The fate of balanced,phenotypic polymorphisms in fragmented metapopulations

In large populations, genetically distinct phenotypic morphs can be maintained in equilibrium (at a 1 : 1 ratio in the simplest case) by frequency‐dependent selection, as shown by Sewall Wright. The consequences of population fragmentation on this equilibrium are not widely appreciated. Here, I use a simple computational model to emphasize that severe fragmentation biases the morph ratio towards the homozygous recessive genotype through drift in very small populations favouring the more common recessive allele. This model generalizes those developed elsewhere for heterostylous plants and major histocompatibility complex alleles, emphasizes one particular outcome and avoids the restricting assumptions of more analytical models. There are important implications for both fundamental evolutionary biology and conservation genetics. I illustrate this with a range of examples but refer particularly to shell polymorphism in snails. These examples show how habitat fragmentation could have a direct and often unappreciated effect on species at the level of their population genetics.     

The distinct effects of habitat fragmentation on population size

We sought to understand how the separation of habitats into spatially isolated fragments influences the abundance of organisms. Using a simple, deterministic model of population growth, we compared analytically exact solutions predicting abundance of consumers in two isolated patches with abundance of consumers in a single large patch where the carrying capacity of the large patch is the sum of the carrying capacities of the isolated ones. For the deterministic model, the effect of fragmentation was to slow the rate of population growth in the fragmented habitat relative to the intact one. We also analyzed a stochastic version of the model to examine the effect of fragmentation on population abundance when resources vary randomly in time. For the stochastic model, the effect of fragmentation was to reduce population abundance. We proved in closed-form, that for a non-equilibrium population exhibiting logistic population growth, fragmentation will reduce population size even when the total carrying capacity is not affected by fragmentation. We provide a theoretical basis for the prediction that habitat fragmentation amplifies the effect of habitat loss on the abundance of mobile organisms.     

Habitat fragmentation and extinction thresholds on fractal landscapes

Habitat fragmentation is a potentially critical factor in determining population persistence. In this paper, we explore the effect of fragmentation when the fragmentation follows a fractal pattern. The habitat is divided into patches, each of which is suitable or unsuitable. Suitable patches are either occupied or unoccupied, and change state depending on rates of colonization and local extinction. We compare the behaviour of two models: a spatially implicit patch-occupancy (PO) model and a spatially explicit cellular automaton (CA) model. The PO model has two fixed points: extinction, and a stable equilibrium with a fixed proportion of occupied patches. Global extinction results when habitat destruction reduces the proportion of suitable patches below a critical threshold. The PO model successfully recreates the extinction patterns found in other models. We translated the PO model into a stochastic cellular automaton. Fractal arrangements of suitable and unsuitable patches were used to simulate habitat fragmentation. We found that: (i) a population on a fractal landscape can tolerate more habitat destruction than predicted by the patch-occupancy model, and (ii) the extinction threshold decreases as the fractal dimension of the landscape decreases. These effects cannot be seen in spatially implicit models. Landscape struc-ture plays a vital role in mediating the effects of habitat fragmentation on persistence.     

Linkage disequilibrium in a population undergoing periodic fragmentation and admixture

Glacial and interglacial cycles are considered to have caused the fragmentation and admixture of populations in many organisms. A simple model incorporating such periodic changes of the population structure is analysed in order to investigate the behaviour of neutral genetic variation at one and two loci. The equilibrium is reached very quickly in terms of cycles if the length of a cycle is long, as would be expected of the glaciation cycles. Heterozygosity and linkage disequilibrium are shown to depend on the length of time of the fragmented and admixed phases, population sizes, and number (n) of subpopulations in the fragmented phase. If the population size is small in the fragmented phase and its duration is long, the squared correlation coefficient of two loci (a measure of linkage disequilibrium) just after the admixture is approximated by 1/(n-1) for n > 1. After admixture, the correlation decays at a rate of approximately twice the recombination rate. Therefore, if post-glaciation admixture created linkage disequilibrium, we expect to observe linkage disequilibrium even between moderately linked loci, and its decay pattern along the chromosome is very different from that in a random mating population at equilibrium. This is especially true in organisms with long generation times such as trees.     

