Population Models - Nc State University · Note: The logistic model and Lotka-Volterra...
Transcript of Population Models - Nc State University · Note: The logistic model and Lotka-Volterra...
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Population Models
“I keep telling people I don’t see explosive population growth. I do see a heck ofa lot of building.” Hart Hodges
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Motivating ExampleMotivation: Use of the mosquitofish Gambusia Affinis is a popular means ofcontrolling mosquitos. This methods of control is a multi-million dollar industryworld wide. Modeling is required to answer the following questions:
• How many fish should be stocked in each paddy?
• How should the fish be initially stocked? All at once or periodically?
• How should they be stocked to augment the control already provided byendemic fish without highly damaging the local fish populations?
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Single Species Model
Notation:
Model 1:
Limitations: Does not incorporate overcrowding, depletion of resources, predators,size effects, etc.
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Single Species Model
Model 2: (Logistic model) Death rate is population dependent
Limitations: Does not incorporate predators or size effects.
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Predator-Prey Model
Notation:
Model: Lotka (1925), Volterra (1926)
Equilibria:
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Predator-Prey Model
Note:
Solution Behavior:
Limitations: Does not incorporate size effects, gender differences, outsideinfluences (e.g., effects of trapping).
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Predator-Prey Model
Example:
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Predator-Prey Model
Example:
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Predator-Prey Model
Example: Lynx-Hare data collected by the Hudson Bay Company
Question: Why does the hare populationlag behind the lynx population? See “DoHares Eat Lynx”, The American Naturalist,107(957), pp. 727-730, 1973.
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Size-Structured Models
Note: The logistic model and Lotka-Volterra predator-prey models are aggregatein the sense that they consider total populations in which individuals are assumedto have identical characteristics. Introducing individual characteristics is muchmore difficulty. Sinko and Streifer balanced the two approaches by assumingthat individuals share common traits.
Note:
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Sinko-Streifer Theory
Assumptions:
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Sinko-Streifer Model
Case 1: No deaths
Flux Balance:
First Principles Derivation:
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Sinko-Streifer Model
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Sinko-Streifer Model
Case 2: Include deaths using flux balance
Births:
Initial Condition:
Model:
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Sinko-Streifer Model
Solution: Determined by characteristics
Simplified Example:
Solution: Recall from the notes on analytic solutions for the wave equation
Total Derivative:
Characteristic Equation: Note:
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Sinko-Streifer Model
Model:
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Sinko-Streifer Model
Notation:
Note:
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Sinko-Streifer Model
Due to InitialPopulation
Due
to B
irths
Case 1:
Case 2:
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Sinko-Streifer Model
Example:
Model:
Characteristic Curve:
Initial Condition Driven Solution:
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Sinko-Streifer Model
Birth Driven Solution:
Example: Consider
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Sinko-Streifer Model
Example: Recruitment function
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Sinko-Streifer Model
Example:
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Sinko-Streifer Model
Example: