The influence of the type 3 (sigmoid) functional response upon the stability of predator-prey difference models

Sigmoid functional responses are found to exert a stabilizing influence upon a discrete-generation predator-prey model in a way analogous to that found in continuous predator-prey models. The precise effect depends upon the degree to which a predator's feeding history influences its reproductive success. The time delay intrinsic in difference equation models imposes constraints not found in differential models, however, it is shown that in an otherwise unstable model the inclusion of a sigmoid functional response can result in local stability. With the addition of prey self-regulation the stabilizing influence of the functional response acts in concert with self-regulation, as it does in continuous models. These results show that the effect of the sigmoid response upon stability is not dependent upon the assumption of continuity, and reinforces the view that sigmoid responses could be an important factor stabilizing natural communities.     

Dominance orders in animal societies: The self-organization hypothesis revisited

In previous papers (Theraulaz et al., 1995; Bonabeau et al., 1996) we suggested, following Hogeweg and Hesper (1983, 1985), that the formation of dominance orders in animal societies could result from a self-organizing process involving a double reinforcement mechanism: winners reinforce their probability of winning and losers reinforce their probability of losing. This assumption, and subsequent models relying on it, were based on empirical data on primitively eusocial wasps (Polistes dominulus). By reanalysing some of the experimental data that was previously thought to be irrelevant, we show that it is impossible to distinguish this assumption from a competing assumption based on preexisting differences among individuals. We propose experiments to help discriminate between the two assumptions and their corresponding models—the self-organization model and the correlational model. We urge other researchers to be cautious when interpreting their dominance data with the ’self-organization mindset’; in particular, ‘winner and loser effects’, which are often considered to give support to the self-organization assumption, are equally consistent with the correlational assumption.     

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.     

Mathematical Elegance with Biochemical Realism: The Covarion Model of Molecular Evolution

There is an apparent paradox in our understanding of molecular evolution. Current biochemically based models predict that evolutionary trees should not be recoverable for divergences beyond a few hundred million years. In practice, however, trees often appear to be recovered from much older times. Mathematical models, such as those assuming that sites evolve at different rates [including a Γ distribution of rates across sites (RAS)] may in theory allow the recovery of some ancient divergences. However, such models require that each site maintain its characteristic rate over the whole evolutionary period. This assumption, however, contradicts the knowledge that tertiary structures diverge with time, invalidating the rate-constancy assumption of purely mathematical models. We report here that a hidden Markov version of the covarion model can meet both biochemical and statistical requirements for the analysis of sequence data. The model was proposed on biochemical grounds and can be implemented with only two additional parameters. The two hidden parts of this model are the proportion of sites free to vary (covarions) and the rate of interchange between fixed sites and these variable sites. Simulation results are consistent with this approach, providing a better framework for understanding anciently diverged sequences than the standard RAS models. However, a Γ distribution of rates may approximate a covarion model and may possibly be justified on these grounds. The accurate reconstruction of older divergences from sequence data is still a major problem, and molecular evolution still requires mathematical models that also have a sound biochemical basis. Received: 13 February 2001 / Accepted: 22 May 2001     

On ‘analytical models for the patchy spread of plant disease’

Epidemiologists are interested in using models that incorporate the effects of clustering in the spatial pattern of disease on epidemic dynamics. Bolker (1999, Bull. Math. Biol. 61, 849–874) has developed an approach to study such models based on a moment closure assumption. We show that the assumption works above a thre shold initial level of disease that depends on the spatial dispersal of the pathogen. We test an alternative assumption and show that it does not have this limitation. We examine the relation between lattice and continuous-medium implementations of the approach.     

HYBRID FITNESS IN THE LOUISIANA IRISES: ANALYSIS OF PARENTAL AND F1 PERFORMANCE

The assumption of hybrid inferiority is central to the two models most widely applied to the prediction of hybrid zone evolution. Both the tension zone and mosaic models assume that natural selection acts against hybrids regardless of the environment in which they occur. To test this assumption, we investigated components of fitness in Iris fulva, I. hexagona and their reciprocal F₁ hybrids under greenhouse conditions. The four cross types were compared on the basis of seed germination, vegetative and clonal growth, and sexual reproduction. In all cases, the hybrids performed as well as, or significantly better than, both of their parents. These results suggest that F₁ hybrids between I. fulva and I. hexagona are at least as fit as their parents. The results of this study are therefore inconsistent with the assumptions of both the tension zone and mosaic models of hybrid zone evolution.     