Allometric growth, disturbance regime, and dilemmas of controlling invasive plants: a model analysis

Disturbed communities are observed to be more susceptible to invasion by exotic species, suggesting that some attributes of the invaders may interact with disturbance regime to facilitate invasion success. Alternanthera philoxeroides, endemic to South America, is an amphibious clonal weed invading worldwide. It tends to colonize disturbed habitats such as riparian zones, floodplain wetlands and agricultural areas. We developed an analytical model to explore the interactive effects of two types of physical disturbances, shoot mowing and root fragmentation, on biomass production dynamics of A. philoxeroides. The model is based on two major biological assumptions: (1) allometric growth of root (belowground) vs. shoot (aboveground) biomass and (2) exponential regrowth of shoot biomass after mowing. The model analysis revealed that the interaction among allometric growth pattern, shoot mowing frequency and root fragmentation intensity might lead to diverse plant ‘fates’. For A. philoxeroides whose root allocation decreases with growing plant size, control by shoot mowing was faced with two dilemmas. (1) Shoot regrowth can be effectively suppressed by frequent mowing. However, frequent shoot mowing led to higher biomass allocation to thick storage roots, which enhanced the potential for faster future plant growth. (2) In the context of periodic shoot mowing, individual shoot biomass converged to a stable equilibrium value which was independent of the root fragmentation intensity. However, root fragmentation resulted in higher equilibrium population shoot biomass and higher frequency of shoot mowing required for effective control. In conclusion, the interaction between allometric growth and physical disturbances may partially account for the successful invasion of A. philoxeroides; improper mechanical control practices could function as disturbances and result in exacerbated invasion.     

Tests of the simple model of Lin and Brandts for the folding kinetics of ribonuclease A

L.-N. Lin and J.F. Brandts recently proposed a simple model for the folding kinetics of ribonuclease A in which folding intermediates are not detectable. We have tested the basic assumption of the simple model for the major unfolded species, which is produced by a slow isomerization (the "X in equilibrium Y reaction" according to Lin and Brandts) after unfolding. The simple model assumes that in refolding the slow Y----X reaction must occur before any folding can take place. We have measured the Y----X reaction during folding. Tyrosine-detected folding occurs before the Y----X reaction; the difference in rate between the Y----X reaction and folding monitored by tyrosine absorbance becomes large when the stabilizing salt 0.56 M (NH4)2SO4 is added. The simple model predicts that the kinetic properties of the X in equilibrium Y reaction in unfolded ribonuclease are the same as those of tyrosine-detected folding. We find, however, that the kinetics of the X in equilibrium Y reaction in unfolded ribonuclease are independent of urea concentration, whereas the rate of tyrosine-detected folding decreases almost 100-fold between 0.3 and 5 M urea, as reported by Lin and Brandts. We point out that the kinetic properties of the X in equilibrium Y reaction in unfolded ribonuclease are characteristic of proline isomerization.     

Histone-DNA binding free energy cannot be measured in dilution-driven dissociation experiments

Despite decades of study on nucleosomes, there has been no experimental determination of the free energy of association between histones and DNA. Instead, only the relative free energy of association of the histone octamer for differing DNA sequences has been available. Recently, a method was developed based on quantitative analysis of nucleosome dissociation in dilution experiments that provides a simple practical measure of nucleosome stability. Solution conditions were found in which nucleosome dissociation driven by dilution fit well to a simple model involving a noncooperative nucleosome assembly/disassembly equilibrium, suggesting that this approach might allow absolute equilibrium affinity of the histone octamer for DNA to be measured. Here, we show that the nucleosome assembly/disassembly process is not strictly reversible in these solution conditions, implying that equilibrium affinities cannot be obtained from these measurements. Increases in [NaCl] or temperature, commonly employed to suppress kinetic bottlenecks in nucleosome assembly, lead to cooperative behavior that cannot be interpreted with the simple assembly/disassembly equilibrium model. We conclude that the dilution experiments provide useful measures of kinetic but not equilibrium stability. Kinetic stability is of practical importance: it may govern nucleosome function in vivo, and it may (but need not) parallel absolute thermodynamic stability.     

Resource limited and equilibrium models of populations

Summary The logistic equation has been used as the basis of two distinct models of population dynamics, a resource limited model and an equilibrium model. It is argued that although the two models are very different, the distinction between them has been obscured by the use of the same terminology for the parameters of both. It is further argued that the simple equilibrium model developed from the logistic equation has inconsistencies which make it untenable as a population model.     

Spatial scale mediates the influence of habitat fragmentation on dispersal success: Implications for conservation

Increasingly, conservationists are seeking insights from ecological theory to choose strategies of habitat management that will best maintain threatened species. Often, these questions revolve around ways of mitigating the dangers posed by habitat fragmentation. Problems involving the scale of both animal movement and spatial heterogeneity inexorably arise when assessing the effects of fragmentation. We present results from a simple spatial model that simulates the dispersal of animals in a landscape of stochastically clustered habitat fragments. Varying the number of clusters and the spatial scale at which clustering occurs illustrates that heterogeneity has different and conflicting effects on animal movement when it occurs at different scales. Indeed, the scale of clustering is the most important feature in determining disperser performance in our model. Seeking to compare our modeling results with actual data, we review empirical studies of fragmented populations and habitats. Surprisingly, we conclude that very few studies have addressed the mechanisms by which fragmentation will influence population dynamics or, in particular, the ways in which spatial scale mediate these effects. We conclude that the explicit consideration of scale is essential in discussions of habitat fragmentation and of optimal conservation strategies.     

