Population Modeling

1
Population Modeling

• Mathematical Biology Lecture 2
• James A. Glazier
• (Partially Based on Brittain Chapter 1)

2
Population Models

• Simple and a Good Introduction to Methods
• Two Types
• Continuum Britton, Chapter 1
• Discrete Britton, Chapter 2

3
Continuum Population Models

• Given a Population, N0 of an animal, cell,
  bacterium, at time t0, What is the Population
  N(t) at time t? Assume that the population is
  large so treat N as a continuous variable.
• Naively
• Continuum Models are Generally More Stable than
  discrete models (no chaos or oscillations)

4
Malthusian Model (Exponential Growth)

• For a Fertility Rate, b, a Death Rate, d, and no
  Migration

?
In reality have a saturation limited food,
disease, predation, reduced birth-rate from
crowding
5
Density Dependent Effects

• How to Introduce Density Dependent Effects?
• Decide on Essential Characteristics of Data.
• Write Simplest form of f(N) which Gives these
  Characteristics.
• Choose Model parameters to Fit Data
• Generally, Growth is Sigmoidal, i.e. small for
  small and large populations ?
• f(0) f(K) 0, where K is the Carrying
  Capacity and f(N) has a unique maximum for some
  value of N, Nmax
• The Simplest Possible Solution is the Verhulst or
  Logistic Equation

6
Verhulst or Logistic Equation

• A Key EquationWill Use Repeatedly
• Assume Death Rate ? N, or that Birth Rate
  Declines with Increasing N, Reaching 0 at the
  Carrying Capacity, K
• The Logistic Equation has a Closed-Form Solution

No Chaos in Continuum Logistic Equation
Click for Solution Details
7
Solving the Logistic Equation
2) Now let
1) Start with Logistic Equation
3) Substitute
4) Solve for N(t)
8
General Issues in Modeling

• Not a model unless we can explain why the death
  rate dN/K.
• Can always improve fit using more parameters.
• Meaningless unless we can justify them.
• Logistic Map has only three parameters N0, K, r
  doesn't fit real populations, very well. But we
  are not just curve fitting.
• Don't introduce parameters unless we know they
  describe a real mechanism in biology.
• Fitting changes in response to different
  parameters is much more useful than fitting a
  curve with a single set of parameters.

9
Idea Steady State or Fixed Point

• For a Differential Equation of Form
• is a Fixed Point ?
• So the Logistic Equation has Two Fixed Points,
  N0 and NK
• Fixed Points are also often designated x

10
Idea Stability

• Is the Fixed Point Stable?
• I.e. if you move a small distance e away from x0
  does x(t) return to x0?
• If so x0 is Stable, if not, x0 is Unstable.

11
Calculating Stability Linear Stability Analysis

• Consider a Fixed Point x0 and a Perturbation e.
  Assume that

• Taylor Expand f around x0

• So, if

Response Timescale, t, for disturbance to grow or
shrink by a factor of e is
12
Example Logistic Equation
Start with the Logistic Eqn.
Fixed Points at N00 and N0K
For N00
unstable
For N0K
stable
13
Phase Portraits

• Idea Describe Stability Behavior Graphically

Arrows show direction of Flow
Generally
14
Solution of the Logistic Equation
Solution of the Logistic Equation
For N0gtK, N(t) decreases exponentially to K For
N0lt2, K/N(t) increases sigmoidally to K For
K/2ltN0ltK, N(t) increases exponentially to K
15
Example Stability in Population Competition
Consider two species, N1 and N2, with growth
rates r1 and r2 and carrying capacities K1 and
K2, competing for the same resource. Both obey
Logistic Equation.
If one species has both bigger carrying capacity
and faster growth rate, it will displace the
other. What if one species has faster growth
rate and the other a greater carrying
capacity? An example of a serious
evolutionary/ecological question answerable with
simple mathematics.
16
Population CompetitionContd.
Start with all N1 and no N2. Represent population
as a vector (N1, N2) Steady state is (K1,0). What
if we introduce a few N2? In two dimensions we
need to look at the eigenvalues of the Jacobian
Matrix evaluated at the fixed point.
Evaluate at (K1,0).
17
Stability in Two Dimensions
Cases
1) Both Eigenvalues PositiveUnstable
2) One Eigenvalue Positive, One NegativeUnstable
1) Both Eigenvalues NegativeStable
18
Population CompetitionContd.
Eigenvalues are solutions of
-r1 always lt 0 so fixed point is stable ?
r2(1-K1/K2)lt0 i.e. if K1gtK2. Fixed Point Unstable
(i.e. species 2 Invades Successfully) ?
K2gtK1 Independent of r2! So high carrying
capacity wins out over high fertility (called
K-selection in evolutionary biology). A
surprising result. The opposite of what is
generally observed in nature.