Mathematical Modeling in
Population Dynamics
Glenn Ledder
University of Nebraska-Lincoln
http://www.math.unl.edu/~gledder1
gledder@math.unl.edu
Supported by NSF grant DUE 0536508
Mathematical Model
Input Data
Math
Problem
Output Data
Key Question:
What is the relationship between input
and output data?
Endangered Species
Fixed
Parameters Mathematical Future
Model
Control
Population
Parameters
Model Analysis:
For a given set of fixed parameters, how
does the future population depend on the
control parameters?
Mathematical Modeling
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
A mathematical model represents a
simplified view of the real world.
• We want answers for the real world.
• But there is no guarantee that a
model will give the right answers!
Example: Mars Rover
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Newtonian physics
• Validation by many experiments
• Result:
Safe landing
Example: Financial Markets
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Conceptual Model:
Financial and credit markets are independent
Financial institutions are all independent
• Analysis:
Isolated failures and acceptable risk
• Validation??
• Result:
Oops!!
Forecasting the Election
Polls use conceptual models
• What fraction of people in each age group vote?
• Are cell phone users “different” from landline users?
and so on
http://www.fivethirtyeight.com
• Uses data from most polls
• Corrects for prior pollster results
• Corrects for errors in pollster conceptual models
Validation?
Most states within 2%!
General Predator-Prey Model
Let x be the biomass of prey.
Let y be the biomass of predators.
Let F(x) be the prey growth rate.
Let G(x) be the predation per predator.
Note that F and G depend only on x.
dx
 F ( x)  G ( x) y
dt
dy
 c G ( x) y  my
dt
c, m : conversion efficiency and starvation rate
Simplest Predator-Prey Model
Let x be the biomass of prey.
Let y be the biomass of predators.
Let F(x) be the prey growth rate.
Let G(x) be the predation rate per predator.
F(x) = rx :
Growth is proportional to population size.
G(x) = sx :
Predation is proportional to population size.
Lotka-Volterra model
x = prey, y = predator
x′ = rx – sxy
y′ = csxy – my
Lotka-Volterra dynamics
x = prey, y = predator
x′ = rx – sxy
y′ = csxy – my
Predicts oscillations of
varying amplitude
Predicts impossibility of
predator extinction.
• Logistic Growth
– Fixed environment capacity
x

F ( x)  rx1  
 K
Relative growth rate
r
F(X )
X
K
Logistic model
x = prey, y = predator
x
x′ = rx 1 – —
K
y′ = csxy – my
(
) – sxy
Logistic dynamics
x = prey, y = predator
x
x′ = rx 1 – —
K
y′ = csxy – my
(
) – sxy
Predicts y → 0 if m too
large
Logistic dynamics
x = prey, y = predator
x
x′ = rx 1 – —
K
y′ = csxy – my
(
) – sxy
Predicts stable xy
equilibrium if m is
small enough
OK, but real systems sometimes oscillate.
Predation with Saturation
• Good modeling requires scientific insight.
• Scientific insight requires observation.
• Predation experiments are difficult to do in
the real world.
• Bugbox-predator allows us to do the
experiments in a virtual world.
Predation with Saturation
The slope decreases from a maximum at x = 0
to 0 for x → ∞.
• Holling Type 2 consumption
– Saturation
Let s be search rate
Let G(x) be predation rate per predator
Let f be fraction of time spent searching
Let h be the time needed to handle one prey
G = fsx and f + hG = 1
sx
qx
G = —–––– = —–––
1 + shx
a+ x
Holling Type 2 model
x = prey, y = predator
x
qxy
x′ = rx 1 – — – —–––
K
a+ x
cqxy – my
y′ = —–––
a+ x
(
)
Holling Type 2 dynamics
x = prey, y = predator
x
qxy
x′ = rx 1 – — – —–––
K
a+ x
cqxy – my
y′ = —–––
a+ x
(
)
Predicts stable xy
equilibrium if m is small
enough and stable limit
cycle if m is even smaller.
Simplest Epidemic Model
Let S be the population of susceptibles.
Let I be the population of infectives.
Let μ be the disease mortality.
Let β be the infectivity.
No long-term population changes.
S′ = − βSI:
Infection is proportional to encounter rate.
I′ = βSI − μI :
Salton Sea problem
• Prey are fish; predators are birds.
• An SI disease infects some of the fish.
• Infected fish are much easier to catch than
healthy fish.
• Eating infected fish causes botulism
poisoning.
C__ and B__, Ecol Mod, 136(2001), 103:
1. Birds eat only infected fish.
2. Botulism death is proportional to bird
population.
CB model
S+I
S′ = rS (1− ——
)
−
βSI
K
qIy
I′ = βSI − —— − μI
a+I
cqIy
y′ = ——
−
my
−
py
a+I
CB dynamics
S+I
S′ = rS (1− ——
)
−
βSI
K
qIy
I′ = βSI − —— − μI
a+I
cqIy
y′ = ——
−
my
−
py
a+I
1. Mutual survival
possible.
2. y→0 if m+p too
big.
3. Limit cycles if m+p
too small.
4. I→0 if β too small.
CB dynamics
1. Mutual survival possible.
2. y→0 if m+e too big.
3. Limit cycles if m+e too small.
4. I→0 if β too small.
BUT
5. The model does not allow the predator
to survive without the disease!
DUH!
The birds have to eat healthy fish too!
REU 2002 corrections
•
Flake, Hoang, Perrigo,
Rose-Hulman Undergraduate Math Journal
Vol 4, Issue 1, 2003
1. The predator should be able to eat healthy
fish if there aren’t enough sick fish.
2. Predator death should be proportional to
consumption of sick fish.
CB model
S+I
S′ = rS (1− ——
)
−
βSI
K
qIy
I′ = βSI − —— − μI
a+I
cqIy
y′ = ——
−
my
−
py
a+I
Changes needed:
1. Fix predator death
rate.
2. Add predation of
healthy fish.
3. Change denominator
of predation term.
FHP model
S+I
qvSy
S′ = rS (1− ——
)
−
————
−
βSI
K
a + I + vS
qIy
I′ = βSI − ———— − μI
a + I + vS
cqvSy + cqIy − pqIy
y′ = ———————
− my
a + I + vS
Key Parameters:
K
R0 

mortality virulence
m
M
cq
FHP dynamics
pp>>cc
pp<<cc
FHP dynamics
FHP dynamics
FHP dynamics
FHP dynamics
FHP dynamics
pp>>cc
pp<<cc