Population Growth in a Structured Population

1
Population Growth in a Structured Population

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

Supported by NSF grant DUE 0536508
2
Population Growth

• Unstructured population model
• a model that counts all individuals together
• (discrete exponential function bt )
• Structured population model
• a model that counts individuals by category
• (not an elementary mathematical function)

3
Outline

1. Introduce mathematical modeling.
2. Introduce the mathematical model concept.
3. Use unstructured population growth as an example.
4. Model structured population growth.

4
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!

5
Mathematical Model
Mathematical Model
Input Data
Output Data
Key Question
What is the relationship between input and
output data?
6
Unstructured Population Growth --Approximation
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation

• Tomorrows population depends only on todays
  population.
• All individuals alive tomorrow are born today or
  survive from today to tomorrow.

7
Unstructured Population Growth -- Derivation
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation
Nt Nt1 todays and tomorrows populations
f s fecundity and survival parameters
Fecundity Survival
Growth Rate Population
8
Unstructured Population Growth -- Analysis
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation
Fecundity Survival
Growth Rate Population
Nt N0 (f s)t
Nt1/Nt f s
9
Unstructured Population Growth --Validation
Real World
Conceptual Model
Mathematical Model
approximation
derivation
analysis
validation

• Misses elements of chance.
• Misses environmental limitations.
• Pretty good for a short-time average.

10
Structured Population Growth

• Some populations have distinct reproductive and
  non-reproductive stages.
• Can we make a model for a structured population?
• Will we find Nt1/Nt f s ?

11
Getting Started

• A conceptual model requires scientific insight.
• We should observe experiments.
• Experiments for structured population growth are
  tricky, expensive, and time-consuming.

12

• Presenting Bugbox-population, a real biology lab
  for a virtual world.
• http//www.math.unl.edu/gledder1/BUGBOX/
• Boxbugs are simpler than real insects
• They dont move.
• Development rate is chosen by the experimenter.
• Each life stage has a distinctive appearance.

larva pupa adult

• Boxbugs progress from larva to pupa to adult.
• All boxbugs are female.
• Larva are born adjacent to their mother.

13
Structured Population Dynamics

• Species 1
• Let Lt be the number of larvae at time t.
• Let Pt be the number of juveniles at time t.
• Let At be the number of adults at time t.

Lt1 f At
Pt1 Lt
At1 Pt
14
Structured Population Dynamics

• Species 2
• Let Lt be the number of larvae at time t.
• Let Pt be the number of juveniles at time t.
• Let At be the number of adults at time t.

Lt1 f At
Pt1 p Lt
At1 Pt
15
Structured Population Dynamics

• Species 3
• Let Lt be the number of larvae at time t.
• Let Pt be the number of juveniles at time t.
• Let At be the number of adults at time t.

Lt1 f At
Pt1 p Lt
At1 Pt a At
16
Structured Population Dynamics

• Species 4
• Let Lt be the number of larvae at time t.
• Let Pt be the number of juveniles at time t.
• Let At be the number of adults at time t.

Lt1 s Lt f At
Pt1 p Lt
At1 Pt a At
17
Computer Simulation Results
A plot of Xt/Xt-1 shows that all variables tend
to a constant growth rate ?
The ratios LtAt and PtAt tend to constant
values.
18
Equation for Growth Rate
Nt1/Nt ? k (constant)
k (k-a) (k-s) pf
There is always a unique k that is larger than
both a and s.