802.96
80.3
%
197.04
31.6434
population per time
0.039409
per time
80.2957
x
802.96
80.3
%
197.04
31.6434
population per time
0.039409
per time
80.2957
x
The Logistic Growth Calculator models density-dependent population growth where the growth rate decreases as the population approaches its carrying capacity. This produces the characteristic S-shaped (sigmoid) growth curve that is widely observed in natural populations from bacteria to large mammals.
Enter the initial population, carrying capacity, intrinsic growth rate, and time to calculate the population size, the percentage of carrying capacity reached, and the instantaneous per capita growth rate at that time. This model is fundamental in ecology, conservation biology, and resource management.
The logistic growth equation is:
N(t) = K / (1 + ((K - N0)/N0) x e^(-r x t))
Where K is the carrying capacity (maximum sustainable population), N0 is the initial population, r is the intrinsic growth rate, and t is time.
The per capita growth rate at any time is: r x (1 - N/K), which decreases linearly as N approaches K. At N = K, growth stops. The population grows fastest at N = K/2 (the inflection point of the sigmoid curve).
Inputs
Results
Starting with 10 deer and K = 1000, after 30 years the population reaches about 291 (29% of carrying capacity), still growing at a per capita rate of 0.14.
Inputs
Results
After 20 time periods with r = 0.5, the bacteria reach 95% of carrying capacity and growth has slowed dramatically to a per capita rate of 0.024.
Carrying capacity (K) is the maximum population size that an environment can sustain indefinitely given the available resources (food, water, habitat, etc.). It is determined by environmental factors and can change over time due to habitat modification, climate change, or resource depletion. At carrying capacity, birth rates equal death rates and population growth stops.
The S-shape results from density-dependent growth regulation. At low population sizes, resources are abundant and growth is nearly exponential (rapid). As the population grows, competition for resources increases, causing the per capita growth rate to decline. The growth rate reaches zero as the population approaches K, producing the characteristic leveling off.
The logistic model assumes a constant carrying capacity, instantaneous response to density, smooth growth without time lags, a single species with no age structure, and that all individuals are identical. Real populations often exhibit time-lagged responses, oscillations around K, age-structured dynamics, and interactions with other species that the simple logistic model does not capture.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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