MATH 2250 Linear Algebra and Differential Equations Chapter 2
Chapter 2 Mathematical Models and Numerical Methods
2.1 Population Models
Birth rate and death rate are defined as follows:
)(t
is the number of births per unit of population per unit of time at time t;
)(t
is the number of births per unit of population per unit of time at time t;
Therefore, the total number of births in a population per unit time is
P
.
If the birth rate
)(t
and death rate
)(t
of a given population are constant, then under ideal
conditions, the time rate of change in population is proportional to the population itself:
kP
dt
dP
.
However, there are a few practical issues for this model.
Conditions are likely not ideal and change in time
If conditions change, then the birth and death rates will likely change in time
More fundamentally, population is not a continuous process (you cannot have 13.675
people)
Nonetheless, a continuous solution allows us to more accurately model the discrete (integer
valued) behavior. It is the dynamic character of change that we want to address in this section.
Specifically, if the birth rate
)(t
and death rate
)(t
are changing, then we have
)()]()([ tPtt
t
P
for small values of
t
.
In the limiting case,
P
dt
dP ][
.
Bounded Populations and the Logistic Model
A common situation is where a population has a limiting capacity due to restricted resources and
space. In this case, the limiting population, M, plays a significant role. For purposes of
illustration, suppose a population satisfies the differential equation
)10( PP
dt
dP
.
We can solve this by separating variables, but before we do, let’s examine the slope field for
this differential equation.
Clearly, if P = 0 or P = 10,
0
dt
dP
, so these values correspond to constant or equilibrium
solutions. That is, if the population is ever zero, it remains zero. Likewise, if the population is
ever 10, it remains 10 for all time.
Furthermore, if 0 < P < 10, the product of P and (10 – P) is positive, so the population is
increasing in this case so solution curves in this region approach P = 10 (not P = 0).
