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Recent studies have increasingly turned to graph theory to model more realistic contact structures that characterize disease spread. Because of the computational demands of these methods, many researchers have sought to use measures of network structure to modify analytically tractable differential equation models. Several of these studies have focused on the degree distribution of the contact network as the basis for their modifications. We show that although degree distribution is sufficient to predict disease behaviour on very sparse or very dense human contact networks, for intermediate density networks we must include information on clustering and path length to accurately predict disease behaviour. Using these three metrics, we were able to explain more than 98 per cent of the variation in endemic disease levels in our stochastic simulations.

Gene banks have a critical role in preserving genetic diversity for many conservation uses. A key challenge in this task is maintaining both the genetic integrity and the viability of a collection during regeneration and long-term storage. In this study we examine the interacting effects of sampling and viability decline from seed aging on genetic diversity. To do this, we developed a model that simulates sampling during regeneration of individual genotypes using two commonly used but contrasting regeneration protocols. Critically, the model simulates a realistic nonlinear decline in viability across time, and the combined effects of sampling and mortality were evaluated for their effect on allelic diversity. In addition, the model was used to contrast the kinetics of allelic richness among rounds of regeneration in accessions with different heterozygosity levels (e.g., cultivars vs. wild) and different breeding systems (outcrossing vs. inbred). Results underscore a significant interaction between regeneration timing and regeneration size. Regeneration procedures should consider viability kinetics and intra-accession diversity to best manage genetic erosion. These changes highlight the trade-off faced by curators between the logistics of increasing regeneration sampling size and improving the technical procedures for seed storage and monitoring.

Malus sieversii (Lebed.) M. Roem. is a wild progenitor species of the domesticated apple. It is found across a mountainous region of central Asia and has been the focus of several collection expeditions by the USDA-ARS-National Plant Germplasm System. This study used microsatellite variation at seven loci to estimate diversity and differentiation within M. sieversii using several complimentary approaches. Multilocus genotypes were amplified from 949 individuals representing seedling trees from 88 half-sib families from eight M. sieversii populations collected in Kazakhstan. Apportioning of genetic variation was estimated at both the family and site level. Analyses using a hierarchical model to estimate F st showed that differentiation among individual families is more than three times greater than differentiation among sites. In addition, average gene diversity and allelic richness varied significantly among sites. A rendering of a genetic network among all sites showed that differentiation is largely congruent with geographical location. In addition, nonhierarchical Bayesian assignment methods were used to infer genetic clusters across the collection area. We detected four genetic clusters in the data set. The quality of these assignments was evaluated over multiple Markov Chain Monte Carlo runs using both posterior likelihood and stability of the assignments. The spatial pattern of genetic assignments among the eight collection sites shows two broadly distributed and two narrowly distributed clusters. These data indicate that the southwestern collection sites are more admixed and more diverse than the northern sites.

Ecology plays an important role in society, informing policy and management decisions across a variety of issues. As such, regularities in processes would indicate higher levels of predictive outcomes and would reduce the amount of research required for specific issues that policy makers need addressed. Scientific laws are considered the pinnacle of success and usefulness in addressing regularities or universal truths. Ecology studies complex interactions of individuals with unique behaviors, making the identification of laws problematic. Two equations, Malthusian growth and the logistic equation, continue to receive attention and are frequently cited as exemplar laws in ecology. However, an understanding of scientific laws shows that neither are good candidates for law status. In this paper, I will discuss why ecology is not well structured for scientific laws, as they are currently understood. Finally, I will consider alternative proposals for the role of laws in ecology and alternate forms of laws that may be applicable.

Laboratory experiments and field trials demonstrated that canola oil, which is rich in fatty acids that are kairomonal cues in cannibalism, is an effective attractant of rangeland grasshoppers and a suitable oil-based carrier for insecticides. Laboratory-based olfactory tests revealed a significant, positive chemotaxis of Melanoplus sanguinipes in response to canola oil and olive oil mixtures. Small-scale field tests made by airbrush applications of 5 ml of canola oil (equivalent to 500 l/ha) to centres of aluminium rings (0.1 m2

