Abstract: Based on the assumptions that the adult could kill and eat the juvenile of the same species and that there is the natural death of the juvenile, a two-stage-structured model with cannibalism is proposed in this paper. In the absence of cannibalism, the conditions ensuring the global stabilities of the population extinction and survival equilibria of the model are obtained by constructing the corresponding Lyapunov functions. In the presence of cannibalism, it is found that the model may have two population survival equilibria and that the saddle-node bifurcation can occur for certain parameter region, and the global dynamics is determined by constructing the Dulac function to rule out the existence of periodic solutions. The existence of two positive equilibria and the occurrence of the saddle-node bifurcation imply that the final state of the population growth depends on the initial condition of the model. The theoretic results obtained are verified by numerical simulation.

Key words: cannibalism, equilibrium, stability, saddle-node bifurcation