. . Population growth Now, we consider the growth of a population (for example, human, an animal, or a bacteria colony) as a function of time t . Let x t ( ) be the size of the population at any time t .
Although x t ( ) is integer-valued, we approximate x t ( ) as a differentiable function and techniques of differential equation can be applied to determine x t ( ) . Assume that population grows at a rate directly proportional to the amount of population present at that time. Then, we obtain dt kx , where k is the constant of proportionality . … ( ) Here k > , since the population always increases.
The solution of the differential equation is x t Ce kt ( ) = , where C is a constant of integration. The values of C and k are determined with the help of initial conditions. Thus, the population increases exponentially with time. This law of population growth is called Malthusian law .
Example . The growth of a population is proportional to the number present. If the population of a colony doubles in years, in how many years will the population become triple? Let x t ( ) be the population at time t .
Then dx dt kx By separating the variables, we obtain dx kdt Integrating on both sides, we get, log kt or x Ce kt , where C is an arbitrary constant. Let x be the population when t = and obtain C . Thus, we get x x e kt Now x = , when t = and thus, k = Hence, x is the population at any time t . - - Ordinary Differential Equations Assume that the population is tripled in t years.
That is, x = , when t = . Thus, t .Therefore, the population is tripled in years.