1. Mathematical Modeling in Population Dynamics
Glenn Ledder
University of Nebraska-Lincoln
Supported by NSF grant DUE

2. Mathematical Model
Math
Problem
Input Data
Output Data
Key Question:
What is the relationship between input
and output data?

3. Endangered Species Fixed Parameters Mathematical Model Future
Population
Control
Parameters
Model Analysis:
For a given set of fixed parameters, how does the future population depend on the control parameters?

4. Mathematical Modeling
Real
World
approximation
Conceptual
Model
derivation
Mathematical
Model
validation
analysis
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!

5. Example: Mars Rover Conceptual Model: Newtonian physics
Real
World
approximation
Conceptual
Model
derivation
Mathematical
Model
validation
analysis
Conceptual Model:
Newtonian physics
Validation by many experiments
Result:
Safe landing

6. Example: Financial Markets
Real
World
approximation
Conceptual
Model
derivation
Mathematical
Model
validation
analysis
Conceptual Model:
Financial and credit markets are independent
Financial institutions are all independent
Analysis:
Isolated failures and acceptable risk
Validation??
Result: Oops!!

7. 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
Uses data from most polls
Corrects for prior pollster results
Corrects for errors in pollster conceptual models
Validation?
Most states within 2%!

8. 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.
c, m : conversion efficiency and starvation rate

9. 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.

10. Lotka-Volterra model x = prey, y = predator x′ = r x – s x y
y′ = c s x y – m y

11. Lotka-Volterra dynamics
x = prey, y = predator
x′ = r x – s x y
y′ = c s x y – m y
Predicts oscillations of varying amplitude
Predicts impossibility of predator extinction.

12. Logistic Growth
Fixed environment capacity
Relative growth rate r K

13. Logistic model x = prey, y = predator x′ = r x (1 – — ) – s x y
y′ = c s x y – m y x K

14. Logistic dynamics x = prey, y = predator x′ = r x (1 – — ) – s x y
y′ = c s x y – m y
Predicts y → 0 if m too large x K

15. Logistic dynamics x = prey, y = predator x′ = r x (1 – — ) – s x y
y′ = c s x y – m y
Predicts stable x y equilibrium if m is small enough x K
OK, but real systems sometimes oscillate.

16. 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.

17. Predation with Saturation
The slope decreases from a maximum at x = 0
to 0 for x → ∞.

18. 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 = f s x and f + h G = 1
G = —–––– = —–––
s x
1 + sh x
q x
a + x

19. Holling Type 2 model x = prey, y = predator x′ = r x (1 – — ) – —–––
y′ = —––– – m y x K
qx y
a + x
c q x y
a + x

20. Holling Type 2 dynamics x = prey, y = predator
x′ = r x (1 – — ) – —–––
y′ = —––– – m y
Predicts stable x y equilibrium if m is small enough and stable limit cycle if m is even smaller. x K
qx y
a + x
c q x y
a + x

21. 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 :

22. 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:
Birds eat only infected fish.
Botulism death is proportional to bird population.

23. CB model S′ = rS (1− ——) − βSI I′ = βSI − —— − μI y′ = —— − my − pyK
qIy
a + I
cqIy
a + I

24. CB dynamics S′ = rS (1− ——) − βSI I′ = βSI − —— − μI y′ = —— − my − pyK
Mutual survival possible.
y→0 if m+p too big.
Limit cycles if m+p too small.
I→0 if β too small.
qIy
a + I
cqIy
a + I

25. CB dynamics y→0 if m+e too big. Mutual survival possible.
Limit cycles if m+e too small.
I→0 if β too small.
BUT
The model does not allow the predator to survive without the disease!
DUH!
The birds have to eat healthy fish too!

26. REU 2002 corrections Flake, Hoang, Perrigo,
Rose-Hulman Undergraduate Math Journal
Vol 4, Issue 1, 2003
The predator should be able to eat healthy fish if there aren’t enough sick fish.
Predator death should be proportional to consumption of sick fish.

27. CB model S′ = rS (1− ——) − βSI I′ = βSI − —— − μI y′ = —— − my − pyK
Changes needed:
Fix predator death rate.
Add predation of healthy fish.
Change denominator of predation term.
qIy
a + I
cqIy
a + I

28. FHP model
S′ = rS (1− ——) − ———— − βSI I′ = βSI − ———— − μI y′ = ——————— − my
S + I K
qvSy
a + I + vS
qIy
a + I + vS
cqvSy + cqIy − pqIy
a + I + vS
Key Parameters:
mortality virulence

29. FHP dynamics
p > c
p > c
p < c
p < c

30. FHP dynamics

31. FHP dynamics

32. FHP dynamics

33. FHP dynamics

34. FHP dynamics
p > c
p > c
p < c
p < c