MATH 164, Spring 2001 Due Date: Name(s):
Extra Project 10.5c: Another Population Model, Equilibria and Stability
Objective
To investigate a population model that is a refinement of the logistic model, equilibria and stability.
Narrative
If you have not already done so, do Project 10.5a on the logistic equation.
While the logistic equation models the growth of a population whose size is limited by the carrying capacity
of its environment, it does not take into account the fact that some populations require a minimal number
of individuals to survive. (Wolves, for example, survive in packs, and if a population of wolves drops below
the size of a pack then it will become extinct.) If P = P(t) denotes the size of a population at time t, L is
the carrying capacity of the environment supporting this population, and m  [0, L] is the minimal number
of individuals the population requires to survive, then the growth of this population can be modeled by the
equation
dP
dt
= -kP(m - P)(L - P) (1)
where k is a positive constant.
Task
1. Using Maple, draw (in one graphic):
a) the direction field associated to the equation P = -0.5P(1 - P)(5 - P) for t  [0, 4] and P  [0, 8],
and
b) the six solutions corresponding to the initial conditions P(0) = 0.25, P(0) = 0.75, P(0) = 1.25,
P(0) = 4.75, P(0) = 5.25, and P(0) = 6.0.
At this point, make a hard-copy of your typed input and Maple's responses. Then, ...
2. On the graph you produced for Task 1, label the coordinate axes, draw and label by hand the lines whose
equations are P = 1 and P = 5 (in this case, m = 1 and L = 5), and label the curves corresponding to
the six initial conditions. (Label the curve corresponding to P(0) = 1.5 by "P(0) = 1.5", for example.)
3. In one or two sentences, justify -- on the basis of population dynamics -- the fact that if the initial
level P(0) of a population whose growth satisfies equation (1), is:
a) between 0 and m then P decreases to 0 as t increases,
b) between m and L then P increases to L as t increases, and
c) greater than L then P decreases to L as t increases.
Comments
An equilibrium for a population is a level of P at which the population does not grow. Since, for the
population modeled by (1), dP/dt = 0 when P = 0, m, and L, this population has three equilibrium levels: if
P ever attains any one of the values 0, m, or L, it will neither increase nor decrease. As the computations you
have performed above illustrate, however, the behavior of P at these equilibria is different. The equiliibria
0 and L are stable because if P = 0 or P = L, and P is then changed by some small amount, P will return
to 0 or L, respectively. The equilibrium m is unstable, however: if P = m and P is decreased slightly, P will
eventually approach 0 (or move away from m), while if P = m and P is increased slightly, P will eventually
approach L (again moving away from m).
One way to think of this is in the context of a ball (see the figure below): If a ball is at rest at the bottom
of a valley and it is moved a small amount one way or another, it will return to its initial position; in this
case we say the ball is at a stable equilibrium. If a ball is at rest on the top of a hill and it is moved a small
amount one way or another, it moves away from its initial position; in this case we say the ball is at an
unstable equilibrium. A third possibility (which does not arise in considering equation (1)) is that a ball is
at rest on a flat surface: in this case, the ball is moved a small amount one way or another it remains where
it was moved; in this case we say the ball is at a neutral equilibrium.
Stable equilibrium Unstable equilibrium Neutrally stable equilibrium
From the point of view of MATH 163, what distinguishes the above situations is whether the graph of the
function f which describes the "landscape" has a second derivative f which is positive (the case of a stable
equilibrium), negative (the case of an unstable equilibrium), or sero (the case of a neutral equilibrium). From
the point of view of population dynamics, the analytic condition for a stable equilibrium is d2
P/dt2
> 0, for
an unstable equilibrium is d2
P/dt2
< 0, and for a neutral equilibrium is d2
P/dt2
= 0. Can you see why 0
and L are stable equilibria of (1), and m is an unstable equilibrium of (1)? Can you think of an example
that illustrates neutral stablity?
The ideas discussed in this project, and extensions of them to multiple interacting populations and stratified
populations, are considered further in courses dealing with population dynamics.