Modeling Population Growth in Excel

The Malthus and Condorcet Equations, simple formulas that model relatively complex trends in population growth, are now accessible with an Excel calculator that allows the user full control over every component of the equations. Students can use the Excel file to model human population growth under the assumption that a human carrying capacity exists.

The Malthus Equation expresses the growth rate of a population as a function of the current population size and current carrying capacity. Specifically, the growth rate of a population is equal to a Malthusian parameter multiplied by the current population size multiplied by the difference between the current carrying capacity and the current population size. This relationship creates a high growth rate once a population is large enough to reproduce at its full potential, but remains a low growth rate when the population is very small or when a population is nearing its carrying capacity and feeling the effect of constrained resources. The Malthusian parameter is almost invariably between zero and one because a negative Malthusian parameter would lead to a population’s gradual extinction while a Malthusian parameter greater than one would lead to explosive population growth that would greatly exceed the carrying capacity. In the latter situation, unrealistically rapid and extreme periods of growth and contraction would ensue.

The Condorcet Equation expresses the growth rate of the carrying capacity of a population as equal to the growth rate of the population multiplied by a constant termed the Condorcet parameter. The logic behind this mathematical relationship is that the carrying capacity of a population increases or decreases proportionally with the growth rate of a population because an additional person in a population can have a positive or negative effect on the carrying capacity. This implies that a Condorcet parameter greater than one results from a society where an additional individual somehow increases the number of people that can be supported even when taking into account the resources that additional individual consumes; this could result from a situation where there are increasing returns to labor. If doctors cure diseases better when more of them work together, this is reflected by a Condorcet parameter greater than one. A Condorcet parameter between zero and one is most realistic for human populations because the contribution of another person will probably grow the carrying capacity but not by more than one. A negative parameter implies that an additional person would actually lower the carrying capacity; perhaps every additional person would consume natural resources at a rate greater than the previous individual’s rate.

As Cohen (1995 Science 269: 341-346) points out, the equations are not necessarily realistic models of human population growth. There is no consensus about whether or not a human carrying capacity exists. In theory, we as a species might be able to continually develop technology at such a rate that we are unable to approach a carrying capacity. A slowdown in overall human population growth is more likely due to a global increase in income per capita that leads to altered reproductive strategies.

With r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

Figure 1: with r=0.1 and c=0.1 as parameters, the population experiences a positive but steadily decreasing growth rate because the carrying capacity increases at 1/10th the rate of population growth, and since population growth slows as the population size approaches the carrying capacity, we observe almost asymptotic behavior. This is a realistic pattern for human population growth if a carrying capacity exists.

The calculator defines the Malthus Equation as dP(t)/dt=rP(t)[K(t)-P(t)] and the Condorcet Equation as dK(t)/dt=c dP(t)/dt (See Cohen 1995: 343). The user may enter values for the initial states of r (the “Malthusian parameter”), P(t), (population size), K(t) (carrying capacity), c (“Condorcet parameter”), t_0 (the starting time for the model) and dt (the length of one interval in time) that determine all of the future changes in population size. The rates of change of population and carrying capacity at time t, dP(t)/dt and dK(t)/dt respectively, are determined by the equations. The Malthusian and Condorcet parameters are constant in a growth model provided that there are no exogenous shocks that affect the nature of population or carrying capacity growth. Because of this, they do not vary as a function of t.

To explore the Malthus-Condorcet calculator, please follow this link to an automatic download of the Excel spreadsheet containing the calculator.

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