Weak persistence of a stochastic delayed competition system with telephone noise and Allee effect

A stochastic competition system with Allee effect and distributed delay is established, which is utilized to investigate combined dynamic effects of telephone noise and Lévy jumps on stochastic population dynamics. Stochastically ultimate boundedness of the positive solution is studied. By constructing appropriate stochastic Lyapunov functions, sufficient conditions for weak persistence of the corresponding competitive population are discussed. •A stochastic dynamical system with Allee effect and distributed delay is proposed.•Combined dynamic effects of telephone noise and Lévy jumps are investigated.•Stochastically ultimate boundedness of solution to stochastic system is discussed.•Sufficient conditions for weak persistence of competitive population are studied.