User Rating 0.0 ★★★★★

Total Usage 0 times

Category Physics & Science

Carrying Capacity (K)

Initial Population (N₀)

Intrinsic Growth Rate (r) per time unit. Values > 2.0 may cause chaotic oscillations.

Time Steps

Model Type

Presets:

Configure parameters and press Calculate to see population dynamics.

Is this tool helpful?

Your feedback helps us improve.

★ ★ ★ ★ ★

About

Carrying capacity (K) defines the maximum population size an environment can sustain indefinitely given available resources. The concept originates from Pierre-François Verhulst's 1838 logistic equation. Misjudging K leads to overshoot-collapse cycles: populations exceed resources, crash, and may not recover. This calculator solves the logistic growth ODE using Euler integration and projects population trajectories across discrete time steps. It accepts initial population (N₀), intrinsic growth rate (r), and either a direct K value or derives it from total resource supply divided by per-capita demand.

The model assumes a closed system with no immigration, emigration, or catastrophic events. It approximates density-dependent growth where per-capita rate declines linearly as N approaches K. Real ecosystems introduce stochastic variation, Allee effects below critical densities, and multi-species competition not captured here. For r values above 2.0 in discrete models, chaotic oscillations emerge. Pro tip: field ecologists typically estimate r from mark-recapture data across at least 3 breeding seasons to reduce sampling bias.

Formulas

The logistic growth model describes density-dependent population change. The continuous form is the Verhulst equation:

dNdt = r ⋅ N ⋅ (1 − NK)

Where N is the population size at time t, r is the intrinsic rate of natural increase (time⁻¹), and K is the carrying capacity. The analytical solution for population at time t is:

N(t) = K1 + (K − N₀N₀) ⋅ e^−rt

Carrying capacity from resource constraints:

K = R_totalR_{per capita}

Doubling time under pure exponential growth (early phase when N << K):

t_d = ln(2)r

Where R_total is the total available resource quantity, R_{per capita} is the resource consumed per individual per time unit, N₀ is the initial population size, and e is Euler's number (≈ 2.71828).

Reference Data

Species / System	Typical r (yr⁻¹)	Estimated K	Habitat	Doubling Time	Notes
E. coli (lab)	69.3	10⁹ cells/mL	Nutrient broth	20 min	Exponential phase only
Paramecium aurelia	1.24	500 individuals	Lab culture	0.56 days	Gause's classic experiment
Daphnia magna	0.70	200 ind/L	Freshwater	1.0 day	Temperature-dependent
House mouse	4.50	Varies	Commensal	56 days	Rapid colonizer
White-tailed deer	0.55	15-30 per km²	Temperate forest	1.26 yr	Density-dependent browse pressure
Grey wolf	0.30	3-5 per 100 km²	Boreal/tundra	2.31 yr	Pack structure limits density
African elephant	0.06	0.5-2 per km²	Savanna	11.6 yr	Long gestation, slow maturation
Bald eagle	0.12	Territory-limited	Riparian	5.8 yr	Nesting site availability is key
Atlantic cod	0.30	Stock-dependent	Marine	2.3 yr	Collapsed from overfishing
Humans (global)	0.011	8-12 billion	Global	63 yr	Debated; technology shifts K
Yeast (S. cerevisiae)	0.50	10⁸ cells/mL	Fermentation	1.4 hr	Ethanol self-inhibition
Loggerhead sea turtle	0.04	Beach-limited	Coastal/pelagic	17.3 yr	Nesting beach loss is limiting
Brown trout	0.40	50-200 per km	Freshwater stream	1.7 yr	Dissolved O&sub2; and riffle area
Barn owl	0.35	Prey-dependent	Agricultural	2.0 yr	Vole population cycles drive K
Giant panda	0.03	<2000 wild	Bamboo forest	23 yr	Habitat fragmentation

Frequently Asked Questions

When N exceeds K, the term (1 − N/K) becomes negative, producing negative growth rate. Population declines toward K. In real ecosystems this overshoot can cause resource depletion, habitat degradation, and a new lower K. The Kaibab Plateau deer irruption of 1924 is a textbook case: population overshot K by roughly 4×, collapsed, and the habitat took decades to recover.

In the discrete logistic map N_t+1 = rN_t(1 − N_t/K), values of r between 0 and 2 produce stable convergence to K. Between 2 and 2.449, damped oscillations appear. Above 2.449, period-doubling cascades lead to deterministic chaos by r ≈ 2.570. This calculator flags r values above 2.0 as potentially unstable.

Yes. K is not a fixed constant. Seasonal resource fluctuation, climate change, technological innovation (in human populations), disease outbreaks, and habitat modification all shift K. This calculator models a static K for simplicity. For dynamic K, you would need coupled differential equations or agent-based models. A practical approach: run the calculator multiple times with different K values representing wet vs. dry season scenarios.

In discrete-time Euler integration with large time steps (Δt) or high r, numerical overshoot can push N below zero. This is an artifact of the numerical method, not the biology. The calculator clamps N to a minimum of 0 at each step. To reduce numerical error, use smaller time steps or the analytical (continuous) solution mode.

The intrinsic rate r can be estimated from life table data: r ≈ ln(R₀) ÷ T, where R₀ is the net reproductive rate and T is generation time. Alternatively, from two census counts: r = ln(N₂ ÷ N₁) ÷ Δt, valid only during exponential phase when N << K.

The Allee effect describes reduced per-capita growth at low population densities due to difficulty finding mates, cooperative defense breakdown, or inbreeding. The standard logistic model does not include it. Below a critical threshold N_c, real populations may decline to extinction even with available resources. This calculator uses the classical Verhulst model which assumes monotonically increasing total growth rate from N = 0 up to N = K/2.