Carrying Capacity Calculator

Populations don't grow infinitely. When food becomes scarce, space is limited and predators appear, a curve that initially appeared to show exponential growth bends into an S-shape. The logistic growth model is the simplest equation to represent this bend, and this calculator can be used from any side of it.

Enter three out of four values and the fourth value will be calculated, as well as the graph of how the rate of change in population changes over time and an explanation of what these numbers mean.

What is carrying capacity?

The carrying capacity is denoted by K and represents the number of individuals that an environment can support in the long term given current resource conditions. It is determined by the least available resource whether it be food, water, nesting sites or oxygen in a tank. K is a property of the habitat not the species so K will change if the habitat changes. A wet spring increases K while a drought decreases it.

Carrying capacity is not the same as maximum population size ever recorded. Under favorable conditions over many years a population may exceed K, but under harsh conditions such as a severe winter it may fall below K. K is the level to which a population will stabilize, and is closer to the long-term average than any one peak value.

The logistic growth equation.

This model is made up of a single equation and four values.

In many ecology textbooks, c is used to represent the rate of change, so the practical form used in this tool is as follows:

The terms in parentheses are crucial. If N is very small relative to K, the fraction (K - N)/K approaches 1, so that growth is approximately rN, which is exponential growth with no limits. As N approaches K, this fraction approaches zero, and the rate of growth slows down. When N equals K, the change is zero. If K is exceeded, the fraction becomes negative, resulting in a decline of the population.

Meaning of individual values:

r and C must use the same time scale. If the unit of r is "per year", then C must also be expressed in terms of "per year". The growth rate r is given as a decimal number. 0.4 means that, without constraints, the population will grow by 40 percent per year. Species with rapid reproduction, such as mice or insects, have higher values for r and can sometimes exceed 1. For species with slow reproduction, such as whales or California condors, the values of r are much lower.

A full example:

Suppose a forest can support 500 deer (K = 500) and the current population is 200 deer (N = 200), with an intrinsic growth rate of 0.4 per year (r = 0.4).

If n is equal to 200, the population is well below maximum so about 48 deer are added in the following year. If n is close to the maximum of 450, then growth rate falls with that same calculation to about 18 per year. At n = 500, there's no growth. This gradual slowing down is what makes this model interesting.

Instructions for using the calculator tool:

Enter three of the four values and leave the value you want to calculate blank. The calculator tool will fill in that value and update results as you change other values. Typically C is left blank to find the growth rate for a current population size. K can be left blank to calculate the appropriate carrying capacity given an observed change. Alternatively r can be left blank to estimate the growth rate, or N can be left blank to determine the population size that will produce a particular growth rate.

Be careful when calculating N. For a given growth rate there are two possible population sizes - one below and the other above half of the maximum (K). This is because the growth curve is symmetrical about K/2. The calculator will give you the larger of the two values.

Growth curve and maximum sustainable yield:

If you plot the change C against population size N, you get a downward-opening parabola. At N = 0 it is zero, then rises to a maximum and falls back to zero at N = K. The maximum lies exactly in the middle of the environmental maximum.

This is where the population can reproduce the most and thus yield the greatest annual surplus. Fishers and wildlife managers refer to this as Maximum Sustainable Yield (MSY) and try to keep populations near K/2. Over many decades, it has become apparent how dangerous it is to strive for this limit, especially if r and K are not actually constant.

Logistic growth versus exponential growth

Exponential growth assumes that the reproduction rate depends only on population size, leading to ever-faster increases in an unbounded J-shaped curve. Logistic growth takes into account the unused fraction of environmental capacity (K - N)/K, which acts as a brake. The two curves are identical for small populations, when the brake has little effect. When N exceeds about one fifth of K, the two curves begin to diverge. In this timeplot both curves are shown simultaneously so that you can see exactly where they diverge.

Factors that limit growth and affect K

The "brake" in the equation represents real factors that depend on density, such as competition for food, diseases that spread more easily when animals are overcrowded, lack of territory or nesting sites and accumulation of waste products in cultures. The higher N is, the stronger these factors act, which leads to a slowing down of growth as it approaches K.

Factors that operate independently of density are different. Floods, cold snaps and forest fires can reduce the population, regardless of how dense it is, but these factors aren't taken into account in the logistic equation. Nor is K itself a fixed quantity. An improvement to habitat increases K, while a deterioration of K reduces its value. What at first seems like a fixed limit is actually an ever-changing target.

Fast strategies and slow strategies.

Species with high r-values are usually small, mature quickly and reproduce successfully without much care. This allows them to recover from near-zero populations within a single season. Weeds, insects, and many rodents follow this strategy. Species with high K-values are typically larger, mature more slowly, and produce few offspring that they invest heavily in. As a result, their populations closely track the carrying capacity of their environment, and it often takes several years for them to recover from population crashes. Whales, elephants, and large birds of prey fall into this category. Two populations with different r-values but both tending toward the same K-value will follow very different paths in getting there.

Fields of application for the model

Wildlife biologists use this model to set hunting quotas and ensure that no more animals are caught than the population can recover. In lab experiments, the same curve represents a batch culture in which growth stops because of nutrient depletion and waste product accumulation. This is one reason why microbiologists attach great importance to this model. Conversely, it can be used to answer questions about recovery. Given a small starting population and a known r-value, it is possible to calculate how long it will take for the reintroduced species to reach its K-value. Most importantly, this is a minimal curve that preserves simplicity while recognizing that nothing can grow indefinitely.

Limitations of the model

Real populations do not always behave as the model predicts. Predators, diseases, weather conditions and delays between good conditions and resulting births can cause a population to overshoot or undershoot K. The model assumes constant r- and K-values and that overpopulation has immediate effects, which are simplifications made for ease of handling. Still it is a good starting point, providing a basic framework for more complex models and a clear method for describing density-dependent growth patterns. Often the most interesting part comes when data do not fit the curve; the deviations can often be revealing.