Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

3An informal introduction to stochastic differential equations

Let be the size of a wildlife population at time and let be the initial population size. Assume there are no food, territorial or other limitations to growth, so that the dynamics of the population size can be described by the Malthusian model

(3.1)

This model simply says that that the (instantaneous) growth rate of the population is proportional to population size, being the constant of proportionality. This is equivalent to saying that the (instantaneous) per capita growth rate is constant and equal to . Remember that is the difference between the (instantaneous per capita) birth and death rates, so it represents the net ...