Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1.  Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity.
2.  We analyse a stochastic environment model of the red kangaroo ( Macropus rufus ), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates.
3.  Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4.  Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates.
5.  Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c . 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c . 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.     

Can life-history traits predict the response of forb populations to changes in climate variability?

1.  Climate change will cause changes in average temperature and precipitation as well as increased fluctuations around the mean, yet few studies have considered the impacts of altered climate variability on plant populations. We tested whether life-history traits (expected life span, generation time and seed size) can predict plant responses to increased environmental variability across similar plant species sharing the same habitat.
2.  We combined long-term demographic data on 10 prairie forb species with stochastic demography techniques to estimate the effects of potential changes in matrix element means and variances on the long-term stochastic population growth rate.
3.  For all 10 species, recruitment had higher contribution and elasticity values than survival, meaning that climate change is more likely to influence population growth through effects on recruitment than on survival for these relatively short-lived forbs. Species with longer generation times had lower elasticities to increases in matrix element variability.
4.   Synthesis. Our analysis of a unique, long-term data set suggests that longer-lived plant species will be less vulnerable to the effects of future increases in climate variability. While this relationship was previously reported for diverse taxa from many locations, our results show that it also applies within a guild of short-lived species from a single community. The generality of the pattern demonstrates the potential for using life-history traits to make predictions about which species may be the most vulnerable to climate change.     

The many growth rates and elasticities of populations in random environments

Despite considerable interest in the dynamics of populations subject to temporally varying environments, alternate population growth rates and their sensitivities remain incompletely understood. For a Markovian environment, we compare and contrast the meanings of the stochastic growth rate (lambdaS), the growth rate of average population (lambdaM), the growth rate for average transition rates (lambdaA), and the growth rate of an aggregate represented by a megamatrix (shown here to equal lambdaM). We distinguish these growth rates by the averages that define them. We illustrate our results using data on an understory shrub in a hurricane-disturbed landscape, employing a range of hurricane frequencies. We demonstrate important differences among growth rates: lambdaS lambdaM. We show that stochastic elasticity, ESij, and megamatrix elasticity, EMij, describe a complex perturbation of both means and variances of rates by the same proportion. Megamatrix elasticities respond slightly and stochastic elasticities respond strongly to changing the frequency of disturbance in the habitat (in our example, the frequency of hurricanes). The elasticity EAij of lambdaA does not predict changes in the other elasticities. Because ES, although commonly utilized, is difficult to interpret, we introduce elasticities with a more direct interpretation: ESmu for perturbations of means and ESsigma for variances. We argue that a fundamental tool for studying selection pressures in varying environments is the response of growth rate to vital rates in all habitat states.     

Buffering and plasticity in vital rates of oldfield rodents

1. Under the hypothesis of environmental buffering, populations are expected to minimize the variance of the most influential vital rates; however, this may not be a universal principle. Species with a life span <1 year may be less likely to exhibit buffering because of temporal or seasonal variability in vital rate sensitivities. Further, plasticity in vital rates may be adaptive for species in a variable environment with reliable cues. 2. We tested for environmental buffering and plasticity in vital rates using stage-structured matrix models from long-term data sets in four species of grassland rodents. We used periodic matrices to estimate stochastic elasticity for each vital rate and then tested for correlations with a standardized coefficient of variation for each rate. 3. We calculated stochastic elasticities for individual months to test for an association between increased reproduction and the influence of reproduction, relative to survival, on the population growth rate. 4. All species showed some evidence of buffering. The elasticity of vital rates of Peromyscus leucopus (Rafinesque, 1818), Sigmodon hispidus Say & Ord, 1825 and Microtus ochrogaster (Wagner, 1842) was negatively related to vital rate CV. Elasticity and vital rate CV were negatively related in Peromyscus maniculatus (Wagner, 1845), but the relationship was not statistically significant. Peromyscus leucopus and M. ochrogaster showed plasticity in vital rates; reproduction was higher following months where elasticity for reproduction exceeded that of survival. 5. Our results suggest that buffering is common in species with fast life histories; however, some populations that exhibit buffering are capable of responding to short-term variability in environmental conditions through reproductive plasticity.     

