Applications to conservation biology

This chapter describes how models can aid in managing populations to prevent extinction, given uncertainty about their state. From previous chapters, it is clear that avoiding extinction requires keeping both abundance and the replacement rate high. However, for both, the question remains, how high? The question of how high abundance should be to achieve a certain risk is addressed by existing population viability analyses (PVA). By contrast, the problem of maintaining high replacement has received little attention. This chapter describes how uncertainty in population parameters and the frequency spectrum of the environment both affect estimates of the probability of extinction, including examples of PVAs that pay greater attention to those complications. Additionally, an example is provided of tracking both abundance and replacement to avoid extinction for many different populations of a single taxon, Pacific salmon. Finally, the role of portfolio effects (diversity in variance among populations) is explored.

Age-structured models in a random environment

Most ecological populations exist in a randomly fluctuating environment, and these fluctuations influence vital rates, thus changing population dynamics. These changes are the focus of this chapter. The primary practical concern about environmental variability is the possibility that it could cause a population to go extinct, so the chapter describes several approaches to estimating the probability of extinction. The first is the small fluctuations approximation (SFA) to describe the growth of a population with a randomly varying Leslie matrix. The results reveal that randomly varying populations grow more slowly on average than the equivalent deterministic population. Further applications of the SFA examine how correlated variation in different vital rates affects the probability of extinction, when variability is too large to use the SFA, and how it has been applied to population time series. Finally, several other approaches to estimating extinction risk—also known as population viability analysis—are compared.

Size-structured models

This chapter begins by revisiting the M’Kendrick/von Foerster model, but using size instead of age as the state variable. It then uses the lessons from that model to describe how individual growth and mortality rates determine both stand distributions (a population of mixed ages) and cohort distributions (all one age). In particular, incorporating variability in growth trajectories is shown to be important in obtaining realistic results—though it is not without pitfalls. Ultimately, the numerical calculations required to model size-structured populations for future projections are more challenging than those needed for age structure, so the chapter closes by discussing some mathematical tools that have been developed to accomplish this. These include the integral projection model, a recent approach that is very useful because, while more complex, it has a lot in common with the age-structured models examined in Chapters 3 and 4.

Simple population models

This chapter introduces basic concepts in population modeling that will be applied throughout the book. It begins with the oldest example of a population model, the rabbit problem, which was described by Leonardo of Pisa (“Fibonacci”) and whose solution is the Fibonacci series. The chapter then explores what is known about simple models of populations (i.e. those with a single variable such as abundance or biomass). The two major classes are: (1) linear models of exponential (or geometric) growth and (2) models of logistic, density-dependent growth. It covers both discrete time and continuous time versions of each of these. These simple models are then used to illustrate several different population dynamic concepts: dynamic stability, linearizing nonlinear models, calculation of probabilities of extinction, and management of sustainable fisheries. Each of these concepts is discussed further in later chapters, with more complete models.

Stage-structured models

This chapter moves to models in which developmental stage is the individual state variable, and abundance at each stage is the population variable. Stage is a period within an individual life history (e.g. juvenile, adult); organisms may survive within a stage or “grow” to other stages. This movement and survival is represented by a projection matrix that describes the transitions between stages over time. The projection matrix is similar to the Leslie matrix for age-structured models (Chapter 3), except it has entries other than just those in the first row and the sub-diagonal. Stage models are conceptually problematic because real population dynamics ultimately depend on the age distribution within each stage category. Stage-based models obscure that age structure, thus stage is not an adequate expression of state (Chapter 1). This chapter demonstrates how this introduces artifacts in model analysis, particularly of transients, and presents some ways to avoid those artifacts.

Spatial population dynamics

This chapter considers populations structured in a different dimension: space. This begins by representing population dynamics with a spatial continuity equation (analogous to the M’Kendrick/von Foerster model for continuity in age or size). If organisms move at random, this motion can be approximated as diffusion. This proves useful for modeling spreading populations, such as the expansion of sea otter populations along the California coast. Adding directional advection represents a population in a flowing stream. Metapopulation models are then introduced using a simple model of the fraction of occupied patches; these are made more realistic by accounting for inter-patch distance using incidence function models. The next level of complexity is models with population dynamics in each patch. These are used to examine how metapopulations can persist as a network even if no patch would persist by itself. Finally, the consequences of synchrony (or lack thereof) among spatially separated populations is described.

Age-structured models with density-dependent recruitment

This chapter examines age-structured models with density-dependent recruitment. In particular, it focuses on populations with over-compensatory density dependence, such as may occur due to cannibalism or some types of space competition. When the slope (at the equilibrium point) of the relationship between egg production and subsequent recruitment is declining in an over-compensatory way, the population may exhibit unstable limit cycles with period twice the generation time (2T). These cycles occur when that slope is steeply negative and the spawning age distribution has a high mean and low width. These results are applied to study the behavior of cycles in the U.S. west coast Dungeness crab fishery, variability in populations of an intertidal barnacle, and cycles in populations of a pest, the flour beetle. Additionally, it is shown how single-sex harvesting and compensatory growth affect population cycles and equilibria.

Thinking about populations

This chapter describes some general ways of thinking about population dynamics as a coherent body of interrelated advances in understanding, as developed in the first eleven chapters. The first generality is that understanding population dynamics depends on the concept of a state variable. It also underscores that dynamics depend on time delays, thus on age, even though demographic rates may ultimately depend more on size than age. Replacement plays a general role in dynamics, influencing both behavior at equilibrium and the time scales at which populations respond to environmental variability. Replacement even retains its importance in spatial models, where it is related to the same replacement threshold as in non-spatial models. Additionally, the way in which the distribution of reproduction (i.e. replacement) over age “filters” environmental variability influences extinction probabilities. The chapter closes by discussing how lessons from this book will guide conservation and management in a changing climate.

Population dynamics in marine conservation

This chapter traces the evolution of models for fishery management, focusing on the problem of maintaining both replacement and a desirable level of yield. Early models from the 1950s led to management for maximum sustainable yield (MSY). Later, recruitment and egg production data from populations at low abundance were used to set critical replacement thresholds (CRT) using age-structured models. Modern fisheries are managed by control rules to avoid both overfishing (replacement too low) and being overfished (abundance too low). With the advent of spatial management through marine protected areas (MPA), strategic models showed that MPAs and conventional management essentially had the same effects on fishery yield. More realistic, spatially explicit, tactical models include more detail but produce results that support the conclusions of the strategic models. Finally, the growing science of adaptive management of MPAs has been based on the understanding of transient population dynamics from Chapter 4.

Age-structured models: Short-term transient dynamics

Linear age-structured models eventually grow geometrically, and reach a stable age distribution (as in Chapter 3). This chapter describes what happens before “eventually.” That is, it describes the short-term, “transient” dynamics that occur when a population is perturbed, then begins to return to its stable distribution. Transients involve eigenvalues other than the largest (real) one, so the chapter begins by showing how complex eigenvalues can produce population cycles. It then addresses factors that make transients shorter or longer. In some cases, frequent environmental disturbances may prevent populations from ever reaching equilibrium. That scenario can be described by switching from linear models to linearized models varying about an equilibrium. The chapter describes temporal characteristics of that variability (such as time scales and frequencies), which require new tools: Fourier transforms and wavelets. These reveal how age-structured populations are more sensitive to certain environmental frequencies than to others, a phenomenon termed cohort resonance.