 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $N_t=a_t+b_t$. Adding the two recurrences gives
$$
N_{t+1}-N_t=(3\mu-k)b_t.
$$
Since $b_t\geq0$, total population can increase only if $3\mu>k$. This is exactly
$$
R_0=\frac{4\mu}{\mu+k}>1,
$$
so the population-growth and offspring criteria agree.

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