QUANTITATIVE GENETICS OF MORPHOLOGICAL DIFFERENTIATION IN PEROMYSCUS. I. TESTS OF THE HOMOGENEITY OF GENETIC COVARIANCE STRUCTURE AMONG SPECIES AND SUBSPECIES

Phenotypic and additive genetic covariance matrices were estimated for 15 morphometric characters in three species and subspecies of Peromyscus. Univariate and multivariate ANOVAs indicate these groups are highly diverged in all characters, P. leucopus having the largest body size, P. maniculatus bairdii the smallest, and P. maniculatus nebrascensis being intermediate. Comparing the structure of P and G within each taxon revealed significant similarities in all three cases. This proportionality was strong enough to justify using P in the place of G to analyze evolutionary processes using quantitative genetic models when G can not be estimated, as in fossil material. However, the similarity between genetic and phenotypic covariance structures is sufficiently low that estimates of the genetic parameters should be used when possible. The additive genetic covariance matrices were compared to examine the assumption that they remain constant during evolution, an assumption which underlies many applications of quantitative-genetic models. While matrix permutation tests indicated statistically significant proportionality between the genetic covariance structures of the two P. maniculatus subspecies, there is no evidence of significant genetic structural similarity between species. This result suggests that the assumption of constant genetic covariance structure may be valid only within species. (It does not, however, necessarily imply a causal relationship between speciation and heterogeneity of genetic covariance structures.) The low matrix correlation for the two P. maniculatus subspecies' genetic covariance matrices indicates G may not be functionally constant, even within species. The lack of similarity observed here may be due partly to sampling variation.     

HOW SIMILAR ARE GENETIC CORRELATION STRUCTURES? DATA FROM MICE AND RATS

An integral assumption of many models of morphometric evolution is the equality of the genetic variance-covariance structure across evolutionary time. To examine this assumption, the quantitative-genetic aspects of morphometric form are examined for eight pelvic traits in laboratory rats (Rattus norvegicus) and random-bred ICR mice (Mus musculus). In both species, all traits are significantly heritable, and there are significant phenotypic and genetic correlations among traits, although environmental correlations among the eight traits are low. The size relations among the pelvic variables are isometric. Three matrix-permutation tests are used to examine similarity of phenotypic, genetic, and environmental covariance and correlation matrices within and between species. Independent patterns of morphometric covariation and correlation arise from genetic and environmental effects within each species and from environmental effects between species. The patterns of phenotypic and genetic covariation and correlation are similar within each species, and the phenotypic and genetic correlations are also similar between these species. However, genetic covariance matrices show no significant statistical association between species. It is suggested that the assumption of equality of genetic variance-covariance structures across divergent taxa should be approached with caution.     

Comparison of six different models describing survival of mammalian cells after irradiation

Summary Six different cell-survival models have been compared. All models are based on the similar assumption that irradiated cells are able to exist in one of three states.S _A is the state of a totally repaired cell, in stateSC the cell contains lethal lesions and in stateS _B the cell contains potentially lethal lesions i.e. those which either can be repaired or converted into lethal lesions. The differences between the six models lies in the different mathematical relationships between the three states. To test the six models, six different sets of experimental data were used which describe cell survival at different repair times after irradiation with sparsely ionizing irradiation. In order to compare the models, a goodness-of-fit function was used. The differences between the six models were tested by use of the nonparametric Mann-Whitney two sample test. Based on the 95% confidence limit, this required separation into three groups.Extended version of an oral presentation held at the 8th International Congress of Radiation Research in Edinburgh     

Equilibrium or not? Modelling potential distribution of invasive species in different stages of invasion

