Understand the relationship
The reasoning behind the result
Continuous and geometric growth use different rates
N(t)=N0 exp(rt); N(k)=N0 λ^k; λ=exp(r)
N0 is the initial population quantity. In the continuous model, r is a log rate per selected time unit and t can be fractional. In the geometric model, λ is a multiplier applied at each complete census period k. Its percentage change is 100(λ−1).
The continuous rate equivalent to 10% growth per period is ln(1.1), about 0.09531 per period. Substituting 0.1 as the continuous rate instead gives about 10.517% growth over one period. The distinction compounds over a long horizon.
Inverse solves identify parameters only under the stated model
N0=Nt exp(−rt); r=ln(Nt/N0)/t
Positive endpoints and positive elapsed time identify one average continuous log rate. Dividing that interval into whole geometric periods gives the corresponding multiplier. Neither inverse proves that the rate was constant between the observations.
Zero initial population stays zero under a finite multiplicative model. Zero-to-zero endpoints do not identify a unique rate; zero-to-positive endpoints cannot be generated without adding a process absent from this model. A finite continuous decline does not reach exact zero.
A geometric target can be crossed without being exactly attained
Log-equivalent t*=ln(target/N0)/ln(λ)
The logarithmic inverse can return a fractional period, but a geometric census model is evaluated at whole periods. The calculator therefore reports the first whole period meeting or passing the target in the direction from the initial value, alongside the real log-equivalent crossing.
At multiplier 1.2, a population of 100 becomes 120, 144 and 172.8. A target of 150 is first exceeded at period three. It is not correct to report the fractional log-equivalent time as an observed generation or to claim the population equals 150 at the third census.
Doubling and halving describe a constant log slope
Interval=ln(2)/|r|
A positive rate has a doubling interval and a negative rate has a halving interval. The interval is independent of the starting positive population in this model. For a geometric multiplier, it is a continuous log-equivalent interval; actual whole-period crossings can occur later.
An unchanged population has no finite doubling or halving interval. A geometric zero multiplier is a separate boundary: all quantity is lost after one complete period and there is no finite equivalent continuous log rate. Reversing that boundary from a final zero does not identify a unique starting quantity.
A projection is not a prediction of biological conditions
The calculation assumes one unchanged rate and no added arrivals, departures or density effects. Fractional outputs are mathematical quantities, not guarantees about whole organisms. Intermediate points are computed from the model and are never labelled as measured observations.
Use a demographic ledger to reconcile recorded births, deaths and migration. Use a logistic scenario when a supplied capacity is part of the intended model, or a stage projection when different groups have different transitions. No organism-specific rate or experimental growth conditions are chosen here.
Follow the numbers
Three discrete periods of ten-percent growth
- The entered initial population is 500 and the geometric multiplier is λ = 1 + 10/100 = 1.1.
- After each complete period the model quantities are 550, 605 and 665.5: 500 × 1.1³ = 665.5.
- The equivalent continuous log rate is ln(1.1) ≈ 0.09531018. Using that rate for three time units gives the same 665.5.
The two conventions agree after converting the rate; using the number 0.1 for both conventions would produce different results.
Quick guide
How to use this calculator
- Choose the quantity to solve and the time unit or complete census period.
- State whether the known rate is continuous, a geometric percentage/multiplier, or a continuous doubling/halving interval.
- Enter the population quantities and elapsed interval required by that solve.
- Read the equivalent rate conventions, target-crossing rule and model timeline. Zero and non-identifiable boundaries are reported explicitly.
Calculation method
Calculation and interpretation
Make the rate convention explicit before projecting population change.
Continuous N(t)=N0 exp(rt); geometric N(k)=N0 λ^k; r=ln(λ); t=ln(target/N0)/r; doubling or halving interval=ln(2)/|r|
Worked example
Three discrete periods of ten-percent growth
The two conventions agree after converting the rate; using the number 0.1 for both conventions would produce different results.
Continuous N(t)=N0 exp(rt); geometric N(k)=N0 λ^k; r=ln(λ); t=ln(target/N0)/r; doubling or halving interval=ln(2)/|r|
Supported inputs
Precision and limits
Entered mathematical model
No species parameters, growth conditions, resource availability, ecological suitability, population viability or guaranteed future count are inferred.
Numerical and time boundaries
Geometric periods are whole safe integers. Positive results outside floating-point range produce an explicit error. Target comparisons allow a small floating-point rounding tolerance at a whole-period boundary.
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