Michael G. Neubert

DEMOGRAPHY AND DISPERSAL: CALCULATION AND SENSITIVITY ANALYSIS OF INVASION SPEED FOR STRUCTURED POPULATIONS

A fundamental characteristic of any biological invasion is the speed at which the geographic range of the population expands. This invasion speed is determined by both population growth and dispersal. We construct a discrete-time model for biological invasions that couples matrix population models (for population growth) with integrodifference equa- tions (for dispersal). This model captures the important facts that individuals differ both in their vital rates and in their dispersal abilities, and that these differences are often determined by age, size, or developmental stage. For an important class of these equations, we demonstrate how to calculate the populations asymptotic invasion speed. We also derive formulas for the sensitivity and elasticity of the invasion speed to changes in demographic and dispersal parameters. These results are directly comparable to the familiar sensitivity and elasticity of population growth rate. We present illustrative examples, using published data on two plants: teasel (Dipsacus sylvestris) and Calathea ovandensis. Sensitivity and elasticity of invasion speed is highly correlated with the sensitivity and elasticity of pop- ulation growth rate in both populations. We also find that, when dispersal contains both long- and short-distance components, it is the long-distance component that governs the invasion speed-even when long-distance dispersal is rare.

ALTERNATIVES TO RESILIENCE FOR MEASURING THE RESPONSES OF ECOLOGICAL SYSTEMS TO PERTURBATIONS

Resilience is a component of ecological stability; it is assessed as the rate at which perturbations to a stable ecological system decay. The most frequently used estimate of resilience is based on the eigenvalues of the system at its equilibrium. In most cases, this estimate describes the rate of recovery only asymptotically, as time goes to infinity. However, in the short term, perturbations can grow significantly before they decay, and eigenvalues provide no information about this transient behavior. We present several new measures of transient response that complement resilience as a description of the response to perturbation. These indices measure the extent and duration of transient growth in models with asymptotically stable equilibria. They are the reactivity (the maximum possible growth rate immediately following the perturbation), the maximum amplification (the largest proportional deviation that can be produced by any perturbation), and the time at which this amplification occurs. We demonstrate the calculation of these indices using previously published linear compartment models (two models for phosphorus cycling through a lake ecosystem and one for the flow of elements through a tropical rain forest) and a standard nonlinear predator-prey model. Each of these models exhibits transient growth of perturbations, despite asymptotic stability. Measures of relative stability that ignore transient growth will often give a misleading picture of the response to a perturbation.

Complexity in Ecology and Conservation: Mathematical, Statistical, and Computational Challenges

Abstract Creative approaches at the interface of ecology, statistics, mathematics, informatics, and computational science are essential for improving our understanding of complex ecological systems. For example, new information technologies, including powerful computers, spatially embedded sensor networks, and Semantic Web tools, are emerging as potentially revolutionary tools for studying ecological phenomena. These technologies can play an important role in developing and testing detailed models that describe real-world systems at multiple scales. Key challenges include choosing the appropriate level of model complexity necessary for understanding biological patterns across space and time, and applying this understanding to solve problems in conservation biology and resource management. Meeting these challenges requires novel statistical and mathematical techniques for distinguishing among alternative ecological theories and hypotheses. Examples from a wide array of research areas in population biology and community ecology highlight the importance of fostering synergistic ties across disciplines for current and future research and application.

Invasion speeds in fluctuating environments

Biological invasions are increasingly frequent and have dramatic ecological and economic consequences. A key to coping with invasive species is our ability to predict their rates of spread. Traditional models of biological invasions assume that the environment is temporally constant. We examine the consequences for invasion speed of periodic and stochastic fluctuations in population growth rates and in dispersal distributions.

When Can Herbivores Slow or Reverse the Spread of an Invading Plant? A Test Case from Mount St. Helens

Here we study the spatial dynamics of a coinvading consumer‐resource pair. We present a theoretical treatment with extensive empirical data from a long‐studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant‐herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low‐density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of “invasion momentum,” beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.

