|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:
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 mortality. Note that it contributes
negatively to the rate of change of the population - it tends to make the
population decrease. We also comment that this term
would be small when
is small, but would grow and
dominate over the other term when
is large. We will see
below that this has the desired effect of preventing population explosion.
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 reproductive rate of the population, , and the carrying capacity of the environment, , by biologists. The first stands for the innate birthrate of the population, under optimal conditions. The second constant, as we shall see below, represents the level of the population that can be sustained by the given environment.
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:
We also notice that the population grows whenever the sign of
is positive, whereas the population decreases
when the sign of this derivative is negative.
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:
We might summarize out discoveries on this page as follows:
Other Sources you may want to consult:
Applications of Differential Equations San Joaquin Delta College,
Population Dynamics University of South Alabama