Gordon A. Fox

Population Numbers Count: Tools for Near‐Term Demographic Analysis

How do unit or proportional changes in vital rates affect populations in the short term? We present a new extension to standard methods of matrix model analysis that allows us to answer this question for the first time. By using the sensitivities of all the eigenvalues/vectors, rather than just the leading eigenvalue/vector pair, we can predict the consequences of unit or proportional changes in vital rates to population size and structure at any arbitrary time, not just when populations have neared their stable distribution. These extensions are particularly important in studying populations subject to frequent disturbance, where stable growth rate and stable distribution do not provide sufficient information about the effects of changes in the vital rates; managed populations in which short‐term goals are defined; and the adequacy of the underlying matrix model for either short‐ or long‐term understanding. We use analysis of empirical data on the cactus Coryphantha robbinsorum to demonstrate this approach and show that short‐term predictions can differ substantially from those based on standard, asymptotic, analysis.

Chaos and evolution

There is growing interest in applying nonlinear methods to evolutionary biology. With good reason: the living world is full of nonlinearities, responsible for steady states, regular oscillations, and chaos in biological systems. Evolutionists may find nonlinear dynamics important in studying short-term dynamics of changes in genotype frequency, and in understanding selection and its constraints. More speculatively, dynamical systems theory may be important because nonlinear fluctuations in some traits may sometimes be favored by selection, and because some long-run patterns of evolutionary change could be described using these methods.

Demographic stochasticity and the variance reduction effect

Demographic stochasticity is almost universally modeled as sampling var- iance in a homogeneous population, although it is defined as arising from random variation among individuals. This can lead to serious misestimation of the extinction risk in small populations. Here, we derive analytical expressions showing that the misestimation for each demographic parameter is exactly (in the case of survival) or approximately (in the case of fecundity) proportional to the among-individual variance in that parameter. We also show why this misestimation depends on systematic variation among individuals, rather than random variation. These results indicate that correctly assessing the importance of demo- graphic stochasticity requires (1) an estimate of the variance in each demographic parameter; (2) information on the qualitative shape (convex or concave) of the mean-variance rela- tionship; and (3) information on the mechanisms generating among-individual variation. An important consequence is that almost all population viability analyses (PVAs) over- estimate the importance of demographic stochasticity and, therefore, the risk of extinction.

Population Dynamic Consequences of Competitive Symmetry in Annual Plants

Asymmetric competition is a form of resource division among plants, in which large plants greatly suppress the growth of smaller neighbors. In annual plants, small size differences between seedlings at the onset of competition are magnified into large differences in seed-set by asymmetric competition. We formulate a novel neighborhood model, which reflects this seedling size effect as modified by the type of competitive symmetry. In the model, competition type is represented by a single, biologically meaningful parameter. We implement the model in a population growth model for two species, one at low density (the invader), and one at high density (the resident). The species are the same, except for their seedling biomass distributions. Under these conditions, we find that asymmetric competition always favors invasion by the species with lager average seedling size, but impairs invasion by the other species. Based on this invasibility criterion, we conclude that asymmetric competition always favors competitive exclusion in our model. However, by modifying some of the model assumptions, we suggest scenarios in which asymmetric competition may promote coexistence

The Underpinnings of the Relationship of Species Richness with Space and Time

Various ecological mechanisms influence the forms of species richness relationships (SRRs). These mechanisms can be gathered under five general categories: more individuals, environmental heterogeneity, dispersal limitations, biotic interactions, and multiple species pools. Often only the first two categories are discussed. In contrast, we examine all five and explore how they can influence the form of SRRs. We discuss how various sampling schemes and methods of SRR construction can be used to gain insight about how various processes influence species richness patterns. The field is ripe for probing these effects through more complex simulation models or more sophisticated mathematical approaches. To facilitate deeper understanding, we need to embrace the full spectrum of SRRs and reconsider the assumed common knowledge about the functional form of SRRs. The relationship between species richness and the space or time over which it is sampled has received increasing attention over the past decade, resultin...

EXTINCTION RISK OF HETEROGENEOUS POPULATIONS

The extinction of small populations is a stochastic process, affected by both environmental variation and chance variation in the fates of individuals (demographic stochasticity). Here I examine how population extinction risk is affected by variation in the underlying individual phenotypes, using a branching-process approach. I define the long-term individual extinction risk as the chance of ultimately leaving no descendants, and the cumulative individual extinction risk as the chance of leaving no descendants by a specified time. I use these to show that if there is a phenotypic correlation between parents and their offspring, variation in these quantities always reduces both the long- and short-term population extinction risk. Such variation in individual extinction risk arises from individual variation in demographic parameters and may have both genetic and environmental causes. Using a well-known approximation of the difference between the log arithmetic and log geometric means, I derive expressions for the sensitivity and elasticity of the approximate log extinction risk to changes in the mean and variance of the individual extinction risk, and to changes in population size. One conclusion is that increasing the variance among individuals in extinction risk can sometimes be at least as important in reducing population extinction risk as increasing the population size itself. These analyses also point to reasons why changes in environmental factors (e.g., toxicants) or management practices may have either larger or smaller effects than would be anticipated by considering the change in the mean risk alone.

