The model for the yeast population size in Equation 7.1.3 of the textbook can also be derived by starting with a model of population growth in discrete time. We start with the recursion for population size given by Equation 1.6.3 of the textbook. This equation is
The variable Nt is the yeast population size at time t, where time is measured in discrete intervals of 1 hour. The simplest assumption for the inflow and outflow is to suppose that each individual produces β offspring in one hour and has probability μ of itself dying. Thus the total inflow of new yeast cells is βNt and the total outflow is μNt. The recursion is given by
Now to model a population exhibiting continuous reproduction and death, consider what happens in a smaller interval of time. In a smaller time interval the number of births and the probability of death will be reduced. The simplest assumption is that both are proportional to the length of the time interval. For example, if we consider an interval of length h (where h is less than one hour) then a parent’s probability of death will be reduced from μ to μh. Likewise, the number of offspring it produces will be reduced from β to βh. Thus, for an interval of length h we have
Our next step is to change notation slightly, writing t and t+h as arguments of a function rather than as subscripts since this is the most common convention for continuous functions. We have
The final step is to rearrange this equation and divide by h to obtain
The left-hand side can be recognized as a difference quotient, and taking the limit of both sides as h goes to zero, making use of the definition of the derivative, gives
This is equation 7.1.3 of the textbook with r = β - μ.