Self-organized criticality (SOC) occurs in systems consisting of a substrate or medium which can be locally stressed to a critical state. When the critical threshold is exceeded, the stress is distributed to the neighborhood around the locale, which can lead to critical states in the neighboring locations thereby propagating the disturbance. The SOC system manifests its behavior in distinctive temporal and spatial relationships. Diagnostic behaviors include conformity of dynamics to a power law (F(s) = βsα) relating the frequency distribution function to the observed behavior (1/f noise) and fractal behavior or spatial self-similarity. We analyzed 40 to 50 years of grasshopper infestations in Idaho and Montana for evidence of SOC. In both states, grasshopper infestations in four of the seven ecoregions statistically fit (the expected frequencies matched the observed with a probability of >95%) the power law relationship. In ecoregions where the model failed to fit the observed behavior, grasshopper infestations were typically rare and small scale. Spatial self-similarity was evident from graphical and statistical analyses. In both states there were statistically significant levels of concordance (among ranks) and correlation (among areas of infestations) between ecoregions. The spatial distributions of grasshopper infestations over time also showed considerable overlaps between ecoregions. However, only in Idaho were the variances in grasshopper infestations among ecoregions homogeneous. If rangeland grasshopper populations are SOC systems, then the management of outbreaks may need to be dramatically restructured. This system appears to be an example of “complex complexity,” in which multiple dynamics (i.e., the appearance of SOC with partial blockage of outbreaks, catastrophic dynamics, and perhaps chaos) give rise to a rich set of behaviors which arise at different scales and certainly necessitate further theoretical developments in order to understand. © 1997 John Wiley & Sons, Inc.

Research reported in this feature identifies a convergence of interpretations regarding the threshold dynamics of complex ecological systems. This convergence has arisen from a diverse set of investigations addressing rangeland ecosystem dynamics, disease transmission, and fluctuations in the populations of insect pests. Effective application of the threshold concept to ecosystem management will require development of more robust linkages between non-equilibrium theory and protocols to identify triggers that initiate threshold conditions, feedback loops that establish system resilience, and developmental trajectories and attributes of potential alternative stable states. Successful implementation of these theory/ application linkages has the potential to underpin an operational framework of resilience-based ecosystem management that is founded upon the identification of structural indicators that are correlated with vulnerability or proximity to thresholds, rather than threshold identification per se. Several investigations indicate that thresholds are strongly influenced by scale; multiple cross-scale interactions demonstrate the need for greater knowledge and analyses to address scale-dependent processes, i.e., critical scales and scaling laws. This feature emphasizes the relevance of thresholds and non-equilibrium dynamics in multiple natural resource management applications and in so doing demonstrates the need for a more comprehensive and integrated ecological framework capable of quantitatively assessing dynamics at multiple scales to inform management and policy recommendations for optimal management and risk assessment.

Catastrophe Theory is a branch of mathematics that provides a framework for studying discontinuous phenomena. Application of the cusp catastrophe, a model with two driving or control variables, indicates that rangeland grasshopper population dynamics are catastrophic. Based on bimonthly temperature and precipitation, the cusp catastrophe modeled changes in grasshopper populations dynamics in four ecotopographic regions of Wyoming with an accuracy significantly better than chance. Although there were differences in model performance among regions, the best predictions generally occurred when the changes were large-scale outbreaks (30–45% of the total area of a region) of economic densities (≥9.6 grasshoppers/m2). Because winter weather, particularly in climatically extreme years, is correlated with spring weather, it appears that the climate in December–January has predictive value with regard to population dynamics in the following spring. However, when temperature and precipitation during hatching and early development of grasshoppers (April–May) are used as the control variables, the cusp catastrophe most accurately models the observed population dynamics.

Summary The origin of cattle on Chirikof Island, off the coast of Alaska, is not well documented. We assessed genetic differentiation of cattle isolated on Chirikof Island from several breeds commonly used for commercial production in North America including breeds popularly believed to have contributed to the Chirikof Island population. A set of 34 microsatellite loci was used to genotype Angus, Charolais, Hereford, Highland, Limousin, Red Angus, Salers, Shorthorn, Simmental, Tarentaise and Texas Longhorn cattle sampled from North America and the Chirikof Island population. Resulting FST statistics for these loci ranged from 0.06 to 0.22 and on average, 14% of total genetic variation was between breeds. Whether population structure was modelled as a bifurcating tree or genetic network, Chirikof Island cattle appeared to be unique and strongly differentiated relative to the other breeds that were sampled. Bayesian clustering for multiple-locus assignment to genetic groups indicated low levels of admixture in the Chirikof Island population. Thus, the Chirikof Island population may be a novel genetic resource of some importance for conservation and industry.

