A single species stage-structured system incorporating partial closure for the populations and non-selective harvesting is proposed and studied in this paper. Local and global stability property of the boundary equilibrium and the positive equilibrium are investigated, respectively. Our study shows that the birth rate of the immature species and the fraction of the stocks for harvesting play a crucial role in the dynamic behaviors of the system. If the birth rate of the immature species is too low, then the species will be driven to extinction; also, with the increase in the fraction of the stocks for harvesting, the speed of driving the species to extinction becomes increasing. If the birth rate of the immature species is large enough, then the system always admits a unique globally asymptotically stable positive equilibrium; however, with the increase in the harvesting area, the final density of the species is decreasing. If the birth rate of the immature species lies in an interval, then there exists a threshold m * such that the species will be driven to extinction for all m ∈ (m * , 1), and the system will admit a unique globally asymptotically stable positive equilibrium for all m ∈ (0, m *); also, with the increase in the parameter m, the system takes much time to reach its steady-state. For this case, though there are some natural protected areas where the harvesting of the species is forbidden, if the area is too small, the species will still be driven to extinction, that is, the small natural protected area has no influence on the protection of the endangered species. Such a finding maybe useful for human beings to design the protected areas for endangered species. Numeric simulations are carried out to show the feasibility of the main results.

A two species stage-structured commensalism model is proposed and studied in this paper. Local and global stability property of the boundary equilibrium and the positive equilibrium are investigated, respectively. If the stage-structured species is extinct, then depending on the intensity of cooperation, the species may still be extinct or become persistent. If the stage-structured species is permanent, then the final system is always globally asymptotically stable, which means the species is always permanent. Our study shows that increasing the intensity of the cooperation between the species is one of very useful methods to avoid extinction of the endangered species. Such a finding may be useful in protecting the endangered species. An example together with its numeric simulations is presented to verify our main results. MSC: 34C25; 92D25; 34D20; 34D40

In this paper, we propose and study a two-species stage structured amensalism model with a cover for the first species. By developing a new analysis technique or, more precisely, by combining the differential inequality theory and the Lyapunov function method, we obtain sufficient conditions ensuring the global attractivity of positive and boundary equilibria, respectively. Our study shows that the final density of the first species is an increasing function of the partial cover, and if the stage structured species is globally asymptotically stable, then there exists a threshold such that if the cover is greater than this threshold, the species can still exist in the long run, whereas if the cover is too small, then the first species is driven to extinction.

This article revisits the stability property of a symbiotic model of commensalism with Michaelis–Menten type harvesting in the first commensal populations. The model was proposed by Nurmaini Puspitasari et al. By constructing some suitable Lyapunov functions, we provide a thorough analysis of the dynamic behaviors of the subsystem composed of the second and third species. After that, by applying the stability results of this subsystem and combining with the differential inequality theory, sufficient conditions which ensure the global attractivity of the equilibria are obtained. The results obtained here essentially improve and generalize some known results.

In this paper, based on the model of Zhao, Qin, and Chen [Adv. Differ. Equ. 2018:172, 2018], we propose a new model of the May cooperative system with strong and weak cooperative partners. The model overcomes the drawback of the corresponding model of Zhao, Qin, and Chen. By using the differential inequality theory, a set of sufficient conditions that ensure the permanence of the system are obtained. By combining the differential inequality theory and the iterative method, a set of sufficient conditions that ensure the extinction of the weak partners and the attractivity of the strong partners and the other species is obtained. Numeric simulations show that too large transform rate will lead to more complicated fluctuation; however, the system is still permanent.