Sensitivity of the population growth rate to demographic variability within and between phases of the disturbance cycle

For species in disturbance-prone ecosystems, vital rates (survival, growth and reproduction) often vary both between and within phases of the cycle of disturbance and recovery; some of this variation is imposed by the environment, but some may represent adaptation of the life history to disturbance. Anthropogenic changes may amplify or impede these patterns of variation, and may have positive or negative effects on population growth. Using stochastic population projection matrix models, we develop stochastic elasticities (proportional derivatives of the long-run population growth rate) to gauge the population effects of three types of change in demographic variability (changes in within- and between-disturbance-phase variability and phase-specific changes). Computing these elasticities for five species of disturbance-influenced perennial plants, we pinpoint demographic rates that may reveal adaptation to disturbance, and we demonstrate that species may differ in their responses to different types of changes in demographic variability driven by climate change.     

Time, transients and elasticity

How does life history affects the short‐term elasticities of population growth rate? We decompose short‐term elasticity as a sum of (i) the effect of the perturbation in rates on the unperturbed population structure and (ii) the effect of the original vital rates on the difference in structure between the original and the perturbed population. We provide exact analytical formulas for these components. In a population at its stable stage distribution (SSD), short‐term elasticity is determined mainly by the SSD and reproductive value. In a non‐stable population, short‐term elasticity depends also on the projection of initial structure on the SSD, equal to population momentum. Non‐stable stage structures matter most to elasticity if stages are missing that take time to fill in. We show how the demographic damping rate of the original population determines the rate at which short‐term elasticity converges to its limiting values.     

Estimating stochastic elasticities directly from longitudinal data

The elasticities of long-run population growth rate with respect to vital rates are useful in studying selection on vital rates, and in evaluating management policy that aims to control vital rates. In temporally varying environments, elasticity is often calculated from simulations that assume a probability distribution for the environmental states. Here we develop a method to estimate elasticities directly from demographic data. Using a time-series of demographic matrices and age-structure we construct a consistent statistical estimator of elasticity that converges to the correct limiting value as the sample length increases. We also construct confidence intervals for elasticities from temporal data and suggest tools for testing hypotheses about the strength of selection. We use data on a natural population to show that our method can indeed accurately estimate elasticities using relatively short time series.     

Proton relaxation mechanisms and the measurements of r phi, r psi and transannular interproton distances in gramicidin S

Monoselective, Rio(SE), biselective, Rio(i,j), and nonselective proton spin-lattice relaxation rates have been measured for dilute solutions of gramicidin S in dimethyl sulfoxide and used to evaluate cross-relaxation rates (sigma ij = Rio(i,j)-Rio(SE)) and Fi ratios (Fi = Ri(NS)/Rio(SE)). The cross-relaxation parameters, sigma, and Fi ratios measured for backbone gramicidin S protons predict that the same correlation time, tau c = 1.2 X 10(-9)s, modulates all the dipolar proton-proton interactions and that these interactions represent the main source for the proton spin-lattice relaxation process. The larger relaxation rates for amide versus alpha-protons of the backbone are attributed to dipolar relaxation between 14N and its directly bonded protons and is an approximate measure of the extent of this. The intrabackbone proton-proton distances, evaluated from sigma values, were consistent with the antiparallel beta-plated sheet/beta II'-turn conformation previously proposed for gramicidin S in solution.     

