Mortality rate is usually measured as the proportion of deaths per unit time in a given cohort of organisms. If the time unit is small enough, it can be interpreted as the instantaneous probability to die per unit time. In mathematical models, in which mortality is the only process that affects population dynamics, the instantaneous mortality rate equals the normalized derivative (dN/dt)/N. In human epidemiology, mortality rate is measured by the annual number of deaths per 1000 people. It is distinct from morbidity rate, which refers to the number ofpeople who have a disease compared to the total number of people in a population. If mortality rates are compared between different groups of people, for example, men and women or smokers and nonsmokers, then the group of individuals with a higher mortality rate is said to have a risk factor (e.g., genotype or behavior) which causes additional mortality. The ratio of mortality rates in the group with the risk factor and control group is called relative risk or odds ratio.
In populations of organisms with a synchronized development through the seasonal cycle, it is often practiced estimating cumulative mortality during specific stages of development (e.g., egg, larvae, or pupae in insects). In contrast to instantaneous mortality rates, cumulative mortality rates are not additive. For example, if predators alone kill 50% of the population, and diseases alone can kill another 50% of the population, then the combined effect ofthese process does not result in a 100% mortality. Instead, the mortality will be only 75%. To make cumulative mortality rates additive, they are transformed into ¿-values which are defined as a negative logarithm of survival, k = —log(1 — m), where m is the proportion of died organisms. If two simultaneous or sequential mortality processes are independent, then the probability to survive is equal to the product of survival probabilities from each individual process. After log transformation, the product of survival probabilities becomes the sum of k-values for individual processes, which proves that ¿-values are additive. Another advantage of ¿-values is that they are proportional to instantaneous mortality rates. The following example shows that the ¿-value represents mortality better than the percentage of dead organisms. If one insecticide kills 99% of cockroaches and another insecticide kills 99.9% of cockroaches, the difference seems very small (<1%). However, the second insecticide is considerably more toxic because the number of survivors is 10 times smaller. This difference is represented much better by ¿-values which are 2 and 3 for the first and second insecticides, respectively. The limitation of the ¿-value concept is that it assumes equal death probabilities for all organisms. In real populations, death probabilities are not equal and depend on local conditions, spatial heterogeneity, and individual variation (both inherited and noninherited).
Analysis of population dynamics requires information not only on the total mortality rate but also on the specific rates of mortality associated with various immediate causes and at different ages or stages of development. This information is summarized in life tables (see below) which show the balance of all mortality processes and can be used for developing mathematical models of population change.
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