where P(0) is the population size at time zero.
This model predicts that a population size will
grow exponentially if the birth rate exceeds the
death rate.
Biologists realize that many ecosystems cannot
support arbitrarily large population sizes (the Earth
can only supply a finite quantity of food resources
per year, for example), and so this simple model is
not considered realistic. Generally, as a population
increases and gets closer to a maximum capacity M,
say, the growth rate kdecreases. One can model this
by no longer assuming that kis constant, but varies
as follows:
k= M– P(t)
for example. (Here, when the population size P(t) is
close to M, the growth rate kis indeed small.) This
leads to the model:
= kP = (M– P)P
This is called the logistic growth model and was intro-
duced by Dutch biologist Pierre-François Verhulst
(1804–49). It is possible to solve this differential equa-
tion and obtain an explicit formula for the population
function P(t).However, one can quickly describe some
features of the population growth without any work.
For example,
1. If a population size starts at value M, then =
(M– M) · M= 0. This means there is no change in
the population size and the function P(t) forever
remains at the value M.
2. If a population size starts at a value greater than M,
then = (M– P) · P< 0. This means that the
population size will decrease. (There are not enough
food resources to support a large population.)
3. If a population size starts at a value less than M,
then = (M– P) · P> 0. This means that the
population size will increase. The rate of increase
decreases as the population size approaches M. The
graph of the function thus looks like an increasing
S-shaped curve trapped above the x-axis and below
the constant line P= M.
The logistic model works well to describe popula-
tion changes for simple biological systems (such as a
yeast culture), and, surprisingly, also worked well to
describe the U.S. population growth between the years
1920 and 1950. However, this is likely coincidental.
Human population growth is very difficult to model,
given unpredictable factors such as advances in medical
technology, wars, and, in modeling a specific country’s
population, immigration.
position vector Given a point Pin the plane, or in
three-dimensional space, the
VECTOR
represented by
the directed line segment
→
OP connecting the origin O
to Pis called the position vector of P. This vector has
the coordinates of Pas its components. For example, if
Pis the point (2,5), then its position vector is the vector
<2,5> = 2i+ 5j.
In physics, the position vector of a particle is
often denoted by r. It is a function of time tand is
usually expressed in the form r(t) = x(t)i+ y(t)j+
z(t)k, where x, y and zare functions of time. It repre-
sents the physical location of the particle at any time
t. The
DERIVATIVE
of the position vector is the
VELOC
-
ITY
vector v(t) = x′(t)i+ y′(t)j+ z′(t)k, and its double
derivative is the acceleration vector a(t) = x′′(t)i+
y′′(t)j+ z′′(t)k.
See also
PARAMETRIC EQUATIONS
.
positive A
REAL NUMBER
xis said to be positive if it
is greater than zero, that is, if x> 0. A real number less
than zero is called negative.
The product of two positive numbers is again posi-
tive. It is surprising that the rules of arithmetic dictate
that the product of two
NEGATIVE NUMBERS
must also
be positive. It follows then that for any nonzero (real)
number xwe must have x2> 0.
No
COMPLEX NUMBER
can be deemed positive or
negative. For example, if the number iis positive then,
by the previous statement, we must have i2> 0, yield-
ing the absurdity –1 > 0. The same conclusion would
follow if we were to deem inegative.
An unspecified real number that is positive or pos-
sibly zero is called nonnegative. One that is negative or
possibly zero is called nonpositive.
See also
ORDER PROPERTIES
.
dP
––
dt
dP
––
dt
dP
––
dt
dP
––
dt
408 position vector