1. Modeling Population Growth Dr. Audi Byrne
MA354
Modeling Population Growth
Dr. Audi Byrne

2. A Population Let N(t) be the size of a population at time t.
By population, we could mean
The number of bacteria dividing in a petri-dish.
The number of rabbits in a predator-prey model.
The number of redwoods in a forest periodically ravaged by fire.
The number of people infected with an infection disease.
With some semantical gymnastics, we might also mean the size of a growing organism. (e.g., total cell population roughly proportional to size)

3. Model 1: Very Simple Linear Growth Leaf Production By a Growing Plant
Initially, there are no leaves on the plant.
Every day, the plant grows a new leaf.
Let N(t) be the population of leaves.

4. Model 2: Simple Exponential Growth: Bacteria Dividing in a Petri Dish
Initially, there is just one bacterial cell.
When the cell divides, it splits into two identical daughter cells.
The cell(s) divide every two hours.

5. Linear versus Exponential
Emphasis on difference in shape, not magnitude

6. Exponential versus Logistic

7. Logistic Growth

8. Models for Population Growth
N = births – deaths (for simplicity)
Models differ in how they describe births and deaths
If birth and death is independent of the population size, then growth will be linear.
If birth and death depend upon the population size:
If birth and death depends proportionally upon the population size, growth will be exponential.
If birth and death depends non-linearly upon the population size, then growth can be much more complex. (A common example is logistic growth.)

9. Models for Population Growth
Very generally,
N =  (increases in the population)
–  (decreases in the population)
Classically,
N = births – deaths + migration
“Conservation Equation”