Buffering of life histories against environmental stochasticity: accounting for a spurious correlation between the variabilities of vital rates and their contributions to fitness

Life-history theory predicts vital rates that on average make large contributions to the annual multiplication rate of a lineage should be highly buffered against environmental variability. This prediction has been tested by looking for a negative correlation between the sensitivities (or elasticities) of the elements in a projection matrix and their variances (or coefficients of variation). Here, we show by constructing random matrices that a spurious negative correlation exists between the sensitivities and variances, and between the elasticities and coefficients of variation, of matrix elements. This spurious correlation arises in part because size transition probabilities, which are bounded by 0 and 1, have a limit to their variability that often does not apply to matrix elements representing reproduction. We advocate an alternative analysis based on the underlying vital rates (not the matrix elements) that accounts for the inherent limit to the variability of zero-to-one vital rates, corrects for sampling variation, and tests for a declining upper limit to variability as a vital rate's fitness contribution increases. Applying this analysis to demographic data from five populations of the alpine cushion plant Silene acaulis, we provide evidence of stronger buffering in the vital rates that most influence fitness.     

Protein matrix and dielectric effect in cytochrome c

The effect of the protein matrix on the standard potential of a buried redox center has been investigated by using a selection of mutants and chemical derivatives in Saccharomyces cerevisiae cytochrome c isoform 1. Assuming only local structural perturbation and no alteration of the iron-ligation chemistry, Delta E(m)(0)' can be regarded as a measure of the difference in polypeptide solvation of the heme charge, which reflects the dielectric properties of the protein. The evaluation of an apparent dielectric constant (U(exp)/U(theo)) yields variable, and sometimes even negative, values if U(exp) = Delta G(0)redox. However, some consistent result are observed if U(exp) = Delta H(0)redox, with a measured epsilon(Delta Delta)(H)(redox) = 19 +/- 6. The variability is thus attributed to an entropic factor (epsilon(Delta Delta)(S)(redox)) that is investigated using a series of substitutions of Asn(52) and/or Tyr(67). In double mutants Y67F/N52I Y67F/N52V, where most of the hydrogen bond network in the heme crevice is eliminated, Delta S(redox) compares to the wild type. This indicates that a fully consistent hydrogen bond network has a similar polarizability as an apolar matrix. We therefore argue that the variability in net dielectric susceptibility arises from conformational polarizability, a factor that is not a function of atomic properties and coordinates and is therefore hard to predict using conventional physical relationships.     

Regulated, stable expression and nuclear presence of retrovirus double-stranded RNA-binding protein sigma3 in HeLa cells.

Reovirus genome segment S4 codes for polypeptide sigma3, a major outer capsid component of virions and a double-stranded RNA (dsRNA)-binding protein implicated in viral cytopathogenesis. We have constructed a stable HeLa cell line (S4tTA) that produces functional sigma3 under tetracycline transactivator control. In the absence of tetracycline, S4tTA cells synthesized stable dsRNA-binding sigma3 that accumulated in the nucleus as well as in the cytoplasm. However, in induced S4tTA cells also expressing reovirus outer shell polypeptide mu1/mu1C, migration of sigma3 into the nucleus was blocked, probably as a result of formation of a complex with mu1/mu1C which was exclusively in the cytoplasm. Mutant analyses indicated a correlation between dsRNA-binding activity and nuclear entry of sigma3, suggesting an additional role(s) for this capsid protein in virus-cell interactions.     

Contributions of high- and low-quality patches to a metapopulation with stochastic disturbance

Studies of time-invariant matrix metapopulation models indicate that metapopulation growth rate is usually more sensitive to the vital rates of individuals in high-quality (i.e., good) patches than in low-quality (i.e., bad) patches. This suggests that, given a choice, management efforts should focus on good rather than bad patches. Here, we examine the sensitivity of metapopulation growth rate for a two-patch matrix metapopulation model with and without stochastic disturbance and found cases where managers can more efficiently increase metapopulation growth rate by focusing efforts on the bad patch. In our model, net reproductive rate differs between the two patches so that in the absence of dispersal, one patch is high quality and the other low quality. Disturbance, when present, reduces net reproductive rate with equal frequency and intensity in both patches. The stochastic disturbance model gives qualitatively similar results to the deterministic model. In most cases, metapopulation growth rate was elastic to changes in net reproductive rate of individuals in the good patch than the bad patch. However, when the majority of individuals are located in the bad patch, metapopulation growth rate can be most elastic to net reproductive rate in the bad patch. We expand the model to include two stages and parameterize the patches using data for the softshell clam, Mya arenaria. With a two-stage demographic model, the elasticities of metapopulation growth rate to parameters in the bad patch increase, while elasticities to the same parameters in the good patch decrease. Metapopulation growth rate is most elastic to adult survival in the population of the good patch for all scenarios we examine. If the majority of the metapopulation is located in the bad patch, the elasticity to parameters of that population increase but do not surpass elasticity to parameters in the good patch. This model can be expanded to include additional patches, multiple stages, stochastic dispersal, and complex demography.     

