Population Growth in a
Structured Population
Glenn Ledder
University of Nebraska-Lincoln
http://www.math.unl.edu/~gledder1
gledder@math.unl.edu
Supported by NSF grant DUE 0536508
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)
Outline
1. Introduce mathematical modeling.
2. Introduce the mathematical model concept.
3. Use unstructured population growth as an
example.
4. Model structured population growth.
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!
Mathematical Model
Input Data
Mathematical
Model
Output Data
Key Question:
What is the relationship between input
and output data?
Unstructured Population Growth -Approximation
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Tomorrow’s population depends only
on today’s population.
• All individuals alive tomorrow are born
today or survive from today to tomorrow.
Unstructured Population Growth -Derivation
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
Nt & Nt+1 : today’s and tomorrow’s populations
f & s : fecundity and survival parameters
Fecundity
& Survival
N t 1  f N t  sN t
Growth Rate
& Population
Unstructured Population Growth -Analysis
Real
World
approximation
validation
Fecundity
& Survival
Conceptual
Model
derivation
analysis
N t 1  f N t  sN t
Nt+1/Nt = f + s
Mathematical
Model
Growth Rate
& Population
Nt = N0 (f + s)t
Unstructured Population Growth -Validation
Real
World
approximation
validation
Conceptual
Model
derivation
analysis
Mathematical
Model
• Misses elements of chance.
• Misses environmental limitations.
• Pretty good for a short-time average.
Structured Population Growth
Some populations have distinct
reproductive and non-reproductive
stages.
1. Can we make a model for a structured
population?
2. Will we find Nt+1/Nt = f + s ?
Getting Started
• A conceptual model requires scientific
insight.
• We should observe experiments.
• Experiments for structured population
growth are tricky, expensive, and timeconsuming.
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 don’t move.
– Development rate is chosen by the experimenter.
– Each life stage has a distinctive appearance.
larva
pupa
• Boxbugs progress from larva to pupa to adult.
• All boxbugs are female.
• Larva are born adjacent to their mother.
adult
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.
Lt+1 = f At
Pt+1 = Lt
At+1 = Pt
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.
Lt+1 = f At
Pt+1 = p Lt
At+1 = Pt
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.
Lt+1 = f At
Pt+1 = p Lt
At+1 = Pt + a At
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.
Lt+1 = s Lt
+ f At
Pt+1 = p Lt
At+1 =
Pt + a At
Computer Simulation Results
A plot of Xt/Xt-1 shows
that all variables tend to
a constant growth rate λ
The ratios Lt:At
and Pt:At tend to
constant values.
Equation for Growth Rate
Nt+1/Nt → k
(constant)
k(k-a)(k-s) = pf
There is always a unique k that is
larger than both a and s.