scholarly journals On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Mehbuba Rehim ◽  
Weixin Wu ◽  
Ahmadjan Muhammadhaji
Keyword(s):  
Time Delay ◽  
Phytoplankton Bloom ◽  
Dynamical Behavior ◽  
Time Lag ◽  
Toxic Substance ◽  
Predation Rate ◽  
Dynamical Behaviour ◽  
Toxic Substances ◽  
Toxic Phytoplankton ◽  
Discrete Time Delay

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

2017 ◽  
Vol 10 (05) ◽  
pp. 1750062 ◽  
Author(s):  
Kalyan Manna
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Dynamical Behavior ◽  
Hbv Infection ◽  
Infection Model ◽  
Direct Lyapunov Method ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Discrete Time Delay ◽  
Number Formula

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

A Predator-Prey Model Incorporating Prey Refuge and Allee Effect

2015 ◽  
Vol 713-715 ◽  
pp. 1534-1539 ◽  
Author(s):  
Rui Ning Fan
Keyword(s):  
Time Delay ◽  
Time Lag ◽  
Functional Responses ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Prey Interaction

The effect of refuge used by prey has a stabilizing impact on population dynamics and the effect of time delay has its destabilizing influences. Little attention has been paid to the combined effects of prey refuge and time delay on the dynamic consequences of the predator-prey interaction. Here, a predator-prey model with a class of functional responses was studied by using the analytical approach. The refuge is considered as protecting a constant proportion of prey and the discrete time delay is the gestation period. We evaluated both effects with regard to the local stability of the interior equilibrium point of the considered model. The results showed that the effect of prey refuge has stronger influences than that of time delay on the considered model when the time lag is smaller than the threshold. However, if the time lag is larger than the threshold, the effect of time delay has stronger influences than that of refuge used by prey.

scholarly journals The dynamics of nutrient, toxic phytoplankton, nontoxic phytoplankton and zooplankton model

2016 ◽  
Vol 5 (1) ◽  
pp. 48
Author(s):  
Hasnaa Fiesal Mohammed Hussien ◽  
Raid Kamel Naji ◽  
Azhar Abbas Majeed
Keyword(s):  
Food Web ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Toxic Substance ◽  
Global Dynamics ◽  
Defensive Strategy ◽  
Toxic Phytoplankton ◽  
System A ◽  
Aquatic Food

<p>The objective of this paper is to study the dynamical behavior of an aquatic food web system. A mathematical model that includes nutrients, phytoplankton and zooplankton is proposed and analyzed. It is assumed that, the phytoplankton divided into two compartments namely toxic phytoplankton which produces a toxic substance as a defensive strategy against predation by zooplankton, and a nontoxic phytoplankton. All the feeding processes in this food web are formulating according to the Lotka-Volterra functional response. This model is represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation and Hopf bifurcation are investigated. Finally, numerical simulation is used to study the global dynamics of this model.</p>

scholarly journals Role of Delay on Planktonic Ecosystem in the Presence of a Toxic Producing Phytoplankton

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Swati Khare ◽  
O. P. Misra ◽  
Chhatrapal Singh ◽  
Joydip Dhar
Keyword(s):  
Hopf Bifurcation ◽  
Time Lag ◽  
Toxic Substance ◽  
Distributed Delay ◽  
Dynamical Behaviour ◽  
Zooplankton Biomass ◽  
State Variables ◽  
Interior Equilibrium ◽  
The Stability

A mathematical model is proposed to study the role of distributed delay on plankton ecosystem in the presence of a toxic producing phytoplankton. The model includes three state variables, namely, nutrient concentration, phytoplankton biomass, and zooplankton biomass. The release of toxic substance by phytoplankton species reduces the growth of zooplankton and this plays an important role in plankton dynamics. In this paper, we introduce a delay (time-lag) in the digestion of nutrient by phytoplankton. The stability analysis of all the feasible equilibria are studied and the existence of Hopf-bifurcation for the interior equilibrium of the system is explored. From the above analysis, we observe that the supply rate of nutrient and delay parameter play important role in changing the dynamical behaviour of the underlying system. Further, we have derived the explicit algorithm which determines the direction and the stability of Hopf-bifurcation solution. Finally, numerical simulation is carried out to support the theoretical result.

