Mathematical models on demography

Various mathematical functions have been used to interpret actuarial senescence, the most prominent of which are the Gompertz and the Weibull. These differ primarily in the manner in which age-independent (often called extrinsic) and age-dependent (intrinsic) death events are handled (Ricklefs and Scheuerlein 2002; Ricklefs 2008), though these sources are not as cleanly separated as may be implied. Both models begin with an assumed basal or initial mortality rate common to relatively young, healthy members of a population before senescent decline begins. As noted above, such mortality events are viewed as being essentially accidents occurring independently of age caused by such factors as predation, starvation, and adverse weather.

In the Gompertz, the relative physiological decline increases and the mortality rate increases exponentially as: m_x = m_oey^x implying that senescence is a multiple of initial mortality. The constant у is the exponential rate of increase in mortality rate with age; m_x (mortality rate at age x) and m_o (initial mortality rate) are instantaneous rates and expressed as time^-1. The mortality rate can range between 0 and infinity (in both models).

In contrast, in the Weibull the age-dependent component is added to the initial mortality rate as: m_x = m_o + ахв implying that the causes of death in a young population are different and independent from those affecting older members. The initial or extrinsic factors are the same as in the Gompertz; this rate is supplemented by a separate category (represented by the term ахв) of age-dependent deaths due to intrinsic dysfunction and disease (cancer, strokes, etc.), as well as deaths from increased vulnerability to the extrinsic factors by virtue of the disabilities of old age (ineffectual escape from predators, etc.; see examples above). Thus, here the relative rate of increase in mortality rate is age-dependent and slows with increasing age (for details on both models, see Ricklefs and Scheuerlein 2002).

These Gompertz and Weibull are examples of relatively simple models and have their proponents and critics. They need to be interpreted in the context of the usual simplifications, caveats, and assumptions inherent in model building (see e.g., Levins 1966; Chap. 1 in Levins 1968; Abrams and Ludwig 1995; Pedersen 1999; Williams 1999). Clearly, these models as applied to senescence are being used to represent the demography of discrete, unitary individuals such as humans and birds. They do not lend themselves well to the population biology of clonal or modular organisms where what constitutes ‘the individual’ is more-or- less abstract and where it is operationally difficult or impossible to separate parents from offspring. Furthermore, the models focus on survival and ignore the fecundity implications of senescence.

In summary, whereas the only complete actuarial data are for humans, substantial demographic evidence exists for certain other higher organisms such as zoo and domestic animals, many annual and some perennial plants (such as a few of the shorter lived trees), and for selected species used in research. These cases are discussed in the following section. Elsewhere, data are few and conclusions have been drawn largely by extrapolation and inference, occasionally based only on age of an organism at death. Life span is not the same as senescence. Although some actuarial evidence can be compiled from existing records or by inference from mortality tallied across all age groups at one time (Comfort 1979 p. 53), the best method would be, where possible, to follow over time survivorship and fecundity of a particular age cohorts under controlled and uncontrolled conditions (however, see Rose 1991, pp. 21-28 on laboratory artifacts).

Usually this is technically difficult and sometimes operationally impossible to do. As will be apparent from the following sections, survivorship and fecundity over time are tracked in some plant studies, but, with notable exceptions (e.g., Gustafsson and Part 1990), rarely for animal populations (or microbial clones) in nature. Variation among and within taxa in the occurrence of senescence is considerable, suggesting genetic variation plays a significant role.