Biology 12

478 MHR • Unit 5 Population Dynamics

equation that produces the J shape: dN/dt=rN.

But recall that in the case of logistic growth, the

rate of population change eventually starts to

decline. Therefore, something must be added to

this basic equation that will make dN/dtsmaller as

the population gets closer to its carrying capacity

(K). In other words, the growth rate should become

slower the closer Ngets to K, until the point where

Nequals Kand dN/dtequals 0 because the growth

rate is zero. This is accomplished by multiplying

the rNterm from the exponential equation by

what is often referred to as the “slowing term” —

(K−N)/K. The equation describing logistic growth

is therefore:

dN

dt

=rN

(

K−N

K

)

The slowing term in this equation reflects the

proportionof the environment’s carrying capacity

that has not been “filled up” by the population —

the space in the environment still available for

population growth. When Nis very small relative

to K, the slowing term will be close to 1. That is, if

Nis a lot less than K, then K−N≅K. Therefore

the slowing term will be approximately the same as

K/K, which equals one. This is what is shown on

the curve in Figure 14.12. When Nis small, the

population grows almost exponentially so that the

bottom portion of the curve is essentially J-shaped.

On the other hand, when Nis very close to K,

the slowing term will be close to zero. If N≅K,

then K−N≅ 0 and the slowing term will be

approximately zero. This is what happens in the

shoulder of the curve, where Napproaches Kand

thus K−Napproaches zero. Multiplying rNby

something so close to zero has the effect of

slowing dramatically what was previously simple

exponential growth.

Finally, as you have seen already, when N=K

the population stops increasing in size. The

slowing term has this effect because if N=K, then

K−N= 0 and (since rN× 0 = 0 ) dN/dt= 0. In

other words, when the population reaches the

carrying capacity of the environment, it theoretically

stops increasing in size because it has reached an

equilibrium point and therefore remains stable.

Do Real Populations Show

Logistic Growth?

The exponential and logistic growth curves shown

in Figures 14.10 and 14.12 are produced by the

mathematical equations you have just seen, and

therefore they are only theoretical. As a result,

these curves are often described as modelsof how

populations might grow under certain conditions.

Ecologists, like scientists in chemistry, physics, and

other fields of biology, use such models to help

them simplify and thus better understand complex

phenomena, such as how populations grow and

what limits their growth. The development of such

models involves a constant interplay between theory

and practice. Theoretical models are devised to

explain or mimic what has been observed in nature,

and are then tested by making more observations.

Lack of agreement between a theoretical model

and real data usually leads to a modification of

the model, so that it better fits (explains) real-life

situations.

The growth of many populations appears to fit

what would be predicted by the exponential model

— at least for a time (see Figure 14.13). Other

populations appear to better fit the logistic model.

This is particularly true of organisms in simplified

laboratory situations, where the environment can

be kept constant and there are no predators or

other species to interact with the population being

studied (see Figure 14.14A). But in more complicated

natural environments, where many abiotic and

biotic factors interact and conditions may fluctuate

unpredictably, the fit to the logistic model is often

not as good. This has led to modifications of the

logistic model, with different terms being added to

the equation to make it explain the growth of some

types of real populations. However, for now, you

will focus on the simplest form of the logistic

equation, since it illustrates well some of the

underlying principles of population growth.

Figure 14.13The growth patterns of forest tent caterpillar

(Malacosoma disstria) populations in Ontario show a series

of J-shaped growth curves followed by rapid declines. (The

area of defoliation is an indicator of caterpillar population

size.) What do you suppose caused the population to crash

in the late 1960s and again 10 years later?

(^019651975)

6

12

18

1985

Area of moderate-to-severe

defoliation (millions of hectares)

Year

Forest tent caterpillar defoliation

1956 – 1988