A MATHEMATICAL MODEL
FOR A
SELF-LIMITING POPULATION
Glenn Ledder
gledder@math.unl.edu
University of Nebraska
THE CONCEPTUAL MODEL
1. A container holds p(t) microorganisms and a
quantity w(t) of waste, with
p(0) = p0 « 1 and w(0) = 0.
2. Waste production is proportional to population.
3. In absence of waste, population growth is
logistic.
4. Relative death rate increases linearly with
waste amount.
THE MATHEMATICAL MODEL
2. Waste production is proportional to population.
dw
dt
—– = kp
w(0) = 0
3. In absence of waste, population growth is
logistic.
4. Relative death rate increases linearly with
waste amount.
dp
p
—– = rp(1– —) – bwp
dt
M
p(0) = p0
NONDIMENSIONALIZATION
dw
—– = kp
dt
w(0) = 0
dp
p
—– = rp(1– —) – bwp
dt
M
Let
Then
p = MP
p(0) = p0
r
w=—
W
W′ = KP
P′ = P(1–P–W)
b
τ= rt
W(0) = 0
P(0)=P0
NULLCLINES
• W is fixed when P = 0 and increasing when P > 0.
• P is fixed when P = 0 and P+W = 1.
P
1
W
1
The DIFFERENTIAL EQUATION
for the TRAJECTORIES
We can combine the differential equations to obtain
the differential equation for the trajectories in the
phase plane.
Given
W′ = KP
and
P′ = P(1–P–W) :
dP P′ 1-P-W
Trajectories satisfy the equation —– = — = ——–
dW W′
K
CHANGE OF VARIABLES
dP = 1-P-W
The trajectory equation, —–
——– , is linear but
dW
K
not autonomous.
Let Z = P + W.
dZ
Then K –— = 1 + K – Z.
dW
This equation is autonomous as well as linear.
PHASE LINE
dZ
K –— = 1 + K – Z
dW
Z = P + W.
Z
1+K
1 + K is a stable equilibrium solution; however,
it is not achieved because W remains finite.
• P + W < 1 + K , and (P+W)′ > 0
TRAJECTORIES
We can solve the trajectory equation to get
P = 1 + K – W + (P0 –1–K) e-W/K
K=0.5
K=1.0
MAXIMUM POPULATION
P = 1 + K – W + (P0 –1–K) e-W/K
The maximum population occurs when P + W = 1.
1+K-P0
P = 1–W = 1–K ln ———
K
LIMITING BEHAVIOR
With P0 = 0 and P = 0, we have W=(1+K)(1-e-W/K).
s
su
(K, W∞) = ( ——, —— ), where u = -ln(1- s)
u-s u-s
EXTINCTION TIME
Let T be the time (in units of 1/r) at which P = 0.
From W′ = KP, we have
T=

W∞/K
0
du
——————————
1+K–Ku+(P0-1-K) e-u
P0=0.01