Begin with bt wild type and mt mutants, but suppose that a fraction 0.2 mutate and that the per capita production of mutants is 1.0. Find the discrete-time dynamical system and the equilibrium fraction of mutants.
This series of problems combines mutation with selection. In one simple scenario, mutations occur in only one direction (wild type turn into mutants but not vice versa), but wild type and mutants have different levels of per capita production. Suppose that a fraction 0.1 of wild type mutate each generation, but that each wild-type individual produces 2.0 offspring while each mutant produces only 1.5 offspring. In each case, find the following.
a. The number of wild-type bacteria that mutate.
b. The number of wild-type bacteria and the number of mutants after mutation.
c. The number of wild-type bacteria and the number of mutants after reproduction.
d. The total number of bacteria after mutation and reproduction.
e. The fraction of mutants after mutation and reproduction.