Aim The assumption of equilibrium between organisms and their environment is a standard working postulate in species distribution models (SDMs). However, this assumption is typically violated in models of biological invasions where range expansions are highly constrained by dispersal and colonization processes. Here, we examined how stage of invasion affects the extent to which occurrence data represent the ecological niche of organisms and, in turn, influences spatial prediction of species’ potential distributions. Location Six ecoregions in western Oregon, USA. Methods We compiled occurrence data from 697 field plots collected over a 9‐year period (2001–09) of monitoring the spread of invasive forest pathogen Phytophthora ramorum. Using these data, we applied ecological‐niche factor analysis to calibrate models of potential distribution across different years of colonization. We accounted for natural variation and uncertainties in model evaluation by further investigating three hypothetical scenarios of varying equilibrium in a simulated virtual species, for which the ‘true’ potential distribution was known. Results We confirm our hypothesis that SDMs calibrated in early stages of invasion are less accurate than models calibrated under scenarios closer to equilibrium. SDMs that are developed in early stages of invasion tend to underpredict the potential range compared to models that are built in later stages of invasion. Main conclusions A full environmental niche of invasive species cannot be effectively captured with data from a realized distribution that is restricted by processes preventing full occupancy of suitable habitats. If SDMs are to be used effectively in conservation and management, stage of invasion needs to be considered to avoid underestimation of habitats at risk of invasion.     

A generalization of the optimal models of arterial branching

Optimal models of arterial branching angles are usually based on the assumption that the equation relating flow and radius is given byf=kr ³, as proposed by Murray in 1926. An exception is the model of Uylings (1977), in which he allowed the exponent ofr to vary from 2.33 to 3.0. Theoretical considerations coupled with empirical evidence suggest that the cubic flow equation may not be appropriate to describe the branching pattern of the arterial tree. The optimal models are modified to accommodate a more general flow equationf=kr ^x . Models that minimize a geometric feature such as surface or volume are sensitive to variations inx in a different way from those which minimize flow-related parameters, such as power loss due to friction and shear stress.     

Illuminating the mysteries of wolf history

One of the most enduring surprises about the genetic history of Late Pleistocene populations is that continuity is often disturbed by upheaval. In fact, studies that support population continuity are increasingly rare in humans, a variety of vertebrate taxa, and vascular plants (Hofreiter & Stewart 2009; Burbrink et al. 2016). Perhaps such continuity should not be expected as the Pleistocene is marked by episodes of climate change, glaciation and the invasions of humans into previously isolated areas. Although fossils are one of the primary sources for inferring population continuity, a problem with fossil material is that, even if similar morphological forms might exist in a place over time, they may not be from the same genetic lineage. There are now readily available methods to assess genetic continuity solely from DNA found in fossil material, provided the record is fairly continuous. In a From the Cover article in this issue of Molecular Ecology, Loog et al. (2020) apply some of these readily available methods to analyse mitochondrial genomes and model the demography of wolves over the last 50,000 years.     

THE EFFECTS OF HOST GENOTYPE AND SPATIAL DISTRIBUTION ON TREMATODE PARASITISM IN A BIVALVE POPULATION

A basic assumption underlying models of host-parasite coevolution is the existence of additive genetic variation among hosts for resistance to parasites. However, estimates of additive genetic variation are lacking for natural populations of invertebrates. Testing this assumption is especially important in view of current models that suggest parasites may be responsible for the evolution of sex, such as the Red Queen hypothesis. This hypothesis suggests that the twofold reproductive disadvantage of sex relative to parthenogenesis can be overcome by the more rapid production of rare genotypes resistant to parasites. Here I present evidence of significant levels of additive genetic variance in parasite resistance for an invertebrate host-parasite system in nature. Using families of the bivalve mollusc, Transennella tantilla, cultured in the laboratory, then exposed to parasites in the field, I quantified heritable variation in parasite resistance under natural conditions. The spatial distribution of outplanted hosts was also varied to determine environmental contributions to levels of parasite infection and to estimate potential interactions of host genotype with environment. The results show moderate but significant levels of heritability for resistance to parasites (h² = 0.36). The spatial distribution of hosts also significantly influenced parasite prevalence such that increased host aggregation resulted in decreased levels of parasite infection. Family mean correlations across environments were positive, indicating no genotype-environment interaction. Therefore, these results provide support for important assumptions underlying coevolutionary models of host-parasite systems.     