Demographic heterogeneity impacts density-dependent population dynamics

Among-individual variation in vital parameters such as birth and death rates that is unrelated to age, stage, sex, or environmental fluctuations is referred to as demographic heterogeneity. This kind of heterogeneity is prevalent in ecological populations, but is almost always left out of models. Demographic heterogeneity has been shown to affect demographic stochasticity in small populations and to increase growth rates for density-independent populations. The latter is due to “cohort selection,” where the most frail individuals die out first, lowering the cohort’s average mortality as it ages. The importance of cohort selection to population dynamics has only recently been recognized. We use a continuous-time model with density dependence, based on the logistic equation, to study the effects of demographic heterogeneity in mortality and reproduction. Reproductive heterogeneity is introduced in three ways: parent fertility, offspring viability, and parent–offspring correlation. We find that both the low-density growth rate and the equilibrium population size increase as the magnitude of mortality heterogeneity increases or as parent–offspring phenotypic correlation increases. Population dynamics are affected by complex interactions among the different types of heterogeneity, and trade-off scenarios are examined which can sometimes reverse the effect of increased heterogeneity. We show that there are a number of different homogeneous approximations to heterogeneous models, but all fail to capture important parts of the dynamics of the full model.

Variation among individuals in cone production in Pinus palustris (Pinaceae)

PREMISE OF THE STUDY Reproductive output varies considerably among individuals within plant populations, and this is especially so in cone production of conifers. While this variation can have substantial effects on populations, little is known about its magnitude or causes. METHODS We studied variation in cone production for 2 years within a population of Pinus palustris Mill. (longleaf pine; Pinaceae). Using hurdle models, we evaluated the importance of burn treatments, tree size (dbh), canopy status (open, dominant, subordinate), and number of conspecific neighbors within 4 m (N(4)). KEY RESULTS Cone production of individuals-even after accounting for other variables-was strongly correlated between years. Trees in plots burned every 1, 2, or 5 years produced more cones than those burned every 7 years, or unburned. Larger trees tend to produce more cones, but the large effects of the other factors studied caused substantial scatter in the dbh-cone number relationship. Among trees in the open, dbh had little explanatory power. Subordinate trees with three neighbors produced no cones. CONCLUSIONS Tree size alone was a weak predictor of cone production. Interactions with neighbors play an important role in generating reproductive heterogeneity, and must be accounted for when relating cone production to size. The strong between-year correlation, together with the large variance in cone production among trees without neighbors, suggests that still more of the variance may be explainable, but requires factors outside of our study.

PERENNATION AND THE PERSISTENCE OF ANNUAL LIFE HISTORIES

Individuals of Eriogonum abertianum that live for more than a year in the wild (perennators) have offspring that live longer under greenhouse conditions than do the offspring of the general population. This suggests some genetic basis to the occasional perennation observed in the wild. Longevity appears to be continuously distributed, with the perennators representing the tail of the populations distribution. Could such apparent genetic variation for life-history traits potentially lead to the evolution of a perennial or biennial population? I examined the persistence of the annual habit using Landes model of life-history evolution and a model of size-dependent demography. Results suggest that for populations with very low adult survival rates, selection generally acts to increase fecundity and the proportion of individuals surviving to maturity, thereby reducing perennation. The annual habit is thus likely to persist even in the presence of genetic variation for perennation.

Sources of variation in growth, form, and survival in dwarf and normal-stature pitch pines (Pinus rigida, Pinaceae) in long-term transplant experiments

Determining the relative contributions of genetic and environmental factors to phenotypic variation is critical for understanding the evolutionary ecology of plant species, but few studies have examined the sources of phenotypic differentiation between nearby populations of woody plants. We conducted reciprocal transplant experiments to examine sources of variation in growth rate, form, survival, and maturation in a globally rare dwarf population of pitch pine (Pinus rigida) and in surrounding populations of normal-stature pitch pines on Long Island, New York. Transplants were monitored over a 6-yr period. The influence of seedling origin on height, growth rate, survival, and form (single-stemmed vs. multi-stemmed growth habit) was much smaller than the effect of transplanting location. Both planting site and seed origin were important factors in determining time to reproduction; seedlings originating from dwarf populations and seedlings planted at the normal-stature site reproduced earliest. These results suggest that many of the differences between dwarf and normal-stature pitch pines may be due more to plastic responses to environmental factors than to genetic differentiation among populations. Therefore, preservation of the dwarf pine habitat is essential for preserving dwarf pine communities; the dwarf pines cannot be preserved ex situ.