On bifurcations of delay single species discrete time population models

Our goal is an study on bifurcations of delay single species discrete time population models. Generally we transform the fixed point to origin. Conditions for existence of transcritical bifurcation, period doubling bifurcation, Neimark-Sacker bifurcation, resonance 1 : 3 bifurcation and resonance 1 : 4 bifurcation are studied. These bifurcations are analyzed by the centre manifold theorem and the normal form theory. In fact we investigate the non degeneracy conditions of generic bifurcations by computing the critical coefficients of the normal form of reduced system to the centre manifold. Then we apply our results for two kinds of population models. First we apply the results for the model x(n+1) = x(n)g(x(n-k)) and we will obtain the conditions for transcritical, period doubling, Neimark-Sacker, resonance 1 : 3 and resonance 1 : 4 bifurcations. Then we apply the results for two examples delay Logistic and delay Ricker models. Second we apply the results alpha for Clark's delayed recruitment model x(n+1) = alpha x(n) + g(x(n-k)) and we will obtain the conditions for mentioned bifurcations. Then we apply the results for the Clark's delayed recruitment model of Ricker type.