However, as the population grows, the ratio P K P K also grows, because K K is constant. If r > 0, r > 0, then the population grows rapidly, resembling exponential growth. Thus, the quantity in parentheses on the right-hand side of Equation 4.8 is close to 1, 1, and the right-hand side of this equation is close to r P. Then P K P K is small, possibly close to zero. Suppose that the initial population is small relative to the carrying capacity.
This differential equation can be coupled with the initial condition P ( 0 ) = P 0 P ( 0 ) = P 0 to form an initial-value problem for P ( t ). The logistic equation was first published by Pierre Verhulst in 1845. See this website for more information on the logistic equation. However, the concept of carrying capacity allows for the possibility that in a given area, only a certain number of a given organism or animal can thrive without running into resource issues. This possibility is not taken into account with exponential growth. Biologists have found that in many biological systems, the population grows until a certain steady-state population is reached.
A natural question to ask is whether the population growth rate stays constant, or whether it changes over time. The growth constant r r usually takes into consideration the birth and death rates but none of the other factors, and it can be interpreted as a net (birth minus death) percent growth rate per unit time. Various factors limit the rate of growth of a particular population, including birth rate, death rate, food supply, predators, and so on. This is unrealistic in a real-world setting. One problem with this function is its prediction that as time goes on, the population grows without bound. Furthermore, it states that the constant of proportionality never changes. Therefore the differential equation states that the rate at which the population increases is proportional to the population at that point in time. The right-hand side is equal to a positive constant multiplied by the current population. The left-hand side represents the rate at which the population increases (or decreases). This differential equation has an interesting interpretation. Figure 4.18 shows a graph of P ( t ) = 100 e 0.03 t. In this function, P ( t ) P ( t ) represents the population at time t, P 0 t, P 0 represents the initial population (population at time t = 0 ), t = 0 ), and the constant r > 0 r > 0 is called the growth rate. An example of an exponential growth function is P ( t ) = P 0 e r t. In Exponential Growth and Decay, we studied the exponential growth and decay of populations and radioactive substances. If P ( t ) P ( t ) is a differentiable function, then the first derivative d P d t d P d t represents the instantaneous rate of change of the population as a function of time. Therefore we use the notation P ( t ) P ( t ) for the population as a function of time. Since the population varies over time, it is understood to be a function of time. The variable P P will represent population. Any given problem must specify the units used in that particular problem. The units of time can be hours, days, weeks, months, or even years. To model population growth using a differential equation, we first need to introduce some variables and relevant terms. In this section, we study the logistic differential equation and see how it applies to the study of population dynamics in the context of biology. A more realistic model includes other factors that affect the growth of the population. We saw this in an earlier chapter in the section on exponential growth and decay, which is the simplest model.
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