WLF 448: Fish & Wildlife Population Ecology

Fall 2004

POPULATION ABUNDANCE

II. POPULATION PARAMETERS: Natality, Survival, Movements

A. Natality

B. Survival

1. Survival as a continuous function of age

• Parametric Survival Analysis

            Survivorship function S(t):  proportion of cohort surviving to time t

• Exponential: Constant death rate through time.  A single parameter p (instantaneous death rate) controls the rate of decline.

• Weibull: Death rate increases or decreases with time.  Two parameters, p controls the rate of decline, and k determines whether, and how much, the rate decreases or increases with time.

                   

If k > 1 then death rate increases with time (Type III survival curve)

If k < 1 then death rate decreases with time (Type I survival curve)

If k = 1 then Weibull reduces to the exponential survival function (Type II survival curve)

From McCallum 2000

            Example curves:  Mallard/Sheep/Human

• Mark-Resight (Band Recovery)

• Based open population models of Jolly-Seber

• Because survival is based on marked individuals, need only to assume that survival is the same for marked and unmarked individuals

• Requires estimates of 2 parameters: 1.) capture probabilities and 2.) survival probabilities. 

• The most important issue using these models is how to choose the appropriate model.

2. Survival as a discrete estimate through each age class or life history stage

LIFE TABLES NOTATION

X = Age interval

n_x = Number of individuals of a cohort alive at the start of age interval X

l_x  = Proportion of individuals surviving at the start of age interval X

d_x = Number of individuals of a cohort dying during the age interval X to X + 1

q_x = Finite rate of mortality during the age interval X to X + 1

p_x = Finite rate of survival during the age interval X to X + 1

e_x = Mean expectation of life for individuals alive at start of age X

Cohort or age-specific or dynamic life tables: data are collected by following a cohort throughout its life. This is rarely possible with natural populations of animals. Note: a cohort is a group of individuals all born during the same time interval.

Static or time-specific life tables: age-distribution data are collected from a cross-section of the population at one particular time or during a short segment of time, such as through mortality data. Resulting age-specific data are treated as if a cohort was followed through time (i.e., the number of animals alive in age class x must be less than alive in age class x-1). Because of variation caused by small samples, data-smoothing techniques may be required (see Caughley 1977).

• Cohort Life Table: Identify or mark a large number of individuals at birth, and then follow these individuals until the last one dies.

Data collected are either:

• d_x The number of individuals dying in each interval X

• n_x The number of individuals alive at the start of each interval X

• Time Specific Life Table:  Takes a snapshot of the population at a particular time and then determines the survival over one time period for each age class.

Methods of data collection are:

1. Age Structure Recorded Directly - The number of individuals aged x in a population is compared with the number of these that died before reaching age x+1. The number of deaths in that age interval, divided by the number alive at the start of the age interval, gives an estimate of q_x directly.

2. Ages at Death Recorded with a Stable Age Distribution and Known Rate of Increase - Often it is possible to find skulls or other remains that give the age at death of an individual. These data can be tallied into a frequency distribution of deaths and thus give d_x directly.  To correct for the fact that the population is growing (or declining), each d_x value is corrected as follows:

                           

where r is the instantaneous rate of population growth

Example: Dall Sheep

3. Age Distribution Recorded for a Population with a Stable Age Distribution and Known Rate of Increase - In this case, the age distribution n_x is measured directly by sampling (e.g., unselective shooting, capture, etc.).  The observed frequency is then corrected for population growth using:

                       

and smoothed to produce a decreasing age distribution

Kaplan-Meier:  Used when an idea of the variability/precision of the survival estimate is needed.  The idea is that each n_x individuals entering age class X is the result of independent binomial trials.  Therefore:

and

              

• Finite and Instantaneous Rates

Why do we need to convert between finite and instantaneous rates?

C. Movements

Revised: October 19, 2004