Age-structured population growth rates in constant and variable environments: a near equilibrium approach

Abstract

General measures summarizing the shapes of mortality and fecundity schedules are proposed. These measures are derived from moments of probability distributions related to mortality and fecundity schedules. Like moments, these measures form infinite sequences, but the first terms of these sequences are of particular value in approximating the long-term growth rate of an age- structured population that is growing slowly. Higher order terms are needed for approximating faster growing populations. These approximations offer a general nonparametric approach to the study of life-history evolution in both constant and variable environments. These techniques provide simple quantitative representations of the classical findings that, with fixed expected lifetime and net reproductive rate, type I mortality and early peak reproduction increase the absolute magnitude of the population growth rate, while type III mortality and delayed peak reproduction reduce this absolute magnitude.

Introduction

The study of age-structured population growth has a long history in population biology. The most common formulation in the present day assumes discrete times and ages. Population growth is then modeled using a population projection matrix, which can be built from a life table (Leslie 1945, Leslie 1948; Bernadelli, 1941). Application of such models has been extensive, facilitated by the ease of numerical computation and the well-developed body of theory on nonnegative matrices. Continuous formulations, however, can achieve the same ends (Caswell, 2001; Charlesworth, 1994), but are not as widely used.

Most commonly, population projection matrices are applied with constant mortality and fecundity rates (vital rates), which means that population growth is density independent (or alternatively, the population is at equilibrium), and is not affected by environmental variability. The dominant eigenvalue of the projection matrix is then equal to the finite rate of increase (in essence the long-run growth rate), which also serves as a fitness measure. In the study of life-history evolution, some authors have alternatively proposed using expected lifetime reproduction as a fitness measure. However, this measure requires that the population under study is at equilibrium (Kozlowski, 1993).

With constant vital rates, a population reaches a stable age distribution and grows exponentially. This simply means that the population will become very large, if the growth rate is positive, and will become extinct, if the growth rate is negative. Although such exponential growth would not be expected to be sustained for long in nature, such situations are nevertheless highly important in understanding life-history evolution and also in understanding species coexistence. For example, the ESS approach to density-dependent life-history evolution relies critically on analyzing the long-term growth rates of variant types at low density in competition with other types. The growth of a low-density variant can be appropriately analyzed as independent of its own density. Moreover, of most interest is the boundary in parameter space between population increase and decrease, and so it is useful to have techniques that are valid when population growth rates are low. A similar situation arises in the study of species coexistence. There, the invasibility approach, which in different circumstances is applicable to establishing stochastically bounded coexistence (Chesson and Ellner, 1989; Ellner, 1989) or permanent coexistence (Law and Morton, 1996), also analyzes populations growing from low density. Density-dependent feedback within the population is minor, and again of most interest are boundaries in parameter space separating increasing and decreasing populations.

The vital rates of most organisms have some degree of age structure, i.e. the age-specific mortality and fecundity rates do in fact depend on age. The critical feature of the standard population projection matrix approach is that it allows this age specificity to be taken into account simply and naturally. The challenge, however, is to obtain general information on the effects of such age dependence on population growth. One approach has been to use sensitivity analysis. The effects of changing any particular vital rate by a small amount can then be assessed (Caswell, 1978). Using such approaches, evolution of senescence, optimum age of maturity, benefits of being annual, biennial, or perennial, and other similar questions have been explored (Stearns, 1992; Roff, 1992; Charlesworth, 1994; Oli and Dobson, 2003).

Questions like those above are indicative of the fact that general features of life histories are more likely to be determined genetically and subject to natural selection than the individual vital rates for particular age classes. Ideally, life-history evolution theory should connect such features to a fitness measure. However, in the absence of a quantitative measure to summarize the details of vital rates in different age classes, the theory cannot proceed in this way directly. Nevertheless, in some cases a trait that is subject to strong selection can be specified by a single parameter. Age at maturity is one example. To date multidimensional traits, such as mortality and fecundity schedules, have not been quantified in such a way that the selection pressure on them can be assessed in a general way. Instead, sensitivity analysis is commonly applied to a single vital rate, such as the death rate of a particular age class, and hence does not lead to general conclusions about the mortality schedule.

The question we address is how one investigates general properties of a vital-rate schedule, such as its shape. For example, mortality schedules are often classified into three general shapes (types I–III) depending on whether the plot of the log fraction surviving to a particular age is a straightline against age, or is a concave or convex function of age. One response to the question of the effects of the shape of a vital-rate schedule on population growth is to use macroparameter analysis. In macroparameter analysis, vital-rate schedules are given by parametric formulae and the effects of changing the parameters of these formulae are examined (Caswell, 1982). Here we take a more direct nonparametric approach that therefore does not depend on specifying vital-rate schedules by particular formulae. We ask, given an arbitrary mortality or fecundity schedule, can we find general measures of shape that characterize the effects of shape on population growth? Our results are a set of shape measures (which we call Δ-measures) that are analogous to moments or cumulants of a probability distribution. These measures allow us to characterize the effects of the shapes of a vital-rate schedule on population growth.

The proposed measures are most useful when population growth rate is low, and so it is a near equilibrium approach. However, it is important to keep in mind that in the ESS approach to life-history evolution, and in the invasibility approach to species coexistence, the chief interest is in determining the boundaries of regions of parameter space delineating positive and negative growth, as discussed above. Hence, a near equilibrium approach is perfectly applicable to this goal.

We anticipate that these shape measures of vital-rate schedules will lead to new insights into life-history evolution and into competitive coexistence (Dewi, 1998; Dewi and Chesson, 2003). We show here that they can be used also in situations where the vital rates change with time due to environmental variation. In that case, where projection matrices follow a Markov process, a quadratic approximation to the long-term population growth rate for small variance has been developed (Tuljapurkar 1982, Tuljapurkar 1990, Tuljapurkar 1997). Our vital-rate shape measures are applied to the situation where the total fecundity of the population varies with the environment, but the fecundity schedule retains its shape. Thus, the fecundities of different age classes are assumed to respond proportionately to their common varying environment, a situation that can be argued as a reasonable first approximation to reality.