Biology · Population Ecology & Life-History Patterns
Exponential and logistic growth as models with assumptions
Suppose a model population changes from 100 to 120 over one time unit. Multiplying by 1.2 reproduces that census change. It does not mean that the continuous rate parameter is 1.2, or that the instantaneous slope at the beginning is 20. The symbols describe different quantities.
Organisms, Environments & Ecological Niches introduces ecological scales, habitat and the distinction between observation and inference. Begin here with the counts and ordinary arithmetic. The mathematics explanations of functions and exponentials are optional preparation for the growth lesson; its multiplier-versus-slope explanation can be read before derivatives. Return to Animal Reproduction, Development & Life Histories and Evolutionary Processes & Adaptation for reproductive outcome stages and the evidence needed to connect demographic differences to selection.
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Distinguish a finite-interval multiplier, continuous per-capita rate and derivative, and interpret logistic change without treating carrying capacity as a hard ceiling.
A discrete model connects census counts
The discrete recursion N(t+1)=λ×N(t) uses a dimensionless multiplier λ for one stated interval. With constant λ=1.2, each census value is twenty per cent larger than the preceding one. Starting at 100 gives 120, 144, 172.8 and 207.36 after four intervals. The absolute additions increase even though the multiplier stays constant.
A multiplier of 1 leaves the model abundance unchanged; a positive multiplier below 1 gives decline. With λ=0.8, the sequence goes 100, 80, 64, 51.2, 40.96. Equal proportional losses do not mean equal numbers lost each time. Smooth or expected model abundance can be noninteger; an actual complete census of individuals is an integer count.
These recursions omit age structure, density feedback, migration and random events. They can describe an assigned finite sequence without showing that the same multiplier would remain valid indefinitely. A growing population need not literally have infinite resources to approximate exponential change over a short interval; sustained unconstrained growth is a stronger assumption.
A multiplier and an instantaneous rate are different quantities
Read the diagram in words
The increasing sequence is 100, 120, 144, 172.8, 207.36 at unit censuses. Its multiplier is 1.2 and matching constant continuous rate is ln(1.2), approximately 0.1823 per time unit. The initial derivative is approximately 18.23 individuals per time unit, while the first full-interval change is 20. The declining sequence uses multiplier 0.8 and negative ln(0.8); it ends at 40.96 after four intervals.
| Dimensionless multiplier per stated interval | 1.2 |
|---|---|
| Instantaneous per-capita rate per time unit | 0.1823215567939546 |
| Change over the first full interval | 20 |
| Instantaneous abundance change per time unit | 18.232155679395458 |
| Assigned elapsed time | Model abundance | Instantaneous abundance change per time unit |
|---|---|---|
| 0 | 100 | 18.232155679395458 |
| 1 | 120 | 21.878586815274552 |
| 2 | 144 | 26.254304178329463 |
| 3 | 172.8 | 31.505165013995356 |
| 4 | 207.36 | 37.806198016794426 |
| Dimensionless multiplier per stated interval | 0.8 |
|---|---|
| Instantaneous per-capita rate per time unit | -0.2231435513142097 |
| Change over the first full interval | -20 |
| Instantaneous abundance change per time unit | -22.31435513142097 |
| Assigned elapsed time | Model abundance | Instantaneous abundance change per time unit |
|---|---|---|
| 0 | 100 | -22.31435513142097 |
| 1 | 80 | -17.851484105136777 |
| 2 | 64 | -14.281187284109421 |
| 3 | 51.2 | -11.424949827287538 |
| 4 | 40.96 | -9.13995986183003 |
N
t
λ
r
N(t+1) = λ × N(t)
dN/dt = r × N
λ = exp(r × Δt)
N(t) = N(0) × exp(r × t)
Assigned time units; Δt = 1
Smooth model values can be noninteger
Original deterministic smooth abundance model and discrete census recursion, fixed one-unit intervals, constant parameters, no migration, density response or age structure. Noninteger values represent model abundance, not fractional counted individuals. Natural logarithms connect the models only under these assumptions. Continuous logarithm rates and derivatives are numerical evaluations; the prose displays rounded approximations.
The slope belongs to an instant
The continuous exponential law is dN/dt=rN, with solution N(t)=N(0)×exp(rt) when r is constant. The derivative dN/dt is the instantaneous slope of abundance against time, in individuals per time unit. The parameter r is the instantaneous net per-capita rate, in inverse time units. Positive r gives increase, negative r gives decline and zero r gives constant abundance in this closed model.
For the same interval length Δt and constant continuous r, the matching finite multiplier is λ=exp(rΔt). When Δt=1, λ=1.2 corresponds to r=ln(1.2), approximately 0.1823 per time unit. At starting N=100, the derivative rN is about 18.23 individuals per time unit. The complete first-interval change is 20; the derivative changes as N changes within that interval.
