Stability of the discrete population model with two delays

Stability of the population dynamics model y(n)=αy(n-m)/(1+βy(n-m)+γy(n-k)) is considered. Here y(n) is the size of the population at time n; k and m are delays (k,m ∈ N), α>1, β>0,γ>0. It appears that if β=0 then the necessary condition for stability of stationary solution is divisibility of k by m. Inequality 1>β> 1/2 is sufficient for asymptotic stability of stationary trajectory of model for every delays k,m. If k, m is mutually prime, k>m and m is even then the condition β> 1/2 is necessary too.