Mathematical Modeling in Population Dynamics

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Mathematical Modeling in Population Dynamics

• Glenn Ledder
• University of Nebraska-Lincoln
• http//www.math.unl.edu/gledder1
• gledder_at_math.unl.edu

Supported by NSF grant DUE 0536508
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Mathematical Model
Math Problem
Input Data
Output Data
Key Question
What is the relationship between input and
output data?
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Endangered Species
Mathematical Model
Fixed Parameters
Future Population
Control Parameters
Model Analysis For a given set of fixed
parameters, how does the future population depend
on the control parameters?
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Mathematical Modeling
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation

• 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!

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Example Mars Rover
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation

• Conceptual Model
• Newtonian physics
• Validation by many experiments
• Result
• Safe landing

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Example Financial Markets
Real World
Conceptual Model
approximation
derivation
Mathematical Model
analysis
validation

• Conceptual Model
• Financial and credit markets are independent
• Financial institutions are all independent
• Analysis
• Isolated failures and acceptable risk
• Validation??
• Result Oops!!

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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!

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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
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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.

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Lotka-Volterra model
x prey, y predator x' r x s x y y' c
s x y m y
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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.

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• Logistic Growth
• Fixed environment capacity

Relative growth rate
r
K
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Logistic model

• x prey, y predator
• x' r x (1 ) s x y
• y' c s x y m y

x K
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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
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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.
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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.

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Predation with Saturation
The slope decreases from a maximum at x 0 to 0
for x ? 8.
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• 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
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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
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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
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Simplest Epidemic Model

• Let S be the population of susceptibles.
• Let I be the population of infectives.
• Let µ be the disease mortality.
• Let ß be the infectivity.
• No long-term population changes.
• S' - ßSI
• Infection is proportional to encounter rate.
• I' ßSI - µI

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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.

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CB model

• S' rS (1- ) - ßSI
• I' ßSI - - µI
• y' - my - py

S I K
qIy a I
cqIy a I
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CB dynamics

• S' rS (1- ) - ßSI
• I' ßSI - - µI
• y' - my - py

• Mutual survival possible.
• y?0 if mp too big.
• Limit cycles if mp too small.
• I?0 if ß too small.

S I K
qIy a I
cqIy a I
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CB dynamics

• Mutual survival possible.
• y?0 if me too big.
• Limit cycles if me too small.
• I?0 if ß too small.
• BUT
• The model does not allow the predator to survive
  without the disease!
• DUH!
• The birds have to eat healthy fish too!

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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 arent enough sick fish.
• Predator death should be proportional to
  consumption of sick fish.

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CB model

• S' rS (1- ) - ßSI
• I' ßSI - - µI
• y' - my - py

S I K

• Changes needed
• Fix predator death rate.
• Add predation of healthy fish.
• Change denominator of predation term.

qIy a I
cqIy a I
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FHP model

• S' rS (1- ) - - ßSI
• I' ßSI - - µI
• y' - my

S I K
qvSy a I vS
qIy a I vS
cqvSy cqIy - pqIy a I vS
Key Parameters
mortality virulence
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FHP dynamics
p gt c
p lt c
p gt c
p lt c
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FHP dynamics
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FHP dynamics
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FHP dynamics
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FHP dynamics
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FHP dynamics
p gt c
p lt c
p gt c
p lt c