**Density-Dependent Population Growth Simulation**

Recall Equation 42.4 for multiplicative population growth:

**Δ N / ΔT = r N**

If the time Δ*T* between breeding episodes in the population becomes vanishingly small
(as in human populations—births occur throughout the year rather than seasonally), this can be written as a
differential equation (see Animated Tutorial 42.1, Multiplicative Population Growth Simulation):

**d N / dt = r N**

If we divide both sides by *N*, the number of individuals in the population (the population
size), we arrive at the average individual's contribution to population growth:

**1/ N dN/dt = r**

This makes it clear that the multiplicative model assumes a constant *per capita* contribution to
population growth, *r*. If *r* is greater than zero the population will grow to infinity, which is
unrealistic. In reality, the per capita contribution to growth will decline as population size increases,
because each individual will have a smaller share of essential resources.

The simplest form of such *density dependence* in the per capita contribution to growth assumes
that this contribution is at the maximum value of *r* only when the population size *N*
is very close to zero, and declines linearly as *N* grows larger. At
some population size *K*, which we call the *carrying capacity*, per capita
birth and death rates are equal, per capita contribution to growth has declined
to zero, and the population stops growing.

Recall that the equation for a straight line is y = *a*x + *b*.
In this case y = 1/*N* d*N*/d*t*, the per capita contribution, and x = *N*, the population size.
The intercept *b* of the straight line describing the decline in per capita
contribution is *r*, and its slope is –(*r*/*K*). Thus the linear decline is described formally as

**1/ N dN/dt = r – (r/K) N**

This is one form of the so-called *Logistic Equation* for density-dependent population growth.
Logistic growth is not universal, but it serves to show general properties of density-dependent population growth.
How does the size of a population growing logistically change through time? To find out click the **Graph**
tab, and try entering *N*_{0} = 100, *r* = 1.0, and *K* = 1000. Click on **New Plot** to see a plot for given values
of *N*_{0}, *r*, and *K*. The dashed horizontal line is the *carrying capacity*. The solid line is the logistic growth plot.
Exponential growth (same values of *N*_{0} and *r*) is shown as "+."
Also try setting *N*_{0} equal to *K* and *N*_{0} larger than *K*. To see the value of
*N* at a given time, click the desired location.

You now see that if resources are limited and growth is density dependent, populations follow an S-shaped growth
trajectory if *N*_{0} < *K* and *r* > 0: growth is rapid initially and then slows as *K* is approached.
This is in stark contrast to the continued acceleration of density-independent growth, which assumes that resources are infinite and
that there is no carrying capacity. If the initial population is far below *K*, however, its initial trajectory does resemble that of
exponential growth. If *N*_{0} > *K* and *r* > 0, however, the population *decreases* rapidly at first,
and then more slowly as it approaches *K*. And if *N*_{0} = *K*, the population size does not change through
time—*K* represents an *equilibrium* population size.
Finally, note that the logistic is only one form of density dependence, because the decline in per capita population
growth might take on any number of nonlinear forms, rather than the linear form
assumed in the logistic. Such nonlinearities are common with populations of real organisms.

**Textbook Reference:** Concept 42.4 Populations Grow Multiplicatively, but the Multiplier Can Change