Qualitative ideas and direction fields
Euler's Method
UBC Calculus Online Course Notes

Limited and Unlimited Growth

We have now seen the differential equation

\frac{dy}{dt} = k y 

and its solutions

y(t) = y_o e^{kt} 

in many contexts. One of these contexts was that of population growth. Although we arrived at it by looking originally at a doubling process involving bacterial growth, it is tempting to use similar ideas to model human populations.

The behaviour of the exponentially growing function and its implications for the population growth of humans made for shocking and sensational controversy two hundred years ago, when it was described by Thomas R. Malthus in his 1798 publication,

An essay on the principle of population as it affects the future improvement of society .

Malthus predicted that unless disasters or plagues limit the population on earth, simple exponential growth would result in unlimited population density whereas food supply could at best increase linearly. He concluded that mass-starvation, strife, and wars would be the lot of mankind.

A simple calculation, below convinces us that this scenario is certainly consistent with the predictions of the exponential growth equation.

Human population under unlimited growth: the Malthus equation

Suppose we let $ y(t) $ represent the earth's population, with $ t=0 $ corresponding to the year 2000. Census figures reveal that $ y(0)= 6 \times 10^9 $ is the expected population in the year 2000. Further, it is currently held that the world population doubles every 50 years. Suppose we use the simple model

\frac{dy}{dt} = ry 

where we have used the constant name $ r $ to signify "reproductive rate" in this equation (i.e. we have just replace the name of the constant $ k $ with $ r $ for convenience). Then we find that

y(t) = 6 \times 10^9~~ e ^{ \ln(2)~ t/50} 

This model would predict that in the year 2800 the population would be $ y = 4 \times 10^{14} $ people on Earth. This corresponds to one person for every square meter (including ocean surface) on the planet, a staggering result!

For your consideration:

  • (1) Carry out the calculations required to arrive at the above results.

  • (2) Do you believe this prediction ? Is there a problem with the way we arrived at it?

Changing the Model

The lack of realism of unlimited growth was evident soon after Malthus spread his message of doom and gloom. In 1838, the Belgian Pierre-Francois Verhulst suggested a revised model which would eliminate the undesirable effect of unlimited growth.

He suggested that when the population gets high, there is a tendency for individuals to spend more time interfereing with each other - fighting for food, or for scarce resources, killing each other over wars for land claims, or somehow increasing the rate of death. Since interference should be low when the density is low, and get more significant for a large population, Verhulst suggested modifying the previous model to read:

\frac{dy}{dt} = ry - \mu y^2 

The term containing the constant $ \mu $ (read "mu" for the Greek symbol) is the mortality. Note that it contributes negatively to the rate of change of the population - it tends to make the population decrease. We also comment that this term would be small when $ y $ is small, but would grow and dominate over the other term when $ y $ is large. We will see below that this has the desired effect of preventing population explosion.

Logistic Growth

Verhulst's equation has come to be known as the Logistic Growth Equation. It was noted that the same equation can be written in a variety of forms which are essentially the same but carry slightly different interpretations. One of these alternate forms is obtained by taking out a factor of $  r y $ from both terms above. We get:

\frac{dy}{dt} = ry (1 - \frac{\mu}{r} y) 

or, letting $ K = r/\mu $ , the equivalent version obtained by plugging in this new constant,

\frac{dy}{dt} = ry (1 - \frac{y}{K}) = ry \frac{(K - y)}{K} 

The latter is the most commonly used varient in the biological literature. In fact, the constants in this equation are named the basic reproductive rate of the population, $ r $ , and the carrying capacity of the environment, $ K $ , by biologists. The first stands for the innate birthrate of the population, under optimal conditions. The second constant, as we shall see below, represents the level of the population that can be sustained by the given environment.

The Logistic equation is used as a fundamental yardstick with which to understand population behaviour, and biologists talk about "r-selection" and "K-selection" to refer to various strategies that species adopt to outwit their competitors.

But before getting carried away with the terminology and the multiple ways of writing the same basic model for growth, let us make a few observations about its behaviour:

We notice that the population does not grow whenever

\frac{dy}{dt} = 0 

This occurs under two circumstances, either:

  • If $ y = 0 $ i.e. there is no population, and consequently no growth, or

  • If $ y = K $ . This happens if the population level is precisely equal to the "carrying capacity".

  • If we stick to the constants that we used in the first version of the model, another way to say the same thing is that at this level of population, the birth term and the death term in the model are delicately balanced.

ry - \mu y ^2 = 0, ~~~~~{\rm that~ is~~} \mu y ^2 = r y , ~~~~~  \mu y  = r 

    Thus $ y = r/\mu $ corresponds to a situation where there is no net growth taking place.

We also notice that the population grows whenever the sign of $ dy/dt $ is positive, whereas the population decreases when the sign of this derivative is negative.

Qualitative analysis

How would we understand the behaviour of this new, more complicated differential equation? We have no recourse to functions that automatically satisfy this new relationship between derivative and quadratic expression, but we can use the qualitative ideas that have been developed in the previous page.

Let us first summarize how the slope of tangent lines to the solutions of this equation would behave. Again we can do this by tabulating values of y and the associated values of the slope of the tangent line, $  dy/dt $ . Now, however, we will refrain from using numerical values, and express only the sizes and signs of the entries in the table.

Value of $ y $ 0 $  0 < y < K  $ K $  y > K $
Sign of $ dy/dt $ 0+0 -

We notice that $  y $ decreases and thus the slope of the tangent lines to be drawn is negative whenever $ y > K $ but that whenever $  0 < y < K  $ , the slopes are positive, and thus $ y $ increases. We show this behaviour in the picture below.

For your consideration:

  • (1) Experiment with the initial value of $ y $ by moving the red dot. What do you notice about the behaviour of the solution $ y(t) $ for various initial values?

  • (2) Under what circumstances does this model predict that there will be no population ?

  • (3) From the information we have discussed, and the labels on this diagram, can you tell what value of the constant $ K $ was used to prepare the diagram ?

  • (4) Is it also easy to figure out the value of the constant $ r $ just by looking at the diagram ? can you try to estimate this value? (Hint: place the red dot on the value y=1, and try to estimate the slope of the tangent line there.)

  • (5) Suppose you start with a positive value of y which is quite small. If we look at the behaviour of the solution over a very short time span, we might think that it is growing exponentially. Can you explain why this might be true?

We might summarize out discoveries on this page as follows:

The Logistic Equation

\frac{dy}{dt}  = ry \frac{(K - y)}{K} 

predicts that a population will grow only up to some limited level, $ K $ , called the Carrying Capacity. Quantitative analysis using the idea of the direction field can give us insight about the behaviour of this (and other) models for which we have no knowlege of the detailed solution of the differential equation.

Other Sources you may want to consult:

International Society of Malthus

Pierre Francois Verhulst

Applications of Differential Equations San Joaquin Delta College,

Population Dynamics University of South Alabama