| UBC Calculus Online Course Notes |

We have now seen the differential equation
and its solutions
in many contexts. One of these contexts was that of population growth. Although
we arrived at it by looking originally at a doubling process involving
bacterial growth, it is tempting to
use similar ideas to model human populations.
A simple calculation, below convinces us that this scenario is certainly consistent with the predictions of the exponential growth equation.
Human population under unlimited growth: the Malthus equation
## For your consideration:- (1)
Carry out the calculations required to arrive at the above results.
- (2) Do you believe this prediction ? Is there a problem with the
way we arrived at it?
He suggested that when the population gets high, there is a tendency for individuals to spend more time interfereing with each other - fighting for food, or for scarce resources, killing each other over wars for land claims, or somehow increasing the rate of death. Since interference should be low when the density is low, and get more significant for a large population, Verhulst suggested modifying the previous model to read:
The term containing the constant
(read "mu" for
the Greek symbol) is the
or, letting , the equivalent version obtained by plugging in this new constant,
The latter is the most commonly used varient in the biological
literature. In fact, the constants in this equation are named
the basic The Logistic equation is used as a fundamental yardstick with which to understand population behaviour, and biologists talk about "r-selection" and "K-selection" to refer to various strategies that species adopt to outwit their competitors.
But before getting carried away with the terminology and the multiple
ways of writing the same basic model for growth, let us make a few observations
about its behaviour:
This occurs under two circumstances, either: - If
i.e. there is no population, and consequently
no growth, or
- If
. This happens
if the population level is precisely equal to the "carrying capacity".
- If we stick to the constants that we used
in the first version of the model,
another way to say the same thing is that at this level of
population, the birth term and the death term in the model
are delicately balanced.
Thus corresponds to a situation where there is no net growth taking place.
We also notice that the population Let us first summarize how the slope of tangent lines to the solutions of this equation would behave. Again we can do this by tabulating values of y and the associated values of the slope of the tangent line, . Now, however, we will refrain from using numerical values, and express only the sizes and signs of the entries in the table.
We notice that
decreases and thus the slope of the tangent lines
to be drawn is negative whenever
but that whenever
, the slopes are positive, and thus
increases. We show this behaviour in the picture below.
## For your consideration:- (1) Experiment with the initial value of
by moving the red
dot. What do you notice about the behaviour of the solution
for various initial values?
- (2) Under what circumstances does this model predict that there
will be no population ?
- (3) From the information we have discussed, and the labels on this
diagram, can you tell what value of the constant
was
used to prepare the diagram ?
- (4) Is it also easy to figure out the value of the constant
just by looking at the diagram ? can you try to estimate this value?
(Hint: place the red dot on the value y=1, and try to estimate the slope of
the tangent line there.)
- (5) Suppose you start with a positive value of y which is quite small. If we look at the behaviour of the solution over a very short time span, we might think that it is growing exponentially. Can you explain why this might be true?
We might summarize out discoveries on this page as follows:
Other Sources you may want to consult: International Society of Malthus Applications of Differential Equations San Joaquin Delta College, Population Dynamics University of South Alabama |