The Mathematics of The Mathematics of Populations… Populations…

……Malthus, Verhulst and Malthus, Verhulst and Logistic GrowthLogistic Growth

Thomas Robert Malthus (1766-Thomas Robert Malthus (1766-

1834)1834) and Limits to Growthand Limits to Growth ""The power of population is so superior to The power of population is so superior to

the power of the earth to produce the power of the earth to produce subsistence for man, that premature death subsistence for man, that premature death must in some shape or other visit the human must in some shape or other visit the human race. The vices of mankind are active and race. The vices of mankind are active and able ministers of depopulation. They are the able ministers of depopulation. They are the precursors in the great army of destruction; precursors in the great army of destruction; and often finish the dreadful work and often finish the dreadful work themselves. But should they fail in this war themselves. But should they fail in this war of extermination, sickly seasons, epidemics, of extermination, sickly seasons, epidemics, pestilence, and plague, advance in terrific pestilence, and plague, advance in terrific array, and sweep off their thousands and array, and sweep off their thousands and tens of thousands. Should success be still tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels in the rear, and with one mighty blow levels the population with the food of the worldthe population with the food of the world."."

An Essay on the Principle of Population, 1798An Essay on the Principle of Population, 1798

Malthus’ conjecture was based Malthus’ conjecture was based on…on…

The belief that populations grew at a The belief that populations grew at a geometric rate (exponentially) while geometric rate (exponentially) while production of food grew at a linear production of food grew at a linear rate. It is inevitable that demand will rate. It is inevitable that demand will always outstrip supply.always outstrip supply.

Compare Unlimited Growth Compare Unlimited Growth with Limited Growthwith Limited Growth

Exponential Growth Exponential Growth is defined as is defined as growth in which growth in which the rate of change the rate of change of the population is of the population is linearly dependent linearly dependent on the size of the on the size of the population:population:

( )( )

dN tN t

dt

Limits to GrowthLimits to Growth Recall a previous example that we did Recall a previous example that we did

on bacterial growth in a sandwich. Here on bacterial growth in a sandwich. Here is the slope field… is the slope field…

A more realistic model should A more realistic model should do this…do this…

The Verhulst Equation…The Verhulst Equation…

How can we modify How can we modify the exponential the exponential growth equation to growth equation to provide a more provide a more plausible model?plausible model?

Pierre Verhulst (1804-1849)

What our model should do…What our model should do…

Initially growth should be nearly Initially growth should be nearly exponentialexponential

Population should reach equilibrium Population should reach equilibrium (“carrying capacity”)(“carrying capacity”)

dN/dt should drop to zerodN/dt should drop to zero

( )dN

N N Mdt

(1 )

dNN N

dt

The Logistic EquationThe Logistic Equation

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

(1 )dN

N Ndt

A Numerical ApproachA Numerical Approach

Wait a minute! The analytic solution Wait a minute! The analytic solution was nice but it really assumes that was nice but it really assumes that the time step “dt” can shrink to zero the time step “dt” can shrink to zero – in real life that can’t happen. So….– in real life that can’t happen. So….

How do we know this works?

Look at Look at logistic.xls

Compare to real Compare to real datadata

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. A mistake has been made in the formal mathematical development;

b. The starting assumptions are incorrect and/or constitute too drastic an oversimplification;

c. One’s own intuition about the biological field is inadequately developed;

d. A penetrating new principle has been discovered.

Harvey J. GoldMathematical Modeling of Biological Systems

The Strange Tale of the Azuki The Strange Tale of the Azuki Bean Weevil!Bean Weevil!

Populations can exhibit chaotic Populations can exhibit chaotic behaviour!behaviour!