$(5$ marks$)$ Suppose the population sizes $of$ two strains $of$ bacteria each grow $as$ described

$by$ the recursion relations:

$a_{t+1}=R_a{a}_t$ and $b_{t+1}=R_b{b}_t,$ respectively.

The frequency $of$ the first strain $at$ time $tis$ defined $as$

$p_t=\frac{a_t}{a_t+b_t}$

Derive a recursion equation for $p_t$ and show that $it$ can $be$ written $in$ terms $of$ a single

constant $C=\frac{R_a}{R_b}.$