The Effect of Habitat Fragmentation on Cyclic Population Dynamics: A Numerical Study

Through four spatially explicit models, we investigate how habitat fragmentation affects cyclic predator–prey population dynamics. We use a Partial Differential Equation (PDE) framework to describe the dispersal of predators and prey in a heterogeneous landscape made of high quality and low quality habitat patches, subject to increasing fragmentation through habitat separation and/or habitat loss. Our results show that habitat fragmentation decreases the amplitude of the predator–prey population cycles while average population density is not as strongly affected in general. Beyond these simple trends however, the four models show differing responses to fragmentation, indicating that when making predictions about population survival and persistence in the face of habitat fragmentation, the choice of model is important. Our results may inform conservation efforts in fragmented habitats for cyclic species such as the snowshoe hare and Canada lynx. Electronic Supplementary Material  The online version of this article () contains supplementary material, which is available to authorised users.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

Determination of dependence of binding parameters on receptor occupancy

Measurements of the binding of ligand to receptors that are macromolecules, either free or components of biomembranes, often show deviation from what is expected of a simple reaction described by an association and a dissociation rate constant. A more versatile model and more discriminating experiments are required for a satisfactory explanation. This paper is based on a general model of the binding reaction in which the rate constants and equilibrium constant are dependent upon occupancy of receptors. The analysis of the model leads to three kinds of experiments: (1) equilibrium measurements which permit quantitative determination of a dissociation equilibrium parameter as a function of receptor occupancy; (2) measurements prior to equilibrium which yield the same information; and (3) measurements prior to equilibrium which reveal quantitatively the dependence of both association and dissociation rate parameters separately, on occupancy.     

The time course of cellular responses to iontophoretically applied drugs.

Simple models of two-species ecosystems are usually analyzed in terms of the existence and stability of a static equilibrium state. We examine the way in which perturbations, in the form of periodic reductions in both species, lead to stable coexistence in a state of dynamic equilibrium. We establish general criteria for the occurrence of such dynamic equilibrium states. We show that coexistence in a dynamic equilibrium occurs for a fairly wide range of model parameters, and that dynamic equilibrium states are a rather robust feature of simple models.     

Heterozygosity in pin-thrum plants or with partial sex linkage

A two locus model is constructed for selection of a gene closely linked to the S locus in pin-thrum plants or to the sex determining part of the Y chromosome. Using this model, conditions for stability at the equilibrium point which is predicted by one-locus theory when there is heterozygotic superiority are derived. If the recombination value is small, it is found that this equilibrium point is unstable and that the gene frequencies go to a new stable equilibrium point at which the population has a higher average fitness. A few simple cases of selection and the implication of these to the theory of the evolution of the Y chromosome are discussed.     

Modelling and analysing evolution of dispersal in populations at expanding range boundaries

Abstract 1. Species would be expected to shift northwards in response to current climate warming, but many are failing to do so because of fragmentation of breeding habitats. Dispersal is important for colonisation and an individual‐based spatially explicit model was developed to investigate impacts of habitat availability on the evolution of dispersal in expanding populations. Model output was compared with field data from the speckled wood butterfly Pararge aegeria, which currently is expanding its range in Britain. 2. During range expansion, models simulated positive linear relationships between dispersal and distance from the seed location. This pattern was observed regardless of quantity (100% to 10% habitat availability) or distribution (random vs. gradient distribution) of habitat, although higher dispersal evolved at expanding range margins in landscapes with greater quantity of habitat and in gradient landscapes. Increased dispersal was no longer evident in any landscape once populations had reached equilibrium; dispersal values returned to those of seed populations. However, in landscapes with the least quantity of habitat, reduced dispersal (below that of seed populations) was observed at equilibrium. 3. Evolutionary changes in adult flight morphology were examined in six populations of P. aegeria along a transect from the distribution core to an expanding range margin in England (spanning a latitudinal distance of >200 km). Empirical data were in agreement with model output and showed increased dispersal ability (larger and broader thoraxes, smaller abdomens, higher wing aspect ratios) with increasing distance from the distribution core. Increased dispersal ability was evident in populations from areas colonised >30 years previously, although dispersal changes were generally evident only in females. 4. Evolutionary increases in dispersal ability in expanding populations may help species track future climate changes and counteract impacts of habitat fragmentation by promoting colonisation. However, at the highest levels of habitat loss, increased dispersal was less evident during expansion and reduced dispersal was observed at equilibrium indicating that, for many species, continued habitat fragmentation is likely to outweigh any benefits from dispersal.     

