The aim of this paper is to investigate the dynamic behaviors of the following single species stagestructured system incorporating partial closure for the populations and nonselective harvesting:
$$ \begin{gathered} \frac{dx_{1}}{dt}=\alpha x_{2}\beta x_{1}\delta_{1} x_{1}q_{1}E mx_{1}, \\ \frac{dx_{2}}{dt}=\beta x_{1}\delta_{2} x_{2} \gamma x_{2}^{2}q_{2}E mx_{2}, \end{gathered} $$
(1.1)
where α, β, \(\delta_{1}\), \(\delta_{2}\), \(q_{1}\), \(q_{2}\), E, and γ are all positive constants, \(x_{1}(t)\) and \(x_{2}(t)\) are the densities of the immature and mature species at time t, the following assumptions are made in formulating model (1.1):

1.
The per capita birth rate of the immature population is \(\alpha> 0\); The per capita death rate of the immature population is \(\delta_{1}> 0\); The per capita death rate of the mature population is proportional to the current mature population with a proportionality constant \(\delta_{2} > 0\); \(\beta>0\) denotes the surviving rate of immaturity to reach maturity; The mature species is density dependent with the parameter \(\gamma>0\);

2.
E is the combined fishing effort used to harvest and m (\(0< m<1\)) is the fraction of the stock available for harvesting.
During the last decades, many scholars investigated the dynamic behaviors of the stagestructured species, see [1–16] and the references cited therein. Among those works, there are two typical ideas used to establish the model.
(1) Assume that the immature species needs time to grown up, and denote this periodic as τ, this leads to the time delay model. For example, Chen, Chen, et al. [1], Chen, Xie, et al. [2], Chen, Wang, et al. [3], and Ma, Li, et al. [4] studied the dynamic behaviors of the following stagestructured predator–prey model:
$$ \begin{gathered} \dot{x}_{1}(t)=r_{1}(t)x_{2}(t) d_{11}x_{1}(t)r_{1}(t\tau_{1})e^{d_{11}\tau_{1}}x_{2}(t \tau_{1}), \\ \dot{x}_{2}(t)=r_{1}(t\tau_{1})e^{d_{11}\tau_{1}}x_{2}(t \tau _{1})d_{12}x_{2}(t) \\ \hphantom{\dot{x}_{2}(t)=}{}b_{1}(t)x_{2}^{2}(t)c_{1}(t)x_{2}(t)y_{2}(t), \\ \dot{y}_{1}(t)=r_{2}(t)y_{2}(t) d_{22}y_{1}(t)r_{2}(t\tau_{2})e^{d_{22}\tau_{2}}y_{2}(t \tau_{2}), \\ \dot{y}_{2}(t)=r_{2}(t\tau_{2})e^{d_{22}\tau_{2}}y_{2}(t \tau _{2})d_{21}y_{2}(t) \\ \hphantom{\dot{y}_{2}(t)=}{}b_{2}(t)y_{2}^{2}(t)+c_{2}(t)y_{2}(t)x_{2}(t), \end{gathered} $$
(1.2)
where \(x_{1}(t)\) and \(x_{2}(t)\) denote the densities of the immature and mature prey species at time t, respectively; \(y_{1}(t)\) and \(y_{2}(t)\) represent the immature and mature population densities of predator species at time t, respectively; \(r_{i}(t)\), \(b_{i}(t)\), \(c_{i}(t)\) (\(i=1,2\)) are all continuous functions bounded above and below by positive constants for all \(t\geq0\). \(d_{ij}\), \(\tau_{i}\), \(i,j=1,2\), are all positive constants. They investigated the persistence, extinction, and stability property of the above system. Li, Chen, et al. [5] investigated the stability property of the following mutualism model in a plantpollinator system with stage structure and the Beddington–DeAngelis functional response:
$$ \begin{gathered} \dot{x}_{i}(t) = \alpha{x_{m}{(t)}} \gamma{x_{i}{(t)}}\alpha{e^{\gamma {\tau}}}x_{m}{(t\tau)}, \\ \dot{x}_{m}(t) = \alpha{e^{\gamma{\tau}}}x_{m}{(t\tau)} \beta {{x^{2}_{m}(t)}}+\frac{mx_{m}(t)y(t)}{1+k_{1}x_{m}(t)+k_{2}y(t)}, \\ \dot{y}(t) = \frac{nmx_{m}(t)y(t)}{1+k_{1}x_{m}(t)+k_{2}y(t)}dy(t), \end{gathered} $$
(1.3)
where \(x_{i}(t)\), \(x_{m}(t)\), \(y(t)\) can be described as the immature, mature plant densities, and the pollinators densities at time t, respectively. The authors investigated the persistence, local and global stability of the above system. Lin, Xie, et al. [10] considered the following stagestructured predator–prey model (stage structure for both predator and prey, respectively) with modified Leslie–Gower and Hollingtype II schemes:
