage Equilibrium (p. 129-148)

Linkage equilibrium is an independence condition among the alleles at a set of gene loci. Equilibrium or disequilibrium only makes sense relative to some reference population of a species. If the loci all occur on the same chromosome, then linkage equilibrium holds provided a random representative of that chromosome from the reference population displays independent alleles at the various loci of the set. Classical deterministic population genetics theory shows that linkage equilibrium is approached asymptotically after many generations of random mating in a reference population of infinite size. The current paper considers a Markov chain model for the establishment of linkage equilibrium in a population of finite size. The states of this Markov chain correspond to counts of chromosomes of various types. Because the chain is reversible, the equilibrium distribution can be explicitly computed. Partial characterization of the geometric rate of convergence of the chain to equilibrium is possible using a strong stationary stopping time. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.36 - English - Thomas L. VINCENT, Aerospace and Mechanical Engineering, University of Arizona, Tucson, Arizona 85721 (U.S.A.), Yosef COHEN, Fisheries and Wildlife, University of Minnesota, St. Paul, Minnesota 55108 (U.S.A.), and Joel S. BROWN, Biological Sciences, University of Illinois at Chicago, Chicago, Illinois 60680 (U.S.A.)

Evolution via Strategy Dynamics (p. 149-176)

Consider a community of various species newly introduced into a stable environment. Evolutionary processes acting on this community will produce, over time, a community of surviving species. Methods for predicting the evolutionarily stable strategies (ESSs) used by the surviving species are now available for a large class of dynamic population models. Here we expand a previously developed evolutionary game theory, which can be used to predict ESSs in a large class of models, by introducing strategy dynamics. By so doing, a more complete description of the evolutionary process is obtained. One not only obtains a convenient way of determining evolutionarily stable strategies, but interesting features about the evolutionary process itself can be observed. Of particular interest here, we show that the number of strategies which are evolutionarily stable can change as certain environmental factors involved with the model change. The process by which the ESS is formed is examined in terms of an "adaptive landscape" formed by our fitness generating function (G-function). The G-function has properties that enhance the likelihood that the various adaptive peaks will be occupied. (BIOLOGY, THEORY, MODELS)

93.16.37 - English - A. GARCIA-DORADO, Departamento de Genetica, Universidad Complutense, Madrid (Spain)

Filling a Gap in the Prediction of the Equilibrium Genetic Variance (p. 177-188)

We derive analytical predictions for the variance and kurtosis of the equilibrium distribution of allelic effects under stabilizing selection and drift, and for any value of the kurtosis of the distribution of the mutational effects. Numerical results relative to the equilibrium genetic variance are compared with other analytical predictions and with simulation results from the literature. Our prediction is superior for relatively weak selection (or relatively low variance of the mutational effects) and not to small population sizes, where previous methods overestimate the equilibrium genetic variance. The behavior of the equilibrium kurtosis is also illustrated. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.38 - English - Kinya NISHIMURA, Institute of Biological Sciences, University of Tsukuba, Ibaraki 305 (Japan)

Local Mate Competition in a Stochastic Environment (p. 189-202)

The optimal sex ratio under local mate competition requires examination of stochastic factors. Numbers of male and female progeny were considered binomially distributed with the following parameters: the total number of progeny (T) and production rate of males (r, strategic parameter). The total number of progeny may be limited, and the progeny numbers may be differ among females by a stochastic factor. It was assumed that the total number of progeny (T) for all females follows an identical probability mass function. When the total number of progeny (T) is limited, the optimum production rate of males (r) is higher than that predicted by Hamilton's classical LMC model, which answers no stochasticity in the sex ratio. As the total number of progeny becomes large, the discrepancy between the stochastic model and Hamilton's model decays. Another stochastic factor also affects the optimum production rate of males (r). A female that colonizes in a group cannot always have precise information on the number of co-foundresses in her group. When the number of co-foundresses in the group is obscure, the optimal production rate of males (r) is lower than that for the exact information. (BIOLOGY, THEORY, MODELS)

93.16.39 - English - Eric FUNASAKI and Mark KOT, Department of Applied Mathematics, University of Washington, Seattle, Washington 98195 (U.S.A.)

Invasion and Chaos in a Periodically Pulsed Mass-action Chemostat (p. 203-224)

This paper contains a simple model for a chemostat with predator, prey, and periodically pulsed substrate. We obtain an exact periodic solution with positive concentrations of substrate and prey. A stability analysis for this solution yields an invasion threshold (the smallest value of the predator's predation constant consistent with invasion of the chemostat). Above this threshold, there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations despite the presence of severe statistical constraints on the behavior of the dependent variables. We determine the average substrate, bacterial, and protozoan concentrations for all periodic orbits, independent of the actual period. We conclude by comparing, the behavior of this system with that of two other model chemostats. (BIOLOGY, THEORY, MODELS)

93.16.40 - English - Uri LIBERMAN, Department of Statistics, Tel Aviv University, Tel Aviv (Israel), and Marcus W. FELDMAN, Department of Biological Sciences, Stanford University, Stanford, California 94305 (U.S.A.)

Recombination Modification with X-Linked Characters (p. 225-245)

In the present paper, we investigate two linked loci that are located on the X-chromosome in an X-Y sex determining system. There is a third locus, also on the X-chromosome, which controls the rate of recombination among the first two. Following Liberman and Feldman (1986), the problem is posed for an arbitrary number of alleles at the modifier locus. We show that there is a strong generalization of Nei's (1969) claim, and that the reduction principle holds for a sex-linked modifier of recombination between sex-linked genes. (BIOLOGY, GENETICS, THEORY, MODELS)

93.16.41 - English - David B. GOLDSTEIN, Aviv BERGMAN and Marcus W. FELDMAN, Department of Biological Sciences, Stanford University, Stanford, California 94305 (U.S.A.)

The Evolution of Interference: Reduction of Recombination among Three Loci (p. 246-259)

Crossover events along chromosomes do not occur independently, but influence the probably of other nearby events. The most common interaction between nearby crossover events is inhibitory: a crossover event tends to reduce the probability of other such events nearby, and this is called positive interference. A crossover event may increase the probability of events nearby, and this rare phenomenon is called negative interference. In this paper, we use numerical methods to investigate how interference among three loci would evolve if it were under the genetic control of a fourth, selectively neutral focus. We first discuss the effect of interference on the overall rate of recombination among the three loci, and then show that, under a variety of conditions, interference evolves in the same way is would be predicted based upon its effect on the overall rate of recombination. That is, the overall rate evolves in the same direction as would the rate at a locus that controls recombination between two loci directly. We then check for the existence of viability-analogous Hardy-Weinberg equilibria in the four-locus model of interference modification. (BIOLOGY, THEORY, MODELS)

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