FIRE INCREASES INVASIVE SPREAD OF MOLINIA CAERULEA MAINLY THROUGH CHANGES IN DEMOGRAPHIC PARAMETERS

We investigated the effects of fire on population growth rate and invasive spread of the perennial tussock grass Molinia caerulea. During the last decades, this species has invaded heathland communities in Western Europe, replacing typical heathland species such as Calluna vulgaris and Erica tetralix. M. caerulea is considered a major threat to heathland conservation. In 1996, a large and unintended fire destroyed almost one-third of the Kalmthoutse Heide, a large heathland area in northern Belgium. To study the impact of this fire on the population dynamics and invasive spread of M. caerulea, permanent monitoring plots were established both in burned and unburned heathland. The fate of each M. caerulea individual in these plots was monitored over four years (1997-2000). Patterns of seed dispersal were inferred from a seed germination experiment using soil cores sampled one month after seed rain at different distances from seed-producing plants. Based on these measures, we calculated projected rates of spread for M. caerulea in burned and unburned heathland. Elasticity and sensitivity analyses were used to determine vital rates that con- tributed most to population growth rate, and invasion speed. Invasion speed was, on average, three times larger in burned compared to unburned plots. Dispersal distances on the other hand, were not significantly different between burned and unburned plots indicating that differences in invasive spread were mainly due to differences in demography. Elasticities for fecundity and growth of seedlings and juveniles were higher for burned than for unburned plots, whereas elasticities for survival were higher in unburned plots. Finally, a life table response experiment (LTRE) analysis revealed that the effect of fire was mainly contributed by increases in sexual reproduction (seed production and germination) and growth of seed- lings and juveniles. Our results clearly showed increased invasive spread of M. caerulea after fire, and call for active management guidelines to prevent further encroachment of the species and to reduce the probability of large, accidental fires in the future. Mowing of resprouted plants before flowering is the obvious management tactic to halt massive invasive spread of the species after fire.

Sex-Biased Dispersal and the Speed of Two-Sex Invasions

Population models that combine demography and dispersal are important tools for forecasting the spatial spread of biological invasions. Current models describe the dynamics of only one sex (typically females). Such models cannot account for the sex-related biases in dispersal and mating behavior that are typical of many animal species. In this article, we construct a two-sex integrodifference equation model that overcomes these limitations. We derive an explicit formula for the invasion speed from the model and use it to show that sex-biased dispersal may significantly increase or decrease the invasion speed by skewing the operational sex ratio at the invasion’s low-density leading edge. Which of these possible outcomes occurs depends sensitively on complex interactions among the direction of dispersal bias, the magnitude of bias, and the relative contributions of females and males to local population growth.

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.

Reactivity and transient dynamics of discrete-time ecological systems

Most studies of ecological models focus exclusively on the asymptotic stability properties of equilibria. However, short-term transient effects can be important, and can in some cases dominate the dynamics seen in experimental or field studies. The reactivity of a stable equilibrium point measures the potential for short-term amplification of perturbations. The reactivity of a fixed point in a discrete-time system is given by the natural logarithm of the largest singular value of the Jacobian matrix of the linear approximation near the fixed point. If the reactivity is positive, the fixed point is said to be reactive. Here we examine the reactivity of discrete-time predator–prey models and density-dependent matrix population models. We find reactivity to be common (but not universal) and sometimes extremely high. Predator–prey or food web models that include a predator whose per-capita growth rate depends on the density of its prey, but not on its own density, are a special case. Any positive equilibrium of such a model must be reactive. Reactivity of discrete-time models depends on the timing of the census relative to the timing of reproduction. Perturbation analysis of singular values can be used to calculate the sensitivity and elasticity of reactivity to changes in model parameters. We conclude that transient amplification of perturbations should be a common ecological phenomenon. The interaction of these transient effects with the asymptotic nonlinear dynamics warrants further study.

A guide to calculating discrete-time invasion rates from data

One measure of biological invasiveness is the rate at which an established invader will spread spatially in its new environment. Slow spread signifies slow increase in ecological impact, whereas fast spread signifies the converse. If one can predict spread rates from life history attributes, such as growth rates and dispersal distances, then potential invasiveness can be assessed before the invasion occurs. A prediction of this sort requires models for population spread. As outlined below, such models have a long and distinguished history in quantitative ecology. Whereas early mathematical models for population spread were primarily conceptual and qualitative in nature, a new generation of realistic models is emerging. These new models are tied directly to the demography and dispersal of individuals. However, there are new challenges in the linking of these models to the biological processes. As we will illustrate in this chapter, spread rate predictions are very sensitive to assumptions about long-distance dispersal. Are there robust methods for estimating spread rates? This is one question we will address. Furthermore almost all mathematical models assume that the spread occurs in one spatial dimension, along a line. This is not because mathematicians have not noticed that most