Chaotic dynamics of a delayed tumor–immune interaction model

2020 ◽  
Vol 13 (02) ◽  
pp. 2050009 ◽  
Author(s):  
Subhas Khajanchi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Tumor Cells ◽  
Discrete Time ◽  
Dynamical Behavior ◽  
Tumor Model ◽  
Host Cells ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Discrete Time Delay

Due to the unpredictable growth of tumor cells, the tumor–immune interactive dynamics continues to draw attention from both applied mathematicians and oncologists. Mathematical modeling is a powerful tool to improve our understanding of the complicated biological system for tumor growth. With this goal, we report a mathematical model which describes how tumor cells evolve and survive the brief encounter with the immune system mediated by immune effector cells and host cells which includes discrete time delay. We analyze the basic mathematical properties of the considered model such as positivity of the system and the boundedness of the solutions. By analyzing the distribution of eigenvalues, local stability analysis of the biologically feasible equilibria and the existence of Hopf bifurcation are obtained in which discrete time delay is used as a bifurcation parameter. Based on the normal form theory and center manifold theorem, we obtain explicit expressions to determine the direction of Hopf bifurcation and the stability of Hopf bifurcating periodic solutions. Numerical simulations are carried out to illustrate the rich dynamical behavior of the delayed tumor model. Our model simulations demonstrate that the delayed tumor model exhibits regular and irregular periodic oscillations or chaotic behaviors, which indicate the scenario of long-term tumor relapse.

Control strategies of avian influenza pandemic model with time delay

2017 ◽  
Vol 10 (07) ◽  
pp. 1750104 ◽  
Author(s):  
U. Roman ◽  
Z. Gul ◽  
I. Saeed ◽  
U. Hakeem ◽  
S. Shafie
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Avian Influenza ◽  
Control Strategies ◽  
Influenza Pandemic ◽  
Dynamical Behavior ◽  
Controlled System ◽  
Exponential Factor ◽  
Discrete Time Delay ◽  
Delay Differential

A mathematical model for avian influenza with optimal control strategies is presented as a system of discrete time delay differential equations (DDEs) and its important mathematical features are analyzed. In alignment to manage this, we develop an optimally controlled pandemic model of avian influenza and insert a time delay with exponential factor. Then we apply two controlled functions in the form of biosecurity of poultry and the education campaign against avian influenza to control the disperse of the disease. Our optimal control strategies will minimize the number of contaminated humans and contaminated birds. We also derive the basic reproduction number to examine the dynamical behavior of the model and demonstrate the existence of the controlled system. For the justification of our work, we present numerical simulations.

DYNAMICAL BEHAVIOR OF A HARVESTED PREY-PREDATOR MODEL WITH STAGE STRUCTURE AND DISCRETE TIME DELAY

2009 ◽  
Vol 17 (04) ◽  
pp. 759-777 ◽  
Author(s):  
CHAO LIU ◽  
QINGLING ZHANG ◽  
JAMES HUANG ◽  
WANSHENG TANG
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Dynamical Behavior ◽  
Stage Structure ◽  
Model System ◽  
Optimal Harvesting ◽  
Gestation Delay ◽  
Interior Equilibrium ◽  
Discrete Time Delay ◽  
Predator Model

A prey-predator model with stage structure for prey and selective harvest effort on predator is proposed, in which gestation delay is considered and taxation is used as a control instrument to protect the population from overexploitation. It is established that when the discrete time delay is zero, the model system is stable around the interior equilibrium and an optimal harvesting policy is discussed with the help of Pontryagin's maximum principle; On the other hand, stability switch of the model system due to the variation of discrete time delay is also studied, which reveals that the discrete time delay has a destabilizing effect. As the discrete time delay increases through a certain threshold, a phenomenon of Hopf bifurcation occurs and a limit cycle corresponding to the periodic solution of model system is also observed. Numerical simulations are carried out to show the consistency with theoretical analysis.

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