Abstract
In this paper, a delayed phytoplankton-zooplankton system with Crowley-Martin functional response is investigated analytically. We study the permanence and analyze the stability of the both boundary and positive equilibrium points for the system with delay as well as the system without delay. The global asymptotic stability is discussed by constructing a suitable Lyapunov functional. Numerical analysis indicates that the delay does not change the stability of the positive equilibrium point. Furthermore, we also show that due to the increase of the delay there occurs a Hopf bifurcation of periodic solutions. It is found that population fluctuations will not appear under the condition of certain parameters. In addition, we determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions by applying a normal form method and center manifold theory. Finally, some numerical simulations are carried out to support our theoretical analysis results.
Highlights
Plankton plays an important role in the ocean and the climate because of their participation in the global carbon cycle at the base of the food chain [ ]
5 Discussion and conclusion Over the two decades, a great deal of research has been devoted to the dynamics of the plankton ecosystem, a clear understanding of the mechanisms that cause the plankton blooms is still lacking and, it has been remained an interesting area of research for many ecologists and mathematical biologists [ ]
An attempt has been made to study the dynamic behaviors of a phytoplankton-zooplankton system with a Crowley-Martin functional response and its corresponding delayed version
Summary
Plankton plays an important role in the ocean and the climate because of their participation in the global carbon cycle at the base of the food chain [ ]. In order to better understand the mechanisms that determine the plankton, it is necessary to study the dynamic behaviors for a phytoplankton-zooplankton system with consider the effect of a time delay. The necessary condition for a change in stability of the interior equilibrium point E∗ is that equation ( ) should have purely imaginary roots, that is, stability switches for increasing τ in I = [ , τ ∗) may occur only with a pair of roots λ = ±iω(τ ). If h ≥ gP∗, the sufficient conditions for global asymptotic stability of the positive solution E∗ for non-delayed system ( ) imply that the interior equilibrium solution E∗ for the delayed system is globally asymptotic stable if E∗ for non-delayed system ( ) is globally asymptotic stable and conditions (i) and (ii) of Theorem . If h ≥ gP∗, the sufficient conditions for global asymptotic stability of the positive solution E∗ for non-delayed system ( ) imply that the interior equilibrium solution E∗ for the delayed system is globally asymptotic stable if E∗ for non-delayed system ( ) is globally asymptotic stable and conditions (i) and (ii) of Theorem . hold
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