## Survivorship Curves and Life Tables

Experimental data on mortality can be collected by following a cohort of individuals from egg (or birth) to death and recording the number of survivors at each age. This method can be easily applied to domestic animals, laboratory populations, and nonmoving organisms in natural populations that can be tagged and followed through life. Studying survival in natural populations of moving animals is more challenging and may require enclosures to prevent migration, capture-recapture methods, or distal tracking equipment. Another approach, which is applicable only to populations with an equilibrium age structure, is to infer survival rates from the age distribution in sampled living or dead organisms if their organs are suitable for identification of age. Examples of such organs are stems of trees, scales of fishes, horns of sheep, roots of canine teeth in bears, and otholits of fishes. The weight ofthe eye lens can be used for age measurement in some animal species. If the population is stationary, then the number of new-born organisms x time units ago was the same as now; and survivors of that group of organisms are of age x. Thus, lx = N(x)/N(0), where lx is the survival probability, and N(x) is the number of live organisms of age x. If available data consist of the age distribution of animal carcasses, and we assume that the probability of finding a carcass does not depend on age, then lx can be calculated as follows. Because animals that survived to age x died at age x or later, lx = M(x)/M(0), where M(x) is the number of carcasses of age >x. In nonstationary populations, these equations can be adjusted by the rate of population increase or decrease.

Survival probability lx is often plotted against age x. These graphs which are called 'survivorship curves' indicate mortality rates throughout the life span. For example, mortality of domestic sheep is low at young ages and increases when animals become old (Figure 1a). Humans have a similar shape of survivorship curve. In contrast, the survivorship curve for lapwings or green plovers (Vanellus vanellus) in Britain declines exponentially which means that mortality rate is almost uniform at all ages. Survivorship curves become more informative if plotted in the log scale (Figure 1b). In this case, mortality rate is equal to the slope of the line with a negative sign, and it is clearer that the mortality of lapwings is constant, whereas sheep mortality increases with age.

Life tables integrate information on survival and reproduction of organisms which is necessary for building dynamical models of age-structured populations (e.g., Leslie model). Here we focus only on the survival/mortality component of life tables and ignore reproduction. The most common are life tables with discrete and uniform time steps. Depending on the life span of an organism, time steps may be years, weeks, or days. Age-specific survival, sx, at age x is defined as the ratio of the number of organisms that survived to age x + 1 to the number of organisms that survived to age x sx = lx + 1/lx. Small time steps are preferred for life tables because then the instantaneous mortality rate can be approximated by (1—sx). Age-dependent life tables usually do not include the rates of individual mortality processes associated with various immediate causes of death because this information is hard to obtain.

Stage-dependent life tables are used if the life cycle is partitioned into distinct stages (e.g., eggs, larvae, pupae, and adults in insects). This approach is justified if the survival and reproduction depend more on organism stage rather than on calendar age, and the age distribution at each particular time does not matter (e.g., if there is only one generation per year). Stage-dependent life tables are used mainly for insects and other terrestrial invertebrates. These organisms are poikilothermous and hence the rate of development may vary with temperature; however, stage-specific survival is relatively independent from the rate of development in most cases. The major advantage of stage-specific life tables is that they usually provide information on individual mortality processes caused by predation, parasitism, or infection. Because insects molt between stages, their exuvium (skin), egg shell, or cocoon can be used as evidence of death or survival and may indicate the immediate cause of death. For example, eggs of the common pine sawfly, Diprion pini, are empty and white after successful hatch (survival), whereas parasitized eggs are black and contain a parasitoid larvae. Parasitoids can be reared in the lab and identified after the emergence of adult. Then it is possible to estimate the proportion of eggs killed by each species of parasitoid. In the same way one can collect gypsy moth (Lymantria dispar) pupae from trees and rear adult moths and parasitoids. These data are then used to estimate the proportion ofpupae killed by each species of parasitoid and by pathogens (dead pupae without parasitoid). Predation rates are usually more difficult to quantify than parasitism. In some cases predators leave evidence of a killed prey. For example, cocoons of the common pine sawfly killed by invertebrate predators have irregular small holes; cocoons killed by mice and shrews have large holes that resemble incision by scissors. However, predation on gypsy moth pupae cannot be detected in this way because the skin of the pupae is often destroyed or displaced by predators (e.g., mice). Thus, mesh enclosures were used to quantify the predation rate on gypsy moth pupae in New England. Nonenclosed pupae were accessible to both vertebrate and invertebrate

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Figure 1 Survivorship curves for sheep and lapwing in (a) probability scale and (b) log scale.

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Figure 1 Survivorship curves for sheep and lapwing in (a) probability scale and (b) log scale.

predators. Enclosures with large mesh size selectively excluded vertebrate predators but allowed access of invertebrates, and enclosures with fine mesh excluded all predators.

If several mortality processes co-occur at the same stage of the life cycle, then identification of a true immediate cause of death may be difficult or impossible. For example, if a cocoon of the common pine sawfly was destroyed by a predator, it is usually impossible to check if the prey was parasitized before destruction. If it was, then the immediate cause of death is the parasitoid rather than the predator because the parasitized sawfly was already destined to die. Then mortality due to predation should be accounted for the parasitoid larvae rather than for the sawfly. Although the true immediate cause of death of an individual sawfly remains unknown, it is still possible to estimate rates of mortality associated with parasitism and predation if we use additional assumption that predators do not distinguish between healthy and parasitized sawflies until they open the cocoon. Then, mortality due to predation equals the ratio of cocoons opened by predators to the total number of cocoons, and mortality due to parasitism equals the ratio of parasitized sawflies in cocoons that were not damaged by predators. These calculations are done as if predation occurred first and was followed by parasitism because the predator destroyed the evidence of parasitism (although the timing of attacks was the opposite). The assumption that predators do not distinguish between healthy and parasitized sawflies was tested and shown to be true. In other cases, it may happen that predators attack only healthy prey, then mortality due to predation should be estimated by excluding parasitized prey, and mortality due to parasitism is estimated by counting those hosts that were attacked by the predator as not parasitized.

Because each organism dies only once due to some immediate cause, the probability of survival during the entire stage of development is equal to the product of survival probabilities associated with all immediate causes of death. After log transformation, the ¿-value of total mortality is equal to the sum of ¿-values for all mortality processes that occur at this stage of development. The same logic is applied to different stages of development which leads to the equation that total mortality during the entire life cycle (from egg to adult) expressed in ¿-value is equal to the sum of ¿-values for all mortality processes in the life cycle. This information together with data on reproduction rates can be incorporated into a model of population dynamics. The log-transformed rate of population increase, R, is then estimated as log(R) = log® - K

where F is the average realized fecundity per emerged female, p is the proportion of females, and K is total mortality (sum of all ¿-values). Depending on whether mortality, K, is smaller, equal, or greater than the log reproduction rate log(Fp), population numbers are expected to increase, remain stable, or decrease in the next generation.

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### Responses

• mikolaj
How to make survivorship curve in log scale?
7 years ago
• Kibra
How does a survivorship curve plotted?
2 years ago