In a resource-limited ecological system, a population of organisms cannot keep growing forever (such as lab bacteria growing inside culture tube). The effective growth rate $g$ (including contributions from births and deaths) depends on the instantaneous abundance of resource $R(t)$, which in this problem we will consider the simple case of linear-dependency: $$\frac{d}{dt}N = gR(n)=\alpha RN,$$ where $N(t)$ is the population size at time $t$. The resources are consumed at a constant rate $\beta$ by each organism: $$\frac{d}{dt}R=-\beta N.$$ Initially, the total amount of resources is $R_0$ and the population size is $N_0$. Given that $\alpha=10^{-9} \text{resource-unit}^{-1} \text{s}^{-1}$, $\beta=1 \text{resource-unit/s}$, $R_0=10^6 \text{resource units}$ and $N_0=1 \text{cell}$, find the total time it takes from the beginning to when all resources are depleted (in hours).
In regular growth and decay problems, I'm good. But in this problem, the number of resources is involved and it's confusing me. Can someone give me a headstart on this?