Dynamical behavior of systems of two phytoplankton and one zooplankton populations with toxin producing phytoplankton

Three‐compartment mathematical models of non‐toxic phytoplankton (NTP), toxin producing phytoplankton (TPP), and zooplankton are proposed to explore the role of TPP in algal blooms. The mutual interference between predator zooplankton and avoidance of TPP by zooplankton are incorporated into the model. The NTP and TPP engage in exploit competition and the toxin produced by TPP has no effect on NTP. Using the concept of uniform persistence, we establish coexistence of NTP, TPP, and zooplankton in certain parameter regimes. We study the effects of mutual interference and avoidance by zooplankton upon the population interactions. In addition to the toxin producing mechanism, it is concluded that mutual interference of zooplankton is an important factor for diminishing harmful blooms. Copyright © 2017 John Wiley & Sons, Ltd.     

Delay driven Hopf bifurcation and chaos in a diffusive toxin producing phytoplankton‐zooplankton model

This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.     

Role of toxin producing phytoplankton on a plankton ecosystem

In this paper, a mathematical model is proposed to study the role of toxin producing phytoplankton on a phytoplankton–zooplankton system with nutrient cycling. The model includes three state variables, viz., nutrient concentration, phytoplankton biomass and zooplankton biomass. It is assumed in the model that phytoplankton biomass is producing toxicant harmful for the zooplankton biomass. All the feasible equilibria of the system are obtained and the conditions for the existence of the interior equilibrium are determined. The local stability analysis of all the feasible equilibria are carried out and the possibility of Hopf-bifurcation of the interior equilibrium is studied. The threshold value in terms of constant input rate of nutrient is determined both analytically and numerically.     

Chaos in a seasonally and periodically forced phytoplankton–zooplankton system

The effect of seasonality and periodicity on plankton dynamics is investigated. Periodic variations are added to two different parameters of the plankton ecosystem: the growth rate of phytoplankton and the death rate of the zooplankton. The dynamic behaviors of the system is simulated numerically. A variety of complex population dynamics including chaos, quasi-periodicity, and periodic resonance are obtained. Our result reinforces the conjecture that seasonality and periodicity are crucial to plankton dynamics.     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Stability and bifurcation analysis in a viral model with delay

In this paper, we consider a three‐dimensional viral model with delay. We first investigate the linear stability and the existence of a Hopf bifurcation. It is shown that Hopf bifurcations occur as the delay τ passes through a sequence of critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit formulaes that determine the stability, the direction, and the period of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the validity of the main results. Finally, some brief conclusions are given. Copyright © 2012 John Wiley & Sons, Ltd.     

A viral infection model with immune circadian rhythms

A viral infection model with immune circadian rhythms is investigated in this paper. By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease. These results can be used to explain the oscillation behaviors of virus population, which were observed in chronic HBV or HCV carriers. Further, numerical simulations indicate that the dynamics of the lytic component of cytotoxicity T cells (CTLs) is crucial to the outcome of a viral infection.     

Analysis of a viral infection model with immune impairment,intracellular delay and general non-linear incidence rate

In this article we study the dynamical behaviour of a intracellular delayed viral infection with immune impairment model and general non-linear incidence rate. Several techniques, including a non-linear stability analysis by means of the Lyapunov theory and sensitivity analysis, have been used to reveal features of the model dynamics. The classical threshold for the basic reproductive number is obtained: if the basic reproductive number of the virus is less than one, the infection-free equilibrium is globally asymptotically stable and the infected equilibrium is globally asymptotically stable if the basic reproductive number is higher than one.     

The role of viral infection in phytoplankton dynamics with the inclusion of incubation class

The role of viral infection in phytoplankton dynamics without and with incubation population class is studied. It is observed that phytoplankton species in the absence of incubated class are unstable around an endemic equilibrium but the presence of delay in the form of incubated class has made it conditionally stable around an endemic equilibrium. We also observe that the dynamical system is very sensitive to the transfer rate from susceptible to incubated class and when it crosses a certain threshold the phytoplankton population start oscillating around the endemic equilibrium, shown both analytically and numerically.     

Bottom up and top down effect on toxin producing phytoplankton and its consequence on the formation of plankton bloom

We consider a plankton-nutrient interaction model consisting of phytoplankton, zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. In this model, it is assumed that phytoplankton releases toxic chemical for self defense against their predators. The model system is studied analytically and the threshold values for the existence and stability of various steady states are worked out. It is observed that if the maximal zooplankton conversion rate crosses a certain critical value, the system enters into Hopf bifurcation. Finally it is observed that to control the planktonic bloom and to maintain stability around the coexistence equilibrium we have to control the nutrient input rate specially caused by artificial eutrophication. In case if it is not possible to control the nutrient input rate, one could use toxic phytoplankton to prevent the recurrence bloom.     

A viral propagation model with a nonlinear infection rate and free boundaries

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties. This model is composed of two ordinary differential equations and one partial differential equation, in which the spatial range of the first equation is the whole space R, and the last two equations have free boundaries. As a new mathematical model, we prove the existence,uniqueness and uniform estimates of the global solution, and provide the criteria for spreading and vanishing, and the long time behavior of the solution components u, v and w. Comparing this model with the corresponding ordinary differential systems, the basic reproduction number R_0 plays a different role. We find that when R_0≤1, the virus cannot spread successfully; when R_0 1, the successful spread of the virus depends on the initial value and varying parameters.     

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.     

Optimal control and optimization algorithm of nonlinear impulsive delay system producing 1,3-Propanediol

According to the controllability of pulse times and the amount of jumps in the states at these times in the process of fed-batch culture producing 1,3-propanediol, this paper proposes a terminal optimal control model, whose constraint condition is the nonlinear impulsive delay system. The existence of optimal control is discussed and an optimization algorithm which is applied to each subinternal over one cycle for this optimal control problem is constructed. Finally, the numerical simulations show that the terminal intensity of producing 1,3-propanediol has been increased obviously.     

The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

一类具有免疫时滞的病毒动力学模型的稳定性分析

建立和分析了一类具有CTL免疫反应且带有免疫时滞的病毒动力学模型.讨论了系统解的有界性,并获得了无病平衡点全局渐近稳定以及正平衡点稳定的条件.最后借助Matlab对模型进行了数值模拟.     

Global stability of an HIV-1 infection model with saturation infection and intracellular delay

In this paper, an HIV-1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.