Epistasis of quantitative trait loci under different gene action models

Modeling and detecting nonallelic (epistatic) effects at multiple quantitative trait loci (QTL) often assume that the study population is in zygotic equilibrium (i.e., genotypic frequencies at different loci are products of corresponding single-locus genotypic frequencies). However, zygotic associations can arise from physical linkages between different loci or from many evolutionary and demographic processes even for unlinked loci. We describe a new model that partitions the two-locus genotypic values in a zygotic disequilibrium population into equilibrium and residual portions. The residual portion is of course due to the presence of zygotic associations. The equilibrium portion has eight components including epistatic effects that can be defined under three commonly used equilibrium models, Cockerham's model, F2-metric, and F(infinity)-metric models. We evaluate our model along with these equilibrium models theoretically and empirically. While all the equilibrium models require zygotic equilibrium, Cockerham's model is the most general, allowing for Hardy-Weinberg disequilibrium and arbitrary gene frequencies at individual loci whereas F2-metric and F(infinity)-metric models require gene frequencies of one-half in a Hardy-Weinberg equilibrium population. In an F2 population with two unlinked loci, Cockerham's model is reduced to the F2-metric model and thus both have a desirable property of orthogonality among the genic effects; the genic effects under the F(infinity)-metric model are not orthogonal but they can be easily translated into those under the F2-metric model through a simple relation. Our model is reduced to these equilibrium models in the absence of zygotic associations. The results from our empirical analysis suggest that the residual genetic variance arising from zygotic associations can be substantial and may be an important source of bias in QTL mapping studies.     

On the possibility of linear modelling the human arm neuromuscular apparatus

It has been widely claimed that linear models of the neuromuscular apparatus give very inaccurate approximations of human arm reaching movements. The present paper examines this claim by quantifying the contributions of the various non-linear effects of muscle force generation on the accuracy of linear approximation. We performed computer simulations of a model of a two-joint arm with six monarticular and biarticular muscles. The global actions of individual muscles resulted in a linear dependence of the joint torques on the joint angles and angular velocities, despite the great non-linearity of the muscle properties. The effect of time delay in force generation is much more important for model accuracy than all the non-linear effects, while ignoring this time delay in linear approximation results in large errors. Thus, the viscosity coefficients are rather underestimated and some of them can even be paradoxically estimated to be negative. Similarly, our computation showed that ignoring the time delay resulted in large errors in the estimation of the hand equilibrium trajectory. This could explain why experimentally estimated hand equilibrium trajectories may be complex, even during a simple reaching movement. The hand equilibrium trajectory estimated by a linear model becomes simple when the time delay is taken into account, and it is close to that actually used in the non-linear model. The results therefore provide a theoretical basis for estimating the hand equilibrium trajectory during arm reaching movements and hence for estimating the time course of the motor control signals associated with this trajectory, as set out in the equilibrium point hypothesis. Received: 17 February 1999 / Accepted in revised form: 22 October 1999     

Among- and within-patch components of genetic diversity respond at different rates to habitat fragmentation: an empirical demonstration

Habitat fragmentation is a ubiquitous by-product of human activities that can alter the genetic structure of natural populations, with potentially deleterious effects on population persistence and evolutionary potential. When habitat fragmentation results in the subdivision of a population, random genetic drift then leads to the erosion of genetic diversity from within the resulting subpopulation, random genetic drift then leads to the erosion of genetic diversity from within the resulting subpopulations and greater genetic divergence among them. Theoretical and simulation analyses predict that these two main genetic effects of fragmentation, greater differentiation among resulting subpopulation and reduced genetic diversity within them, will proceed at very different rates. Despite important implications for the interpretation of genetics data from fragmented populations, empirical evidence for this phenomenon has been lacking. In this analysis, we carry out an empirical study in population of an alpine meadow-dwelling butterfly, which have become fragmented increasing forest cover over five decades. We show that genetic differentiation among subpopulations (G(ST)) is most highly correlated with contemporary forest cover, while genetics diversity within subpopulation (expected heterozygosity) is better correlated with the spatial pattern of forest cover 40 years in the past. Thus, where habitat fragmentation has occurred in recent decades, genetic differentiation among subpopulation can be near equilibrium while contemporary measures of within subpopulation diversity may substantially overestimate the equilibrium values that will eventually be attained.