Modeling bird species distribution change in fire prone Mediterranean landscapes: incorporating species dispersal and landscape dynamics

Species distribution models (SDMs) have traditionally been founded on the assumption that species distributions are in equilibrium with environmental conditions and that these species–environment relationships can be used to estimate species responses to environmental changes. Insight into the validity of this assumption can be obtained from comparing the performance of correlative species distribution models with more complex hybrid approaches, i.e. correlative and process‐based models that explicitly include ecological processes, thereby accounting for mismatches between habitat suitability and species occupancy patterns. Here we compared the ability of correlative SDMs and hybrid models, which can accommodate non‐equilibrium situations arising from dispersal constraints, to reproduce the distribution dynamics of the ortolan bunting Emberiza hortulana in highly dynamic, early successional, fire driven Mediterranean landscapes. Whereas, habitat availability was derived from a correlative statistical SDM, occupancy was modeled using a hybrid approach combining a grid‐based, spatially‐explicit population model that explicitly included bird dispersal with the correlative model. We compared species occupancy patterns under the equilibrium assumption and different scenarios of species dispersal capabilities. To evaluate the predictive capability of the different models, we used independent species data collected in areas affected to different degree by fires. In accordance with the view that disturbance leads to a disparity between the suitable habitat and the occupancy patterns of the ortolan bunting, our results indicated that hybrid modeling approaches were superior to correlative models in predicting species spatial dynamics. Furthermore, hybrid models that incorporated short dispersal distances were more likely to reproduce the observed changes in ortolan bunting distribution patterns, suggesting that dispersal plays a key role in limiting the colonization of recently burnt areas. We conclude that SDMs used in a dynamic context can be significantly improved by using combined hybrid modeling approaches that explicitly account for interactions between key ecological constraints such as dispersal and habitat suitability that drive species response to environmental changes.     

Origins of life: An operational definition

Two very different models are used for the scientific study of life's origins: in the Troland-Muller model, life is molecular and its defining characteristic is gene function; in the Oparin-Haldane model, life is cellular and its defining characteristic is metabolic function. While each of these models implicitly defines the living, neither provides criteria by which theemergence of life could be recognized in the laboratory.Anoperational definition of the living makes explicit the system logic of metabolic self-production: (1) that whatever form it may take, life is a function of its biochemical processes; (2) that no single biochemical process has integrity apart from an entire network of processes; (3) that a network of processes can have continuity only by being enclosed within a boundary structure, i.e., by the selective partition of a microenvironment as a domain for the bioenergetic-biosynthetic network; and (4) that life is a single phenomenon, distinct in its continuity of capture and storage of energy in such networks, driving the processes that produce its material constituents.This paper presentsautopoiesis as life-defining and discusses the utility of its criteria in our search for the origins of life on Earth. Enactment of the autopoietic criteria would result in aminimal cell and would demonstrate the experimental recapitulation of life's Archaean origins.     

A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data

Summary Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions is one approach to mixture model identification ( Little, 1995 , Journal of the American Statistical Association 90 , 1112–1121; Little and Wang, 1996 , Biometrics 52 , 98–111; Thijs et al., 2002 , Biostatistics 3 , 245–265; Kenward, Molenberghs, and Thijs, 2003 , Biometrika 90 , 53–71; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis) and is a natural starting point for missing not at random sensitivity analysis ( Thijs et al., 2002 , Biostatistics 3 , 245–265; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random (MAR) may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g., baseline covariates with time‐invariant coefficients). In this article, we explore conditions necessary for identifying restrictions that result in MAR to exist under a multivariate normality assumption and strategies for identifying sensitivity parameters for sensitivity analysis or for a fully Bayesian analysis with informative priors. In addition, we propose alternative modeling and sensitivity analysis strategies under a less restrictive assumption for the distribution of the observed response data. We adopt the deviance information criterion for model comparison and perform a simulation study to evaluate the performances of the different modeling approaches. We also apply the methods to a longitudinal clinical trial. Problems caused by baseline covariates with time‐invariant coefficients are investigated and an alternative identifying restriction based on residuals is proposed as a solution.     

