The Sensitivity of Population Growth Rate: Three Approaches
The essence of stable population theory is the fact that a population subject to time invariant vital rates will (with a few exceptions not of interest here) converge to a stable structure and grow exponentially at a constant rate (the population growth rate, or intrinsic rate of increase).
The essence of stable population theory is1 the fact that a population subject to time-invariant vital rates will (with a few exceptions not of interest here) converge to a stable structure and grow exponentially at a constant rate (the population growth rate, or intrinsic rate of increase). The calculation of the population growth rate from the vital rates is one of the most important accomplishments of formal demography (Sharpe and Lotka 1911).2 Ecologists recognized early on that, by integrating survival and fertility over the life course, the population growth rate provided a powerful tool for describing the population consequences of environmental conditions (e.g., Birch 1953). For the same reason, evolutionary biologists recognized it as a measure of fitness (Fisher 1930), although that concept requires careful consideration of both demographic and genetic processes (Charlesworth 1994; de Vries and Caswell 2018).
This makes the sensitivity analysis of population growth rate an important problem. It has been approached in three ways. The earliest approach (Hamilton 1966) is specific to age-classified models, and relies on differentiation of the characteristic equation. The second (Caswell 1978) applies to stage-classified as well as age-classified models, and uses eigenvalue perturbation theory. The third is based on matrix calculus and is more flexible than its predecessors.
3.2 Hamilton’s Equation for Age-Classified Populations
Sensitivity of r
Hamilton’s results are obtained by implicit differentiation of the Euler-Lotka equation (3.2). We will derive Hamilton’s original formulation and then show how it reduces to the relation between the stable age distribution and the reproductive value distribution in (3.7) and (3.8).
3.2.1 Effects of Changes in Mortality
3.2.2 Effects of Changes in Fertility
3.2.3 History and Perspectives
Hamilton (1966) obtained the relationship (3.22) in his analysis of the evolution of senescence. From (3.22) and (3.8) it is apparent that (provided r ≥ 0) the magnitudes of the sensitivities of r to mortality and fertility decline with age. These sensitivities measure the selection gradients on age-specific mortality and fertility. Thus Hamilton concluded that the strength of selection against deleterious mutations would necessarily decline with their age of action, that small positive effects at early ages could easily compensate for much larger negative effects at later ages, and that the evolution of senescence was therefore inevitable.
In the years that followed Hamilton’s paper, several other authors developed perturbation analysis for r, using related methods. Demetrius (1969) used a discrete age-classified model, and Emlen (1970) used Hamilton’s results to derive the dynamics of gene frequencies resulting from the selection gradients on age-specific survival and fertility.
Keyfitz (1971) in a remarkable paper, used implicit differentiation to obtain the sensitivity of population growth rate, life expectancy, birth rates, death rates, and the stable age distribution, apparently independently of Hamilton. He noted the appearance of reproductive value in the sensitivity of r to mortality. Goodman (1971) was apparently the first to note that the sensitivities of r to mortality and fertility could be expressed in terms of the stable age distribution and reproductive value.
When Hamilton’s paper appeared, it was regarded as difficult and esoteric, but it had a great impact. It provided the analytical machinery for examining trade-offs between opposing demographic traits, known as antagonistic pleiotropy (Williams 1957; Rose 1991). It also describes the accumulation of deleterious mutations due to the balance between mutation and selection (e.g., Steinsaltz et al. 2005). These ideas are fundamental to the analysis of human aging (e.g., Rose 1991; Wachter and Finch 1997; Carey and Tuljapurkar 2003; Baudisch 2008) and, more generally, the analysis of life history evolution in humans and other species (e.g., Charlesworth 1994; Stearns 1992).
3.3 Stage-Classified Populations: Eigenvalue Perturbations
Implicit in Hamilton’s analysis is the assumption that the vital rates are functions of age. In many cases, they are not. In humans, characteristics such as education, marital status, health status, or spatial location, may provide important information in addition to age. In other species, the vital rates may depend on developmental stage or body size more than on age. Such populations are described by stage-classified demographic models, of which the age-classified theory is a special case.
Sensitivity of λ
3.3.1 Age-Classified Models as a Special Case
3.3.2 Sensitivity to Lower-Level Demographic Parameters
I first encountered the basis for this perturbation expansion in a paper by C.A. Desoer in the proceedings of an engineering conference (Desoer 1967).3 Eigenvalue perturbations were of particular interest to engineers in the 1960s as part of a shift from frequency-domain methods to state-space methods in the study of linear systems (Zadeh and Desoer 1963). However, the result dates back to Jacobi (1846), and has been independently rediscovered many times (e.g., Faddeev 1959; Papoulis 1966; Franklin 1968). In population biology, this perturbation approach has been extended to many other sensitivity problems, including the sensitivity of subdominant eigenvalues and transient behavior, of growth rates in periodic and stochastic environments, of the eigenvectors, and of the spreading speed in biological or demographic invasions (see Caswell (2001) for reviews and references).
3.4 Growth Rate Sensitivity via Matrix Calculus
Matrix calculus provides a still more general approach to the sensitivity analysis of the population growth rate. Equation (3.29) perturbs only a single entry of A; derivatives with respect to other parameters are assembled by summing their effects over all the entries of A, as in (3.41). Using matrix calculus, we now consider λ as a scalar function of A and A as a matrix-valued function of a parameter vector θ.
Sensitivity of λ
3.5 Second Derivatives of Population Growth Rate
The second derivatives of λ measure the curvature of the response to changes in parameters. They have important applications in evolutionary demography, where they indicate the action of stabilizing, disruptive, or correlational selection on fitness-related traits (e.g., Phillips and Arnold 1989; Caswell 2001), in adaptive dynamics, where they help determine the stability of evolutionary singular strategies (e.g., Diekmann 2004), and in extending sensitivity analysis to second-order effects.
Since the first derivatives of λ are written, in Eqs. (3.29) and (3.46), in terms of the right and left eigenvectors of A, the second derivatives of λ require the first derivatives of those eigenvectors. Caswell (1996) derived the second derivatives of λ to entries of A by an extension of the method in Sect. 3.3. However, a more general and rigorous method is available using matrix calculus.
Each of the three approaches to growth rate sensitivity, leading to Eqs. (3.7), (3.8), (3.29), and (3.42), uses its own analytical methods. They agree, however, in showing how the sensitivity of population growth rate can be written in terms of the stable stage distribution and the reproductive value. In general, the effect of a change in the rate at which individuals move from stage j to stage i is proportional to the abundance of the origin stage (j) and the reproductive value of the destination stage (i). If a transition yields individuals with low reproductive value, or if few individuals are available to experience the change in the rate of transition, the effect on population growth will be small.
Chapter 3 is modified, under the terms of a Creative Commons Attribution License, from Caswell, H. 2010. Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates. Demographic Research 23:531–548, ⒸHal Caswell.
Leonard Euler had obtained the result in 1760, but his derivation rediscovered until 1970 (Keyfitz and Keyfitz 1970).
By a fortunate accident; I was searching for something completely different. We may wonder whether the chances of such coincidences are higher or lower in the internet search era.
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