Hopf bifurcations in a predator-prey system of population allelopathy with a discrete delay and a distributed delay

摘要: A delayed Lotka–Volterra predator-prey system of population allelopathy with discrete delay and distributed maturation for the predator described by an integral a strong kernel is considered. By linearizing at positive equilibrium analyzing associated characteristic equation, asymptotic stability investigated Hopf bifurcations are demonstrated. Furthermore, direction bifurcation bifurcating periodic solutions determined normal form theory center manifold theorem functional differential equations. Finally, some numerical simulations carried out illustrating theoretical results.