Abstract
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 min(m^{*},1), and the system will admit a unique globally asymptotically stable positive equilibrium for all min(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.
Highlights
IntroductionXiao and Lei Advances in Difference Equations (2018) 2018:245 selective harvesting: dx dt
His study showed that depending on the range of the parameter m, the system may collapse, or partially survive, or the two species could coexist in a stable state
Though Chakraborty et al [20] could not give a distinct analysis of the parameter m, both the works [21] and [22] showed that, depending on the range of the parameter m, the system they considered could collapse, or partially survive, or the two species could coexist in a stable state
Summary
Xiao and Lei Advances in Difference Equations (2018) 2018:245 selective harvesting: dx dt. By using the iterative technique method and fluctuation lemma, sufficient conditions which guarantee the global stability of the positive equilibrium and boundary equilibrium are obtained Their results indicate that for a stage-structured predator–prey community, both stage structure and the death rate of the mature species are the important factors that lead to the permanence or extinction of the system. By constructing a suitable Lyapunov function, the authors obtained a set of sufficient conditions which ensure the uniform persistence and global asymptotic stability of the system They showed that the constant prey refuge plays an important role in the coexistence of stage-structured predator–prey species. Lin [22] investigated the dynamic behaviors of the following two species commensal symbiosis model with non-monotonic functional response and non-selective harvesting in a partial closure: dx =x dt a1.
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