You can first read this distinction as snapshot multiplier versus momentary slope. Logarithms and derivatives explain the exact connection, but the biological questions still concern the boundary, births, deaths and omitted influences. A fitted net r is not automatically the species’ maximal potential rate under ideal conditions.
The logistic model changes the per-capita rate
A simple continuous logistic law writes dN/dt=rN(1−N/K). Here we stipulate positive constant r and positive constant K. The factor 1−N/K reduces the net per-capita rate as N increases. K is the abundance at which this closed model has zero net instantaneous change; it is often called a carrying-capacity parameter.
At N below K, the stipulated positive-r law has positive growth; at N=K it has zero growth; above K it has negative growth. K is therefore not a hard ceiling that makes N>K impossible. Starting above it is allowed, with decline towards it in this fixed continuous law. Real environmental support can change with season, resources and interactions, so one constant K is a model simplification.
Exactly N=0 is also an equilibrium in the closed deterministic equation: zero times any per-capita rate is zero. The model cannot spontaneously colonise an empty habitat. At small positive abundance it increases under the stated positive r, but it does not represent the discrete chance of every member dying.
For this no-delay, constant-parameter continuous law, a positive trajectory approaches K from its starting side. Oscillations, delayed responses or an abrupt changing environment require additional structure. A smooth S-shaped picture is not a universal outcome of population growth, and its upper flattening does not show that competition was absent earlier.
The logistic derivative changes sign above the assigned equilibrium
Read the diagram in words
At sizes 0, 20, 50, 80, 100, 120, instantaneous changes are 0, 3.2, 5, 3.2, 0,−4.8. At N=120 the per-capita rate is−0.04 and multiplying by 120 gives−4.8. The maximum derivative for this fixed law is five at N=50.
| Model population size | Assigned net per-capita rate (Assigned time units⁻¹) | Instantaneous abundance change per time unit |
|---|---|---|
| 0 | 0.2 | 0 |
| 20 | 0.16 | 3.2 |
| 50 | 0.1 | 5 |
| 80 | 0.04 | 3.2 |
| 100 | 0 | 0 |
| 120 | -0.04 | -4.8 |
dN/dt = r × N × (1 − N/K)
r × (1 − N/K); r = 0.2; K = 100
Zero instantaneous change in this closed model: 0, 100
Instantaneous abundance change per time unit: 5; Model population size: 50
Assigned time units
Above the assigned K, the stipulated positive-r model declines
A derivative is not the next population size
K is an equilibrium parameter, not a physical hard ceiling
Original continuous logistic law with constant r=0.2 per assigned time unit and K=100 model individuals, no migration, delay, age structure or environmental change. Listed sizes are separate evaluations of the derivative, not successive population censuses.
Worked example
Evaluate a derivative without advancing the census
Set r=0.2 per time unit and K=100. At N=20, the derivative is 0.2 × 20 × (1 − 20/100) = 3.2 individuals per time unit. At N=80 it is also 3.2: a larger population is multiplied by a smaller per-capita modifier.
At N=50 the derivative is 5, the maximum for this fixed law. At N=120, the modifier is−0.2, the net per-capita rate is−0.04, and the derivative is−4.8. That says the curve is declining at that instant; it does not make the next census population−4.8 or automatically give a full-interval loss of 4.8.
The horizontal axis in this figure is population size, not elapsed time. Its six rows are separate evaluations of one law. To calculate a future N, use the time-dependent solution or an appropriately checked numerical method with its time step; do not read a derivative table as a trajectory.
Check your understanding
In the constant increasing model, is λ=1.2 the same parameter as r=0.2 per time unit?
Show reasoning
No. λ is the full-interval multiplier. For interval length 1 and constant continuous growth, the matching r is ln(1.2), approximately 0.1823 per time unit. Using r=0.2 would give multiplier exp(0.2), approximately 1.2214.
Why is the initial derivative about 18.23 while the change from 100 to 120 is 20?
Show reasoning
The derivative is the slope at the starting instant. Under positive continuous growth the population and slope increase during the interval. The full-interval change combines that changing slope; it is not the initial slope multiplied once without approximation.
With positive r=0.2 and K=100, can the logistic model be evaluated at N=120, and what does it say?
Show reasoning
Yes. K is not a forbidden-count ceiling. The derivative is 0.2 × 120 × (1 − 1.2)=−4.8 individuals per time unit, so the stipulated continuous model declines at that size. This is an instantaneous change, not a negative population count.
What to carry forward
A multiplier connects census values, a continuous rate acts per member at an instant, and a derivative describes the resulting slope.
References
Resources supporting this lesson’s explanations and examples.