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We structure a phytoplankton zooplankton interaction system by incorporating (i) Monod-Haldane type functional response function; (ii) two delays accounting, respectively, for the gestation delay $ \tau $ of the zooplankton and the time $ \tau_1 $ required for the maturity of TPP. Firstly, we give the existence of equilibrium and property of solutions. The global convergence to the boundary equilibrium is also derived under a certain criterion. Secondly, in the case without the maturity delay $ \tau_1 $, the gestation delay $ \tau $ may lead to stability switches of the positive equilibrium. Then fixed $ \tau $ in stable interval, the effect of $ \tau_1 $ is investigated and find $ \tau_1 $ can also cause the oscillation of system. Specially, when $ \tau = \tau_1 $, under certain conditions, the periodic solution will exist with the wide range as delay away from critical value. To deal with the local stability of the positive equilibrium under a general case with all delays being positive, we use the crossing curve methods, it can obtain the stable changes of positive equilibrium in $ (\tau, \tau_1) $ plane. When choosing $ \tau $ in the unstable interval, the system still can occur Hopf bifurcation, which extends the crossing curve methods to the system exponentially decayed delay-dependent coefficients. Some numerical simulations are given to indicate the correction of the theoretical analyses. [ABSTRACT FROM AUTHOR]

In this paper, the system of describing the interactions between poisonous phytoplankton and zooplankton is presented. It focuses on the effects of two delays on the dynamic behavior of the system. At first, the properties of solutions including positivity and boundedness are given. Next, the stability of equilibria and the existence of local Hopf bifurcation are established when delays change and cross some threshold values. Especially, the existence of global periodic solutions is discussed when the two delays are equal. Furthermore, the implicit algorithm is derived for deciding the properties of the branching periodic solutions by using center manifold theory. Some numerical simulations are performed for supporting the theoretical results. Finally, some conclusions are given. [ABSTRACT FROM AUTHOR]

In this paper, a model with time delay describing biodegradation of Microcystins (MCs) is investigated. Firstly, the stability of the positive equilibrium and the existence of Hopf bifurcations are obtained. Furthermore, an explicit algorithm for determining the direction and the stability of the bifurcating periodic solutions is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]

A toxin-producing phytoplankton and zooplankton system is investigated. Considering that zooplankton can be harvested for food in some bodies of water, the harvesting term is introduced to zooplankton population. Firstly, from the ordinary differential equation (ODE) system, we obtain the global asymptotic stability of equilibrium and optimal capture problem. Secondly, based on the ODE system, the diffusion term is introduced and the global asymptotic stability of the steady state solution is obtained. As a result, the diffusion cannot affect the global asymptotic stability of equilibrium, and Turing instability cannot occur. Once again, a delayed differential equation (DDE) system is put forward. The global asymptotic stability of boundary equilibrium and the existence of local Hopf bifurcation at positive equilibrium are discussed. Furthermore, it is proved that there exists at least one positive periodic solution as delay varies in some region by using the global Hopf result of Wu for functional differential equations. Lastly, some numerical simulations are carried out for supporting the theoretical analyses and the positive impacts of harvesting effort, and the release rate of toxin is given. The unstable interval of the positive equilibrium becomes smaller and smaller with the increase of harvesting effort or the release rate of toxin. [ABSTRACT FROM AUTHOR]

Although caring and raising children is the duty of all parents, some animals relieve themselves of this labored work by relying on others to raise their young. After the host has made a nest and laid eggs, the parasite locates the nest and lays a few eggs in the host’s nest. Several possible interactions occur between the host and parasite in brood parasitism: the parasite does not destroy the host’s eggs and the host does not abandon her nest with eggs. Here, we establish a mathematical framework for studying an ecological model on brood parasitism with stage structure of the parasite, where time delay represents the incubation period of the parasite. For the proposed model, we studied the impact of incubation time delay on the coexistence equilibrium. By using corresponding characteristic equations, the local stability behavior of each of the feasible equilibria of the model and existence of Hopf bifurcations at the coexistence equilibrium are established. Further, the global properties of some equilibria are established using Lyapunov–LaSalle invariance principle. The incubation time delay was found to be crucial for establishing stability behavior. When the incubation time delay increased and crossed some critical values, Hopf bifurcation may occur at the coexistence equilibrium. Further, by using normal form and center manifold theories, formulas were established to determine the stability and direction of the Hopf bifurcation. Numerical simulations were conducted to support the theoretical analysis. Our mathematical model could help lay a foundation for further studies on brood parasitism by the cuckoo bird. [ABSTRACT FROM AUTHOR]