CropPest DSS
Age Specific Fecundity Table
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Age Specific Fecundity Calculation Table

Life table parameters

The two most basic parameters of a population are an individual?s likelihood of surviving and an individual?s likelihood of breeding. Both of these parameters depend on the individual?s age, in most species (very young ones and very old ones do not breed and very young individuals often have high odds of mortality). These basic parameters are combined in a life-table, as age-specific survivorship and age-specific fecundity. From these two parameters, we can derive demographic information that allows measurement of the rate of population growth and projection of future population sizes.

Continuous vs. Discrete events. Many life-history processes are continuous, but are broken into discrete units for the purpose of demographic analysis. Example is aging. Age is a continuous variable, but it is generally broken into discrete age-classes in demographic analysis, each potentially including a bout of reproduction.

A set of individuals (a cohort) are observed through time, from birth to death, recording how many are still alive in each age class (at beginning of class usually, but can also be at mid-point of age class). E.g. Start with 500 newborns. N0 = 500. 400 still alive at age 1, so N1 = 400. 200 still alive at age 2 so N2 = 200. Continue until all are dead, Nω = 0. ω (omega) is typical symbol for oldest age attained.

Age class Number in age class Survivorship from birth Age-specific survival
x Nx lx Sx
0 500 1.00 0.8
1 400 0.8 0.5
2 200 0.4 0.25
3 50 0.1 0
4 0 0

Survivorship from birth to age-class x, is denoted lx. (l for life)
lx = Nx/N0 (N for number)
This is the likelihood of living to a given age. Age-specific survival is denoted sx. (s for survival)
sx = Nx+1/Nx (= lx+1/lx) lx decreases continually through age classes.


mx is usually measured as female offspring per female of age x (m for maternity).

mx = 1/2 number of offspring born to parent of age x.

For each offspring produced, male and female parent each credited with 1/2 of an
offspring produced.

Age class Number in age class Survivorshop from birth Fecundity
x Nx lx mx lxmx xlxmx
0 500 1.00 0 0 0
1 400 0.8 2 1.6 1.6
2 200 0.4 3 1.2 2.4
3 50 0.1 1 0.1 0.3
4 0 0 0 0 0
Sum 6 2.9 4.3

Gross reproductive rate = Σmx. Total lifetime reproduction in the absence of mortality. This is the average lifetime reproduction of an individual that lives to senescence, useful in considering potential population growth if all ecological limits (predation, competitors, disease, starvation) were removed for a population. GRR is rarely if ever attained in nature, but useful to consider how far below this a population is held by ecological limits.

Net reproductive rate, R0 = Σ lxmx. Average number of offspring produced by an individual in its lifetime, taking normal mortality into account. lx is the odds of living to age x, mx is the average # of kids produced at that age, so the product lxmx is the average number of kids produced by individuals of age x. Summed across all ages, this is average lifetime reproduction.

R0 is also called the replacement rate:
R0 < 1 individuals not fully replacing themselves, population shrinking
R0 = 1 individual exactly replacing themselves, population size stable
R0 > 1 individuals more than replacing themselves, population growing
The schedule of reproduction (mx curve) can be used to determine the generation time, T.

T = Σxlxmx / Σlxmx� = 4.3/2.9 = 1.48

Where, lxmx is the average number of offspring born to female at age x, as discussed above. If we weight each offspring by the age of the mother, x, and then sum across all ages, then we have the mother's age when each offspring was born, summed across all offspring born in her life.

The denominator (Σlxmx) is equal to R0. In a stable population, R0 = 1, so the denominator has no
effect on generation time. In a growing population, R0 > 1 and T is decreased, because it takes less time for a cohort to ?replace? itself. In a shrinking population, R0 <1 and T is increased, because it takes longer for a cohort to ?replace? itself.

Relationship of net reproductive rate (R0) to intrinsic rate of increase, (r):

R0 measures reproduction on the basis of individual lifetimes (offspring produced per individual per lifetime), and most models of population growth measure growth on the basis of births - deaths per unit time, where time doesn?t have to be a generation - often years or days are the units. The most common measure of population growth is the intrinsic rate of increase, r.

r = lnR0/T = ln(2.9)/1.48 = 0.72 individuals/individual/year (growing rapidly)

when R0 = 1 and r = 0, stable population
R0 < 1 and r < 0, shrinking
R0 > 1 and r > 0, growing
remember that r = b - d where b = births/unit time and d = deaths/unit time. So units of r
are individuals produced per unit time.

The equation r ≈ lnR0/T only give accurate results when R0 ≈1 (r ≈ 0). The exact solution
comes from Euler?s equation.
1 = Σ e-rx lx mx

This is solved by iteration

1. Use the approximate solution to get a close estimate of r.
r = lnR0/T = ln(2.9) /1.48 = 0.72 individuals/individual/year

2. Make an e-rx column.
3. Make an e-rxlxmx column and sum it.
4. Σe-rxlxmx = 1.075, so first estimate of r =0.72 is too small. (We are summing terms
raised to a negative exponent of r, so ↑r will ↓ sum). Adjust estimate of r and try again.
As a guess, try r = 0.75

Σ =1.034
5. Σe-rxlxmx = 1.034, so estimate of r (0.75) is still too small. Increase it and try again.
By repeating this process, will home in on r = 0.776, which gives Σe-rxlxmx = 1
Σ =1.000

The true intrinsic rate of increase is r = 0.78, compared with original estimate of r = 0.72.
This is a difference of 8%, which is big enough to matter in practical applications.