Parasites detect host spatial pattern and density: a field experimental analysis

Summary A common assumption in mathematical models of parasitism is that the susceptibility to parasitism of an individual host increases both with host density and the degree of host spatial aggregation. To determine whether this assumption is correct in nature, we developed a factorial field experiment with the parasitic marine isopod Hemioniscus balani and its barnacle host Chthamalus dalli. Our factorial design enabled evaluation of the separate effects on parasitism of the two factors (host density and host spatial pattern) and also to assess the host density-spatial pattern interaction effect. Both host density and spatial aggregation were found to lead to increased parasitism, and the interaction effect was nonsignificant. These findings are the first experimental field demonstration that these processes occur in nature, as widely assumed in ecological theory.     

On the importance of habitat continuity for delimiting biogeographic regions and shaping richness gradients

The formation and maintenance of biogeographic regions and the latitudinal gradient of species richness are thought to be influenced, in part, by the spatial distribution of physical habitat (habitat continuity). But the importance of habitat continuity in relation to other variables for shaping richness gradients and delimiting biogeographic regions has not been well established. Here, we show that habitat continuity is a top predictor of biogeographic structure and the richness gradient of eastern Pacific rocky shore gastropods (spanning c. 23 000 km, from 43°S to 48°N). Rocky shore habitat continuity is generally low within tropical/subtropical regions (compared to extratropical regions), but particularly at biogeographic boundaries where steep richness gradients occur. Regions of high rocky shore habitat continuity are located towards the centres of biogeographic regions where species turnover tends to be relatively low. Our study highlights the importance of habitat continuity to help explain patterns and processes shaping the biogeographic organisation of species.     

Species richness, area and climate correlates

Aim Species richness–area theory predicts that more species should be found if one samples a larger area. To avoid biases from comparing species richness in areas of very different sizes, area is often controlled by counting the numbers of co‐occupying species in near‐equal area grid cells. The assumption is that variation in grid cell size accrued from working in a three‐dimensional world is negligible. Here we provide a first test of this idea. We measure the surface area of c. 50 × 50 km and c. 220 × 220 km grid cells across western Europe. We then ask how variation in the area of grid cells affects: (1) the selection of climate variables entering a species richness model; and (2) the accuracy of models in predicting species richness in unsampled grid cells. Location Western Europe. Methods Models are developed for European plant, breeding bird, mammal and herptile species richness using seven climate variables. Generalized additive models are used to relate species richness, climate and area. Results We found that variation in the grid cell area was large (50 × 50 km: 8–3311 km²; 220 × 220: 193–55,100 km²), but this did not affect the selection of variables in the models. Similarly, the predictive accuracy was affected only marginally by exclusion of area within models developed at the c. 50 × 50 km grid cells, although predictive accuracy suffered greater reductions when area was not included as a covariate in models developed for c. 220 × 220 km grid cells. Main conclusions Our results support the assumption that variation in near‐equal area cells may be of second‐order importance for models explaining or predicting species richness in relation to climate, although there is a possibility that drops in accuracy might increase with grid cell size. The results are, however, contingent on this particular data set, grain and extent of the analyses, and more empirical work is required.     

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov functions for two-dimension SIR and SIRS compartmental epidemic models with non-linear transmission rate of a very general form f(S,I) constrained by a few biologically feasible conditions are constructed. Global properties of these models including these with vertical and horizontal transmission, are thereby established. It is proved that, under the constant population size assumption, the concavity of the function f(S,I) with respect to the number of the infective hosts I ensures the uniqueness and the global stability of the positive endemic equilibrium state. AMS Classification 92D30 (primary), 34D20 (secondary)