The model that underlies the sterile insect technique in this exercise is relatively straightforward to understand using the ideas of exponential growth and decay from Section 3.6 of the textbook as a starting point. Recall that a population of size P(t) grows exponentially if it satisfies Equation 3.6.3 of the textbook - that is
where k is a positive constant thought of as the per capita birth rate. Likewise P(t) decays exponentially if k is a negative constant, thought of as the per capita death rate.
With these ideas in mind, suppose a population is subject to both a constant per capita birth rate r and a constant per capita death rate d. Then the net per capita rate of growth will be r - d, and the population size P(t) will satisfy the equation
Let’s suppose the above equation applies to a normal population of insect pests, and let’s now consider how to modify this model if sterile males are introduced.
We start by viewing P(t) as representing the population size of females only, rather than the entire population. Further, we assume that prior to the introduction of sterile males, the sex ratio is equal (that is, the number of males equals the number of females).
Now if r is rate at which females give birth to other females in a normal population (that is, when they mate with fertile males) then once sterile males are released into the population this birth rate will decline. In particular, females will now mate with fertile males only a fraction M/(M+S) of the time, where M is the number of fertile males and S is the number of sterile males. Therefore, their birth rate r will be reduced to r M/(M+S). Furthermore, since the sex ratio prior to the release is equal, we know that M = P and therefore this can be written as r P/(P+S). Substituting this for r into the equation that P must satisfy gives the new model
As one last simplification, let’s now assume that d = 1. Then the new model can be rearranged to give
The population size P(t) must satisfy this new equation, and this will no longer correspond to exponential growth or decay. To find the relationship between P and t we will therefore need some new techniques. These will be developed later in the textbook. In particular the technique of separable equations in Section 7.4 shows that, to obtain a relationship between P and t, we need to integrate the reciprocal of the right side of the above equation. That is, we need to integrate
This is the integrand given in Exercise 5.6.24. The exercise asks you to integrate this function from the population size of females at initial release (that is, P=10,000) up to the population size at time t (that is, P=P(t)). The resulting quantity is then equal to the elapsed time t. This gives a relationship between the population size of females and time after the sterile males have been released.