Understand the relationship
The reasoning behind the result
The density factor changes the instantaneous slope
dN/dt = rN(1 − N/K)
N is the current model population, K is a positive constant and r is a continuous coefficient per time unit. The factor 1−N/K modifies the unrestricted slope rN. With positive r, a population below K increases, while one above K decreases.
K is an equilibrium parameter, not a hard barrier that prevents an initial value above it. Requiring N0<K would discard a valid branch of the same differential equation. A population exactly at zero or K remains at that equilibrium.
The trajectory approaches a stable capacity without reaching it
N(t)=K/[1+A exp(−rt)], where A=(K−N0)/N0
For positive N0 below K and positive r, the denominator decreases toward one and N approaches K from below. Starting above K gives a negative A and approaches K from above. The model remains on the same side of K for every finite valid time.
The calculation uses stable exponential and logarithmic forms near an equilibrium. A displayed value may round to K even though the mathematical distance is nonzero; that distance is reported separately when representable. An asymptotic target is not assigned a finite arrival time.
Two compatible endpoints can identify a coefficient or time
r Δt = ln[N1/N0] − ln[|K−N1|/|K−N0|]
The formula applies to positive endpoints on the same side of K, with neither equal to K. With entered duration it solves r; with entered nonzero r it solves duration. A negative solved duration places the target in the reverse-time direction.
Equilibrium endpoints are different. Zero-to-zero or K-to-K records do not identify r because every coefficient preserves them. Endpoints on opposite sides of K cannot be joined by one finite trajectory of this model.
Inferred capacity is conditional and can be poorly determined
K = N1(1−exp(−rΔt)) / [1−(N1/N0)exp(−rΔt)]
This rearrangement uses two positive population quantities, a supplied coefficient and positive elapsed time. It infers the K required by those inputs under the logistic equation. It does not measure resources, habitat quality, density tolerance or long-term sustainability.
A zero denominator corresponds to the unrestricted exponential limit rather than a finite K. Near that limit, or very near equilibrium, small input changes can cause large changes in the inferred parameter. Incompatible or numerically unresolved inputs are reported explicitly.
Negative r changes the meaning of the model
For negative r, K is unstable: populations below K decline toward zero and populations above K increase. The above-K branch reaches a finite-time singularity; horizons reaching or passing that point are rejected rather than producing a negative population.
This branch is retained as transparent mathematics and is labelled separately from the usual stable-capacity interpretation. No negative coefficient is turned into a habitat or biological recommendation.
Follow the numbers
A capacity of 100 gives a final model quantity of 50
- Enter N0=20, K=100, r=ln(4) per year and t=1 year. The initial ratio term is A=(100−20)/20=4.
- exp(−rt)=exp(−ln4)=1/4, so N(1)=100/[1+4×(1/4)]=50.
- For capacity inference using N0=20 and N1=50, K=50(1−1/4)/[1−(50/20)(1/4)]=37.5/0.375=100.
The forward and inverse arithmetic agree for the supplied model; neither calculation independently establishes an environmental carrying capacity.
Quick guide
How to use this calculator
- Choose forward population, target time, coefficient inference or conditional capacity inference.
- Supply a common time unit and distinguish the continuous coefficient from a per-period percentage.
- Read equilibrium and inverse-identifiability messages rather than substituting an arbitrary parameter.
- Inspect the trajectory, the signed distance from K and the final instantaneous model slope. These are model quantities, not an environmental assessment.
Calculation method
Calculation and interpretation
Keep a mathematical capacity parameter separate from an observed environmental limit.
dN/dt=rN(1−N/K); N(t)=K/[1+((K−N0)/N0)exp(−rt)]; log-odds shift = rt for positive endpoints on the same side of K
Worked example
A capacity of 100 gives a final model quantity of 50
The forward and inverse arithmetic agree for the supplied model; neither calculation independently establishes an environmental carrying capacity.
dN/dt=rN(1−N/K); N(t)=K/[1+((K−N0)/N0)exp(−rt)]; log-odds shift = rt for positive endpoints on the same side of K
Supported inputs
Precision and limits
No environmental certification
K is an entered or conditionally fitted parameter. It is not a species husbandry minimum, stocking rule, enclosure limit, sustainability guarantee or population-viability conclusion.
Constant-parameter model
The model omits changing resources, migration, delays, age structure and stochastic variation. A fit from two endpoints does not validate its biological assumptions.
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