Parameters
Adjust scenario A parameters. Graph, metrics, and data tables update synchronously in real time.
Population vs Time
Scenario AExperiments & Comparative Analysis
All curves share identical axes and scaling for direct visual comparison. CSV exports provide exact daily resolution without rounding.
How the Model Works
N is population at time t. N₀ is starting population. r is intrinsic per-capita growth rate per day. K is carrying capacity: the maximum sustainable population under environmental resources.
When N is low, resources are abundant and reproduction is rapid. As N rises, intraspecific competition for food, water, and territory increases. The environmental resistance factor (1 − N/K) decelerates net growth until population reaches K, where births equal deaths.
Why an S-curve (Sigmoid Growth)? [+]
For 0 < N₀ < K/2 and r > 0, growth progresses through four ecological phases:
- Lag Phase: Few individuals are present to reproduce, so initial absolute increase is small.
- Log / Exponential Phase: Abundant resources allow rapid multiplying. Daily net additions peak at N = K/2 (inflection point).
- Deceleration Phase: Crowding and resource depletion restrict reproduction and increase mortality.
- Carrying Capacity Plateau: Population levels off at K in dynamic equilibrium.
Ecological Assumptions & Limitations [+]
This continuous model simplifies real ecosystems for educational clarity:
- Closed Population: Assumes zero immigration and emigration.
- Constant r and K: Real carrying capacity shifts with weather, seasons, and droughts.
- Instant Density Feedback: Assumes no gestational delay; real species often experience time-lag overshoots and crashes.
- Homogeneous Individuals: Age structure, genetic diversity, and sex ratios are not modeled.
- No Interspecific Dynamics: Predator-prey cycles, epidemics, and competitors are omitted.
Mathematical Solution & Special Cases [+]
We graph the closed-form analytical integration of the differential equation:
N(t) = K / [1 + ((K − N₀) / N₀) · e^(−rt)] (for N₀ > 0)
- N₀ = 0: Population remains 0 (no organisms to reproduce).
- r = 0: Net growth is zero; population remains constant at N₀.
- N₀ = K: Factor (1 − N/K) = 0; population is stable at capacity.
- N₀ > K: The habitat is over-saturated; net growth is negative, producing an asymptotic decline down to K.
- Time to 95% K: Derived analytically as t₉₅ = ln(19 · (K/N₀ − 1)) / r.