Consider $m$ populations with densities $n_{k}(t)$ at time $t$ that compete for a unique food resource that is renovated at constant rate, $K > 0$, and consumed by the members of each population at a rate $a_{k} > 0$. The the total consumption is $ \sum_{j=1}^{m}a_{j}n_{j}(t)$ and the lost is $ K - \sum_{j=1}^{m}a_{j}n_{j}(t)$, suppose that the rate of reproduction is given by $ \frac{1}{n_{k}} \frac{dn_{k}}{dt} = c_{k}( K - \sum_{j=1}^{m}a_{j}n_{j}(t)), c_{k} > 0$.
Show for two populations that if $n_{k}(0)$ is such that $K - \sum_{j=1}^{m}a_{j}n_{j}(0) \ge 0$ then the populations tend to an equilibrium.
Calculate $n_{k}(\infty)$ for the populations.
Can somebody help? I have no idea what to do.