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.
Highlights
In nature world, the growth of species experiences different stages
The paper is arranged as follows: we study the existence and local stability of the equilibria of system (1.7)
2.2.2 Stability of the positive equilibria Ei∗(x∗i, y∗i ) (i = 1, 2, 3) To discuss the stability of Ei∗(x∗i, y∗i ) (i = 1, 2, 3), we simplify the determinant of Ei∗: Det J Ei∗
Summary
The growth of species experiences different stages. For example, the growth of frogs can be divided into fertilized eggs, tadpoles, and adult frogs; the growth of silkworm can be divided into eggs, larvae, pupae, and adults. Considering the time delay due to pregnancy effect on the growth of the predator, Zhang and Zhang [6] studied the stage structure model with time delay and density-dependent juvenile birth rate, and they gave conditions for uniform persistence and extinction of the system in the following model:. To the best of our knowledge, still no scholars proposed and investigated the dynamic behavior of the single-species stage structure model with Michaelis–Mententype harvesting. This motivated us to propose the following system: dx dt = αy – βx – δ1x,. 2.2.2 Stability of the positive equilibria Ei∗(x∗i , y∗i ) (i = 1, 2, 3) To discuss the stability of Ei∗(x∗i , y∗i ) (i = 1, 2, 3), we simplify the determinant of Ei∗: Det J Ei∗.
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