A Derivation of the Logistic Equation

Joseph F. Heiser and Allen Martin

Abstract: The Verhulst-Pearl logistic equation, dN/dt = rN(1-N/K), is frequently utilized in modeling of population growth. The factor (1-N/K) is present in the equation to attenuate unrestricted growth, but the details of its origin are often presented without explanation. This article will attempt to derive the logistic equation from a set of basic assumptions regarding population growth.

It is believed that a population, N, will grow at a rate which is somehow dependent upon its size. If we ignore the effect of immigration and emigration, then one view is to express such growth as a differential equation, dN/dt = rN, where r = the difference between the birth rate, b, and the death rate, d, and t is a convenient unit of time. The solution yields an exponential function, N = N₀e^rt. If r is positive, one can readily deduce that the population will unrealistically grow to an unlimited size. While populations may manifest such growth patterns during periods of short duration, it is clear that there must be factors which attenuate growth in the long run. One model that takes these factors into account is the Verhulst-Pearl equation.

Since the resources of a population are generally limited, it is reasonable to assume that as its density increases and it approaches the carrying capacity of its environment, the birth rate will decline and the death rate increase. Thus we could redefine the birth rate as b = b₀ - k_bN, where b₀ is the initial growth rate and k_b is the rate at which the birth rate declines as N grows. Similarly, the death rate could be redefined as d = d₀ + k_dN, where d₀ is the initial death rate and k_d is the rate at which the death rate increases as N grows. In other words, the birth and death rates can be interpreted as linearly related to the size of the population. N will stabilize, or reach a fixed point, if b = d, or b₀ + k_bN = d₀ + k_dN. Solving for N, we get a carrying capacity K = (b₀ - d₀)/(k_b + k_d). Let r = b₀ - d₀. Then (k_b + k_d) = r/K. We will put this relationship aside for future reference.

Let us now return to our original differential equation, modify the rate of growth in terms of our revised ideas about b and d, and perform some algebraic simplification:

dN/dt = rN

= (b-d)N

= [(b₀ - k_bN) - (d₀ + k_dN)]N

= [(b₀ - d₀) - (k_b + k_d)N]N

= [r - (k_b + k_d)N]N Let (k_b + k_d) = r/K (from earlier calculation)

= [r - rN/K]N

= rN(1 - N/K) The Logistic Equation!

Thus, the net rate of growth in the logistic equation is continually changing. If N is small relative to its carrying capacity, K, the rate will be near to the constant rate, r, of the exponential model. As N nears K, the rate will shrink toward 0. Theoretically, when N reaches K a situation of equilibrium will be reached and the population will be stable. If N were to somehow exceed K, the rate would become negative and the population would decrease toward the carrying capacity. The following graphic illustrates the negative linear slope of the net rate relative to population size.

Woodrow Wilson Leadership Program in Mathematics lpt@www.woodrow.org
The Woodrow Wilson National Fellowship Foundation webmaster@woodrow.org
CN 5281, Princeton NJ 08543-5281 Tel:(609)452-7007 Fax:(609)452-0066