$$ \begin{gathered} x'_{1}(t)= r_{1} x_{2}(t)d_{11}x_{1}(t)r_{1}e^{d_{11}\tau_{1}}x_{2}(t \tau_{1}), \\ x'_{2}(t)= r_{1}e^{d_{11}\tau_{1}}x_{2}(t \tau_{1})d_{12}x_{2}(t) bx_{2}^{2}(t) \frac{a_{1}y_{2}(t)x_{2}(t)}{x_{2}(t)+k_{1}}, \\ y'_{1}(t)= r_{2}y_{2}(t)d_{22}y_{1}(t)r_{2}e^{d_{22}\tau_{2}}y_{2}(t \tau_{2}), \\ y'_{2}(t)= r_{2}e^{d_{22}\tau_{2}}y_{2}(t \tau_{2})d_{21}y_{2}(t)\frac {a_{2}y^{2}_{2}(t)}{x_{2}(t)+k_{2}}, \end{gathered} $$
(1.4)
where \(d_{12}\) and \(d_{21}\) represent the death rate of mature prey \(x_{2}\) and mature predator \(y_{2}\), respectively; \(\tau_{1}\) is the time length of prey species from immature ones to mature ones, \(\tau _{2}\) is the time length of predator from immature ones to mature ones. 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 stagestructured 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. For more works in this direction, one could refer to [1–16] and the references cited therein. We mention here that the topic of the stability of the equilibrium and the extinction property of the ecosystem are the most important topics in the study of mathematics biology, one could refer to [17–35] for more works in this direction.
(2) Assume that the surviving rate of immaturity to reach maturity is proportional to the number of immature species. For example, Wu and Chen [15] studied the following singe species stagestructured ecosystem with both toxicant effect and harvesting:
$$ \begin{gathered} x_{1}^{'}(t) = ax_{2}d_{1}x_{1}d_{2}x_{1}^{2} \beta x_{1}r_{1}x_{1}^{3}, \\ x_{2}^{'}(t) = \beta x_{1}b_{1}x_{2}c_{2}Ex_{2}, \end{gathered} $$
(1.5)
where \(x_{1}(t)\), \(x_{2}(t)\) represent the population density of the immature and the mature at time t, respectively, \(r_{1}x_{1}^{3}\) is the effects of toxicant on the immature, E is the harvesting effort, \(c_{2}\) is the catchability coefficient. They assumed that the immature is density restricted, toxicant affects the immature population and only harvesting the mature species. They showed that toxicant has no influence on the persistence property of the system. They also considered the system with variable harvest effect, and sufficient conditions which ensure the global stability of bionomic equilibrium were obtained. Chen [14] studied the existence and stability of the strictly positive (componentwise) almost periodic solution of the following nonautonomous almost periodic competitive twospecies model with stage structure in one species:
$$ \begin{gathered} \dot{x}_{1}(t) = a_{1}(t)x_{1}(t)+b_{1}(t)x_{2}(t), \\ \dot{x}_{2}(t) = a_{2}(t)x_{2}(t)b_{2}(t)x_{2}(t)c(t)x_{2}^{2}(t) \beta _{1}(t)x_{2}(t)x_{3}(t), \\ \dot{x}_{3}(t) = x_{3}(t) \bigl(d(t)e(t)x_{3}(t) \beta_{2}(t)x_{2}(t)\bigr). \end{gathered} $$
(1.6)
Here \(x_{1}(t)\) and \(x_{2}(t)\) are immature and mature population densities of one species, respectively, and \(x_{3}(t)\) represents the population density of another species. Khajanchi and Banerjee [16] proposed the following stagestructured predator–prey model with ratiodependent functional response:
$$ \begin{gathered} \frac{dx_{i}}{dt} = \alpha x_{m}(t) \beta x_{i}(t)\delta_{1}x_{i}(t), \\ \frac{dx_{m}}{dt} = \beta x_{i}(t)\delta_{2}x_{m}(t) \gamma x_{m}^{2}(t)\frac {\eta(1\theta)x_{m}(t)y(t)}{g(1\theta)x_{m}(t)+hy(t)}, \\ \frac{dy}{dt} = \frac{u\eta(1\theta)x_{m}(t)y(t)}{g(1\theta )x_{m}(t)+hy(t)}\delta_{3}y(t). \end{gathered} $$
(1.7)
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 stagestructured predator–prey species. For more works in this direction, one could refer to [14–16] and the references cited therein.
