"I think I may fairly make two postula. First, That food is necessary to the existence of man. Second, That the passion between the sexes is necessary and will remain nearly in its present state. Assuming then, my postula as granted, I say, that the power of population is indefinitely greater than the power in the earth to produce subsistence for man" Thomas Robert Malthus, 1789.
What Malthus realized in his Essay on the Principle of Population was that populations of humans, and in fact any self-replicating entity, grow geometrically, and that this kind of growth can overwhelm finite or arithmetically increasing resources. Geometric growth is most easily visualized by considering a single amoebae which, we assume, reproduces by division once every day. After two days there will be 2 amoebae, after 3 days 4, then 8, then 16, then 32, then 64, then 128 and so on. In 10 days there will be 1024 amoebae, in 20 one million, and in a month one trillion. Geometric growth is indeed a powerful force.
Geometric growth (also called exponential growth, as shown below) will be known, henceforth, as the first principle of population dynamics. By using the term "principle" we imply that geometric growth is a fundamental, if somewhat obvious, property of all living systems.
Let N0 be the original density of amoebae (1 individual) and N1 be the density of the population after one division. Signify the geometric rate of reproduction by G; in the case of amoebae, which reproduce by binary fission, G = 2. After one division the population of amoebae will have
individuals, and after 2 generations
and so on. However, we could have calculated the number of amoebae after 2 divisions by substituting the right-hand side of the first equation for N1 in the second
In fact we could just as easily write the general geometric growth equation
where Nt is the number of amoebae after t divisions.
It should be obvious that the geometric growth parameter G measures, not only the reproductive potential of the organism, but also its survival rate. In fact we can write G = 1+ B - D, where B and D the per-capita birth and death rates, respectively. G is often called the finite per-capita rate of change, or rate of growth, of the population over a unit period of time, usually a year or generation. Notice that this equation can also be used to calculate the growth of your compound interest savings account, with G the interest rate.
Taking the natural logarithm (logarithm to the base e = 2.71828..) of the finite growth equation gives the straight line relationship
(remember that the equation for a straight line is y = mx + a). This transformation tells us that populations grow at a constant rate when measured on the logarithmic scale. In addition, if we let R = ln G, then the equation becomes
and, taking antilogs, we have the exponential growth equation
where R is the instantaneous or exponential per-capita rate of change of the population.
Forest managers are often interested in forecasting the number or density of a pest population to be expected next year from the number present this year. In this case the time step in the geometric and exponential growth equations is one and so the equations simplify to
This model, which we will call the step-ahead forecasting equation, will be used throughout this course for predicting pest population changes. For a start let us apply the exponential step-ahead forecasting equation to see how populations grow under the action of the 1st principle of population dynamics. This is sometimes called transient dynamics because populations growing under the 1st principle are going eleswhere, sometimes towards very large numbers (R > 0), sometimes towards extinction (R < 0).
<Use BACK key to return to Sessions>
© 1998 Alan A. Berryman