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

Matrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage-structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross-taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.     

Monoclonal antibodies to reovirus reveal structure/function relationships between capsid proteins and genetics of susceptibility to antibody action.

Thirteen newly isolated monoclonal antibodies (MAbs) were used to study relationships between reovirus outer capsid proteins sigma 3, mu 1c, and lambda 2 (core spike) and the cell attachment protein sigma 1. We focused on sigma 1-associated properties of serotype specificity and hemagglutination (HA). Competition between MAbs revealed two surface epitopes on mu 1c that were highly conserved between reovirus serotype 1 Lang (T1L) and serotype 3 Dearing (T3D). There were several differences between T1L and T3D sigma 3 epitope maps. Studies using T1L x T3D reassortants showed that primary sequence differences between T1L and T3D sigma 3 proteins accounted for differences in sigma 3 epitope maps. Four of 12 non-sigma 1 MAbs showed a serotype-associated pattern of binding to 25 reovirus field isolates. Thus, for reovirus field isolates, different sigma 1 proteins are associated with preferred epitopes on other outer capsid proteins. Further evidence for a close structural and functional interrelationship between sigma 3/mu 1c and sigma 1 included (i) inhibition by sigma 3 and mu 1c MAbs of sigma 1-mediated HA, (ii) enhancement of sigma 1-mediated HA by proteolytic cleavage of sigma 3 and mu 1c, and (iii) genetic studies demonstrating that sigma 1 controlled the capacity of sigma 3 MAbs to inhibit HA. These data suggest that (i) epitopes on sigma 3 and mu 1c lie in close proximity to sigma 1 and that MAbs to these epitopes can modulate sigma 1-mediated functions, (ii) these spatial relationships have functional significance, since removal of sigma 3 and/or cleavage of mu 1c to delta can enhance sigma 1 function, (iii) in nature, the sigma 1 protein places selective constraints on the epitope structure of the other capsid proteins, and (iv) viral susceptibility to antibody action can be determined by genes other than that encoding an antibody's epitope.     

Spike trains in a stochastic Hodgkin-Huxley system

We consider a standard Hodgkin-Huxley model neuron with a Gaussian white noise input current with drift parameter mu and variance parameter sigma(2). Partial differential equations of second order are obtained for the first two moments of the time taken to spike from (any) initial state, as functions of the initial values. The analytical theory for a 2-component (V,m) approximation is also considered. Let mu(c) (approximately 4.15) be the critical value of mu for firing when noise is absent. Large sample simulation results are obtained for mumu(c), for many values of sigma between 0 and 25. For the time to spike, the 2-component approximation is accurate for all sigma when mu=10, for sigma>7 when mu=5 and only when sigma>15 when mu=2. When mumu(c), most paths show similar behavior and the moments exhibit smoothly changing behavior as sigma increases. Thus there are a different number of regimes depending on the magnitude of mu relative to mu(c): one when mu is small and when mu is large; but three when mu is close to and above mu(c). Both for the Hodgkin-Huxley (HH) system and the 2-component approximation, and regardless of the value of mu, the CV tends to about 1.3 at the largest value (25) of sigma considered. We also discuss in detail the problem of determining the interspike interval and give an accurate method for estimating this random variable by decomposing the interval into stochastic and almost deterministic components.