How to satisfy a logistic growth equation?

Two bacterial populations, Q(t) and Z(t) are umed to grow according to Euler exponential growth model, with parameters a and b. Suppose that the two populations are grown together in a beaker. W(t) = Q(t) / (Q(t) + Z(t)) to be the fraction of the total population of Q type. Using differential equations for Q and Z, show that W(t) satisfies a logistic growth equation.