The concept of marine reserves as a method of improving management of fisheries is gaining momentum. While the list of benefits from reserves is frequently promoted, precise formulations of theory to support reserve design are not fully developed. To determine the size of reserves and the distances between reserves an understanding of the requirements for persistence of local populations is required. Unfortunately, conditions for persistence are poorly characterized, as are the larval dispersal patterns on which persistence depends. With the current paucity of information regarding meroplanktonic larval transport processes, understanding the robustness of theoretical results to larval dispersal is of key importance. From this formulation a broad range of dispersal patterns are analyzed. Larval dispersal is represented by a probability distribution that defines the fraction of successful settlers from an arbitrary location, the origin of the distribution, to any other location along the coast. While the effects of specific dispersal patterns have been investigated for invasion processes, critical habitat size and persistence issues have generally been addressed with only one or two dispersal types. To that end, we formulate models based on integrodifference equations that are spatially continuous and temporally discrete. We consider a range of dispersal distributions from leptokurtic to platykurtic. The effect of different dispersal patterns is considered for a single isolated reserve of varying size receiving no external larvae, as well as multiple reserves with varying degrees of connectivity. While different patterns result in quantitative differences in persistence, qualitatively similar effects across all patterns are seen in both single- and multiple reserve models. Persistence in an isolated reserve requires a size that is approximately twice the mean dispersal distance and regardless of the dispersal pattern the population in a patch is not persistent if the reserve size is reduced to just the mean dispersal distance. With an idealized coastline structure consisting of an infinite line of equally spaced reserves separated by regions of coastline in which reproduction is nil, the relative settlement as a function of the fraction of coastline and size of reserve is qualitatively very similar over a broad range of dispersal patterns. The upper limit for the minimum fraction of coastline held in reserve is about 40%. As the fraction of coastline is reduced, the minimum size of reserve becomes no more than 1.25 times the mean dispersal distance.

Rangeland ecologists have elucidated 2 apparently distinct processes underlying rangeland dynamics. In some cases, disturbed or recovering rangelands move through a gradual, continuous series of changes which has been termed succession. In other instances, rangeland dynamics are typified by sudden, discontinuous changes in the vegetation, and this has been cared state-and-transition. Catastrophe theory is a mathematical framework designed for the study of discontinuous phenomena, but it also generates models that permit continuous dynamics. Based on available literature, it appears that rangeland ecosystems conform to the mathematics of catastrophe theory. Rangelands exhibit the 5 essential symptoms of catastrophe systems: modality (distinct conditions or states of existence), inaccessibility (conditions which are very unstable), sudden changes (relatively rapid movement between states), hysteresis (processes associated with degradation or recovery are not readily reversible by simply inverting the sequence of events), and divergence (relatively small changes in initial conditions can result in dramatically different outcomes with time). Catastrophe theory has been successfully used to model rangeland grasshopper population dynamics, and it appears that many of the same control variables affecting insects (e.g., temperature and precipitation) may also underlie vegetative community dynamics. Application of catastrophe theory to empirical data sets will require relatively long-term but low-intensity research efforts.

Catastrophe theory is a form of non-linear mathematics which allows predictions of discontinuities within systems that are governed by a relatively small number of control variables and are derivable from smooth potentials. Grasshopper population dynamics appear to possess properties which can be described by a cusp catastrophe. Previous work has shown that weather conditions at the time of hatching and early development appear to mediate outbreaks and crashes of grasshopper populations. Using several measures of temperature and precipitation and various expressions of grasshopper infestations in Wyoming, we developed a series of models using Catastrophe Theory. The cusp catastrophe based on outbreak levels of infestation (> 9.6 grasshoppers per m2) and monthly (April and May) weather data generally provided the most appropriate descriptions of grasshopper population dynamics; the best models were up to 76% accurate based on a priori criteria. Catastrophic changes in grasshopper populations occur on a very large scale (state-wide), and it appears that different weather parameters affect high and low density outbreaks.