Science
Life Table R0, Cohort Generation Time and a Leslie Projection
Calculate lifetime recruitment, a cohort-weighted generation interval and an approximate growth rate, then project three age classes using an explicit discrete census convention.
Use a consistent census and recruitment convention
This discrete three-class example uses one equal time step per class. x=1,2,3 is the lag from a newborn entrant to its recruited daughters. l=[1,0.8,0.5] is survival to the corresponding parent class, and F=[0,1.25,0.8] is female recruits entering the first class at the next census per current female in that class. F already includes the recruitment convention; it is not a raw birth rate to copy from a different census system.
The first class is normalized to l=1. It does not assert that every real newborn survives to age one. Subsequent transition survival is 0.8/1=0.8 and 0.5/0.8=0.625. There is no final-class persistence or reproduction beyond the listed classes. Choose and document the census timing before adapting real data.
Net reproductive rate and the cohort mean interval
E3 calculates R0=SUMPRODUCT(l,F)=1.4 female recruits per initial female over her modeled lifetime. E4 calculates Tc=SUMPRODUCT(x,l,F)/R0=3.2/1.4=2.285714 steps. The fractions l*F/R0 weight the recruitment lags.
R0 is a per-generation replacement measure, not the population multiplier for a single time step. Under fixed vital rates and a closed population, its relation to 1 indicates the asymptotic growth threshold. The initial age structure can still produce transient growth or decline.
If R0=0, the generation interval is undefined because there is no modeled reproduction. Existing individuals need not already be extinct. E3 returns zero and E4:E5 give an explanatory message.
=IF(E2="OK",SUMPRODUCT(B2:B4,C2:C4),E2)Approximate growth is not the exact Leslie eigenvalue
E5 computes r_approx=LN(R0)/Tc, about 0.147207 per step. This is a cohort-interval approximation. It is not automatically the exact intrinsic growth rate. Keep unrounded E3 and E4 when calculating it.
For this convention, the exact dominant growth factor lambda satisfies SUM(l_x*F_x/lambda^x)=1; r=LN(lambda) per step. A matrix eigensolver or an Euler-Lotka root calculation is needed for that exact result. Do not report LN(R0)/Tc as an eigenvalue.
=IF(E2<>"OK",E2,IF(E3=0,"No reproduction: undefined",LN(E3)/E4))A working one-step Leslie projection
H2:J4 contains the matrix: first row [0,1.25,0.8], second row [0.8,0,0], third row [0,0.625,0]. L2:L4 holds the current class vector [100,80,50]. M2:M4 calculates the next vector [140,80,50].
The next first class is the sum of recruits from all current classes. The older classes receive survivors from the preceding class. These expected populations may be fractional; do not round them before projecting another step.
The initial total of 230 becomes 270, not 230*1.4. Enter [140,80,50] into L2:L4 for the second step: the result is [140,112,50]. Each projection is one step, whereas R0 summarizes a lifetime.
| Class | Current | Next step |
|---|---|---|
| 1 | 100 | 140 |
| 2 | 80 | 80 |
| 3 | 50 | 50 |
Validate the schedule and try a replacement case
The worksheet checks exactly three consecutive lags 1,2,3; numeric inputs; nonnegative fertility; and survivorship starting at 1, within [0,1] and never increasing. Class populations must be numeric and nonnegative. Extending the example requires extending all ranges and the projection matrix consistently.
Change F for class2 in C3 to 0.75: R0 becomes 1 and r_approx becomes zero. Set every fertility to zero: no new recruits are generated, but the surviving older classes still appear in the next projection.
When survival to class2 is zero, survival to class3 must also be zero. The sheet sets the otherwise undefined second transition to zero as an explicit convention. Real cohorts with such unreachable classes should normally be simplified rather than interpreted as measured transitions.
Try the example
Use the original values or change them to test your understanding.