Life Tables: (Campbell; p. 1098). Life tables are a convenient way to summarize age-specific information about a population. Originally developed for actuarial purposes (e.g. life insurance), a life table, strictly speaking, only lists l_x (age-specific survivorship). Life tables used for most demographic studies also include b_x (age-specific fecundity). A typical format for a life table is:

Population growth is commonly approached using mathematical models. There are three major reasons for using models:

The simplest model of population growth is a purely empirical description of changing numbers. It can be written in a number of ways:

This model can describe any change in any population, but only what has already happened. It has no generality and no projective or predictive value, and is unique for one population at one time.

This is the simplest model for projections or predictions.

This model describes a "J-shaped" curve in which growth continues at a constant per capita growth rate ( r ). It describes unlimited growth in an unchanging environment with unlimited resources and fixed values for l_x and b_{x .}

r = the intrinsic rate of natural increase. This is the constant per capita growth rate, independent of population density (does not change when N changes). The value of r does vary if the environment changes, but has a unique fixed for a specified, constant environment:

b > d	r > 0	positive growth	population increasing
b = d	r = 0	no growth	Zero Population Growth
b < d	r < 0	negative growth	population declining

T

he limiting maximum value of r is r_max which is possible only under ideal physical/chemical conditions, with unlimited ideal resources, and without other species in the system (i.e. competitors, predators, pathogens etc). r_max is usually determined experimentally under laboratory conditions simulating the "optimal niche". It expresses the maximum physiological potential for that population.

I

n age-structured populations, the intrinsic rate of increase for a given set of conditions can be calculated from life table data, using the "Euler Expression": There is a only one value of r which will satisfy this expression for a particular life table.

:Every population has an age distribution, which simply describes the proportion of individuals present in each age class:

Age distributions contain three kinds of information; about:

Both long-term exponential growth and "memory" are properties of the Stable Age Distribution (S.A.D), a special case that must exist before a population can grow at a constant r. The stable age distribution ( C_x )for a specified environment can be calculated from the life table.

In practice virtually any population will achieve its stable age distribution within 2- 5 generations in a constant environment. Therefore, while the exponential model's descriptions of unlimited, density-independent, population growth are unrealistic in the long-term, it frequently is a valid description of short-term growth (over a few generations). Deviations of the observed age distribution ( c_x ) from the expected Stable Age Distribution ( C_x ) constitute the population "memory" of changes during the previous 2-5 generations.

This is the simplest model describing growth in a finite environment. It assumes that a maximum population size is determined by finite resources. The maximum sustainable population size on the resources available in a specified environment is the carrying capacity ( K ) of that environment. K is a property of the environment, rather than of the species.

The population is limited by a density-dependent process of intra-specific competition, in which l_x and b_x change as N increases, and the observed per capita growth rate r declines at a constant rate as N increases.

In the third arrangement, growth is separated into two terms. The first describes potential exponential growth at r in an unlimited environment. The second is a negative term describing the reduction in growth caused by intra-specific competion. Its effects are such that as N increases by 1 individual, the observed growth rate declines by 1/K of r. This model only describes positive population growth: as N = 0 to N = K, the observed per capita growth rate declines from r to 0.

Logistic growth defines a smooth S-shaped curve that is followed in practice by many natural populations when simple resource limitation is the primary factor regulating population density.

Deviations from a simple "S-shaped" (sigmoidal) curve are caused by time lags between the change in N, the depletion of resources, and the modifications of l_x and b_x that alter the observed r.

Common deviations from logistic growth include: