1. The Mathematics of Populations…
…Malthus, Verhulst and Logistic Growth

2. Thomas Robert Malthus (1766-1834) and Limits to Growth
"The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction; and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague, advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world."
An Essay on the Principle of Population, 1798

3. Malthus’ conjecture was based on…
The belief that populations grew at a geometric rate (exponentially) while production of food grew at a linear rate. It is inevitable that demand will always outstrip supply.

4. Compare Unlimited Growth with Limited Growth
Exponential Growth is defined as growth in which the rate of change of the population is linearly dependent on the size of the population:

5. Limits to Growth
Recall a previous example that we did on bacterial growth in a sandwich. Here is the slope field…

6. A more realistic model should do this…

7. The Verhulst Equation…
How can we modify the exponential growth equation to provide a more plausible model?
Pierre Verhulst ( )

8. What our model should do…
Initially growth should be nearly exponential
Population should reach equilibrium (“carrying capacity”)
dN/dt should drop to zero

9. The Logistic Equation
This is called the logistic equation or Verhulst’s equation and is a very good description of simple populations. Let’s solve it…

10. A Numerical Approach
Wait a minute! The analytic solution was nice but it really assumes that the time step “dt” can shrink to zero – in real life that can’t happen. So….
How do we know this works?

11. Look at logistic.xls
Compare to real data

12. The result of mathematical development should be continuously checked against one’s own intuition about what constitutes reasonable biological behavior. When such a check reveals disagreement, then the following possibilities must be considered:
A mistake has been made in the formal mathematical development;
The starting assumptions are incorrect and/or constitute too drastic an oversimplification;
One’s own intuition about the biological field is inadequately developed;
A penetrating new principle has been discovered.
Harvey J. Gold Mathematical Modeling of Biological Systems

13. The Strange Tale of the Azuki Bean Weevil!

14. Populations can exhibit chaotic behaviour!