Example 5 The mathematical model of the population dynamics of four types of chicks, where the young can become mature within the same year, estimates the number of a female population after 5 years.
(1) Each female breeds once a year;
(2) Each female breeds only once in the first year of development, and does not breed again afterward, but still survives;
(3) Each female breeds once in the first two years, and then is eliminated (or dies).
(1996, "Mathematical Knowledge Application" Summer Camp)
Solution: Let $a_{n}$ represent the number of female birds in the $n$th generation.
(1) According to the problem, $a_{0}=1, a_{n}=2 a_{n-1}, n \geqslant 1$.

It is easy to get $a_{n}=2^{n} a_{0}=2^{n}$.
Therefore, $a_{5}=2^{5}=32$.
(2) According to the problem, $a_{0}=1, a_{n}=a_{n-1}+1$.

Thus, $a_{5}=6$.
(3) Let $b_{n}$ represent the number of young birds in the first year of development in the $n$th year, and $c_{n}$ represent the number of mature birds in the second year of development in the $n$th year. According to the problem,
$$
a_{n}=b_{n}+c_{n}, b_{n}=b_{n-1}+c_{n-1}, c_{n}=b_{n-1}.
$$

From the above equations, we get $a_{n}=a_{n-1}+a_{n-2}, n=2$. Initial conditions are $a_{0}=1, a_{1}=2$.
It is easy to see that $a_{5}=13$.