Stability and bifurcation analysis in a single-species stage structure system with Michaelis–Menten-type harvesting

Abstract

In this paper, we prpose a single-species stage structure model with Michaelis–Menten-type harvesting for mature population. We investigate the existence of all possible equilibria of the system and discuss the stability of equilibria. We use Sotomayor’s theorem to derive the conditions for the existence of saddle-node and transcritical bifurcations. From the ecological point of view, we analyze the effect of harvesting on the model of mature population and consider it as a bifurcation parameter, giving the maximum threshold of continuous harvesting. By constructing a Lyapunov function and Bendixson–Dulac discriminant, we give sufficient conditions for the global stability of boundary equilibrium and positive equilibrium, respectively. Our study shows that nonlinear harvesting may lead to a complex dynamic behavior of the system, which is quite different from linear harvesting. We carry out numeric simulations to verify the feasibility of the main results.

Dynamic Behaviors of a Single Species Stage Structure Model with Michaelis–Menten-TypeJuvenile Population Harvesting

“…According to the analysis, we can get, compared with [33], that the model exhibits rich dynamic behaviors, and there are many kinds of positive steady states. The first type of bistability phenomenon shows that if b < c(e − a), ∆(b) < 0, then the model is persistent when the initial value lies in the first quadrant.…”

“…In addition, in order to meet their own needs, human beings develop and utilize natural resources. Therefore, to ensure the sustainable development of the ecosystem and maximize the economic benefits, the harvest model has attracted the attention of many scholars [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. There are two types of harvesting proposed by May et al [34]: constant [14]: it is a constant independent of the number of harvested populations; and linear harvesting [15][16][17][18][19][20][21][22][23][24]: it is proportional to the number of harvested populations, namely its expression is h = qEx.…”

“…With linear harvesting, if x is fixed, E → ∞ or E is fixed, and x → ∞, then we have h → ∞; this is contrary to the facts. In order to overcome this shortcoming, nonlinear harvesting [25][26][27][28][29][30][31][32][33] was proposed by Clark [35]. Among them, the nonlinear harvesting, which is also called Michaelis-Menten-type harvesting, can be expressed as…”

“…From the perspective of fishery production, etc., we basically harvest the adult fish and keep the juvenile fish. Therefore, Yu, Zhu, Lai, et al [33] proposed the single species stage structure system with Michaelis-Menten-type adult species harvesting. They obtained sufficient conditions for the global asymptotic stability of the unique boundary equilibrium and positive equilibrium of the model.…”

Dynamics of the Stage Structured Population Model, Predator Accompanied by Michaelis Menten Holling Type Functional Responseand Delay and Prey Taking Refuge

“…Prey, predator and top predator with functional response Michaelis Menten kind and two unequal delays is attempted by Dai et al in 2014. They used normal form method and center manifold theorem to evaluate the properties of periodic solution [13]. Research have also been performed in Volterra type functional response, linear functional response, bilinear functional response [14][15][16] incorporated with stage structure and harvesting.…”