Sterile Male Mathematics




Let's imagine a hypothetical insect pest with an initial population of 2,000,000 individuals.   The sex ratio is 1:1, so there are one million males and one million females.   If each female produces an average of five daughters that live to reproduce, then the value of "R" (the population's replacement rate) equals 5.   This is a rapidly growing population!   In six generations, it will grow from one million females to 3.125 billion:

Generation
Number of Females
1
       1,000,000
2
       5,000,000
3
     25,000,000
4
   125,000,000
5
   625,000,000
6
3,125,000,000

Now let's try to control this population by releasing sterile males each generation.   If we can release 9 million sterile males during the first generation, then there will be a total of 10 million males competing for one million females.   Females will have only a 10% chance (1 in 10) of mating with a fertile male.   (Assume females mate only once and sterile males are equally competitive with fertile males for unmated females).   Continue to release 9 million males each generation and the population heads quickly toward extinction:

Number of Females in Population
 

Generation

If No Sterile Males

If Sterile Males Present
Ratio of Sterile
Males to Females
1
       1,000,000
1,000,000
           9:1
2
       5,000,000
   500,000
         18:1
3
     25,000,000
   131,000
         68:1
4
   125,000,000
       9,535
       944:1
5
   625,000,000
            50
180,000:1
6
3,125,000,000
              0

  1. What is the lowest ratio of males to females needed (in generation #1) to prevent this population from growing?
  1. If the intrinsic rate of increase equals 2.5, what is the lowest ratio of males to females needed to prevent growth?