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

Population Models Simple and a Good Introduction to Methods Two Types Continuum [Britton, Chapter 1] Discrete [Britton, Chapter 2]

Continuum Population Models Given a Population, N 0 of an animal, cell, bacterium,… at time t =0, 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)

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…

Density Dependent Effects How to Introduce Density Dependent Effects? 1)Decide on Essential Characteristics of Data. 2)Write Simplest form of f(N) which Gives these Characteristics. 3)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, N max The Simplest Possible Solution is the Verhulst or Logistic Equation

Verhulst or Logistic Equation A Key Equation—Will 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

Solving the Logistic Equation 2) Now let: 3) Substitute: 1) Start with Logistic Equation: 4) Solve for N(t):

General Issues in Modeling Not a model unless we can explain why the death rate d~N/K. Can always improve fit using more parameters. Meaningless unless we can justify them. Logistic Map has only three parameters N 0, 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.

Idea: Steady State or Fixed Point For a Differential Equation of Form is a Fixed Point  So the Logistic Equation has Two Fixed Points, N =0 and N = K Fixed Points are also often designated x *

Idea: Stability Is the Fixed Point Stable? I.e. if you move a small distance e away from x 0 does x ( t ) return to x 0 ? If so x 0 is Stable, if not, x 0 is Unstable.

Calculating Stability: Linear Stability Analysis Consider a Fixed Point x 0 and a Perturbation . Assume that: Taylor Expand f around x 0 : So, if Response Timescale, , for disturbance to grow or shrink by a factor of e is:

Example Logistic Equation Start with the Logistic Eqn. Fixed Points at N 0 =0 and N 0 =K For N 0 =0 For N 0 =K stable unstable

Phase Portraits Idea: Describe Stability Behavior Graphically Arrows show direction of Flow Generally:

Solution of the Logistic Equation Solution of the Logistic Equation: For N 0 >K, N(t) decreases exponentially to K For N 0 <2, K/N(t) increases sigmoidally to K For K/2<N 0 <K, N(t) increases exponentially to K

Example Stability in Population Competition Consider two species, N 1 and N 2, with growth rates r 1 and r 2 and carrying capacities K 1 and K 2, 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.

Population Competition—Contd. Start with all N 1 and no N 2. Represent population as a vector ( N 1, N 2 ) Steady state is (K 1,0). What if we introduce a few N 2 ? In two dimensions we need to look at the eigenvalues of the Jacobian Matrix evaluated at the fixed point. Evaluate at (K 1,0).

Stability in Two Dimensions 1) Both Eigenvalues Positive—Unstable Cases: 2) One Eigenvalue Positive, One Negative—Unstable 1) Both Eigenvalues Negative—Stable

Population Competition—Contd. - r 1 always K 2. Fixed Point Unstable (i.e. species 2 Invades Successfully)  K 2 >K 1 Independent of r 2 ! 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. Eigenvalues are solutions of