On the other hand, as was pointed out by Chakraborty et al. [20], the study of resource management, including fisheries, forestry, and wildlife management, has great importance, it is necessary to harvest the population but harvesting should be regulated, so that both the ecological sustainability and conservation of the species can be implemented in a long run. Chakraborty et al. [20] proposed the following predator–prey model:
$$ \begin{gathered} \frac{dx}{dt} = rx\biggl(1\frac{x}{K} \biggr)\frac{\alpha xy}{a+bx+cy}q_{1}mE x, \\ \frac{dy}{dt} = sy\biggl(1\frac{y}{L}\biggr)+\frac{\beta xy}{a+bx+cy}q_{2}mEx. \end{gathered} $$
(1.8)
They tried to investigated the existence and stability property of the equilibria of the system; however, since the system is too complicated, they could not give detailed analysis of the influence of parameter m. Recently, many scholars investigated the dynamic behaviors of the nonselective harvesting ecosystem incorporating partial closure. Lin [22] investigated the dynamic behaviors of the following two species commensal symbiosis model with nonmonotonic functional response and nonselective harvesting in a partial closure:
$$ \begin{gathered} \frac{dx}{dt} = x\biggl(a_{1}b_{1}x+ \frac{c_{1}y }{d_{1}+y^{2}}\biggr)q_{1}Emx, \\ \frac{dy}{dt} = y(a_{2}b_{2}y)q_{2}Emy, \end{gathered} $$
(1.9)
where \(a_{i}\), \(b_{i}\), \(q_{i}\), \(i=1,2\), \(c_{1}\), E, m (\(0< m<1\)), and \(d_{1}\) are all positive constants, where E is the combined fishing effort used to harvest and m (\(0< m<1\)) is the fraction of the stock available for harvesting. 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. He also showed that if the system admits a unique positive equilibrium, then it is globally asymptotically stable. Chen [21] studied the influence of nonselective harvesting on a Lotka–Volterra amensalism model incorporating partial closure for the populations, and he also found that the dynamic behaviors of the system become complicate.
As was shown above, though there are many works on a stagestructured ecosystem [1–16], seldom did they consider the influence of harvesting [15]. Also, though there are several scholars who investigated the dynamic behaviors of the nonselective harvesting ecosystem incorporating partial closure for the populations (see [20–22, 32, 33, 35]), to this day, still no scholars investigated the influence of nonselective harvesting stagestructured ecosystem incorporating partial closure for the populations. This motivated us to propose system (1.1). We will try to give a thorough analysis of the dynamic behaviors of system (1.1).
The paper is arranged as follows. We investigate the existence and locally stability property of the equilibria of system (1.1) in the next section. In Sect. 3, by constructing some suitable Lyapunov function, we are able to investigate the global stability property of the equilibria. Section 4 presents some numerical simulations to show the feasibility of the main results. We end this paper with a brief discussion.