EFFECT OF TIME-DELAY ON A PREY–PREDATOR MODEL WITH MICROPARASITE INFECTION IN THE PREDATOR

2011 ◽  
Vol 19 (02) ◽  
pp. 365-387 ◽  
Author(s):  
SWETA PATHAK ◽  
ALAKES MAITI ◽  
SHYAM PADA BERA
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Predator Population ◽  
Dynamical Characteristics ◽  
Dynamical Behaviors ◽  
Discrete Time Delay ◽  
Mathematical Analyses ◽  
Predator Model ◽  
Living Organisms

To increase a prey population that is attacked by a predator it is more convenient and economical to choose the living organisms to control the predator. In this paper, the dynamical behaviors of a prey–predator model with microparasitic infection in the predator have been discussed. In this epidemiological model the microparasite is horizontally transmitted and attacks the predator population only. The infected population does not recover or become immune. The dynamical characteristics of the system are studied through mathematical analyses. The role of discrete time-delay has been discussed to show that time-delay can induce instability and oscillation. Numerical simulations are carried out. Biological implications have been discussed.

A PREY–PREDATOR MODEL WITH MICROPARASITE INFECTION IN THE PREDATOR

2008 ◽  
Vol 16 (02) ◽  
pp. 219-239 ◽  
Author(s):  
A. MAITI ◽  
S. P. BERA ◽  
G. P. SAMANTA
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Predator Prey ◽  
Dynamical Behaviors ◽  
Prey System ◽  
Discrete Time Delay ◽  
Predator Model

This paper aims to study the dynamical behaviors of a predator–prey system where the predator is affected by a microparasite infection. The effect of discrete time-delay is investigated. It has been shown that the time-delay can induce instability and oscillations via Hopf bifurcation. Also delay of suitable range may keep the populations at a desired level. Computer simulations are carried out to illustrate our analytical findings. The biological implications of our analytical and numerical findings are discussed critically.

scholarly journals Stability analysis of an eco-epidemiological model incorporating a prey refuge

2010 ◽  
Vol 15 (4) ◽  
pp. 473-491 ◽  
Author(s):  
A. K. Pal ◽  
G. P. Samanta
Keyword(s):  
Stability Analysis ◽  
Time Delay ◽  
Computer Simulations ◽  
Discrete Time ◽  
Predator Population ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Discrete Time Delay ◽  
The Stability

The present paper deals with the problem of a predator-prey model incorporating a prey refuge with disease in the prey-population. We assume the predator population will prefer only infected population for their diet as those are more vulnerable. Dynamical behaviours such as boundedness, permanence, local and global stabilities are addressed. We have also studied the effect of discrete time delay on the model. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.

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.

scholarly journals Immunotherapy with Interleukin-2: A Study Based on Mathematical Modeling

2008 ◽  
Vol 18 (3) ◽  
pp. 389-398 ◽  
Author(s):  
Sandip Banerjee
Keyword(s):  
Mathematical Modeling ◽  
Time Delay ◽  
Numerical Analysis ◽  
Discrete Time ◽  
Delay Differential Equations ◽  
Interleukin 2 ◽  
Tumor Cell Population ◽  
Discrete Time Delay ◽  
Delay Differential

Immunotherapy with Interleukin-2: A Study Based on Mathematical ModelingThe role of interleukin-2 (IL-2) in tumor dynamics is illustrated through mathematical modeling, using delay differential equations with a discrete time delay (a modified version of the Kirshner-Panetta model). Theoretical analysis gives an expression for the discrete time delay and the length of the time delay to preserve stability. Numerical analysis shows that interleukin-2 alone can cause the tumor cell population to regress.

Dynamical Behaviors of a Delayed Prey–Predator Model with Beddington–DeAngelis Functional Response: Stability and Periodicity

2020 ◽  
Vol 30 (16) ◽  
pp. 2050244
Author(s):  
Xin Zhang ◽  
Renxiang Shi ◽  
Ruizhi Yang ◽  
Zhangzhi Wei
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Response ◽  
Discrete Time ◽  
Numerical Continuation ◽  
Positive Equilibrium ◽  
Local Hopf Bifurcation ◽  
Discrete Time Delay ◽  
Predator Model ◽  
The Stability

This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL.

A stage-structured prey–predator model with discrete time delay

2006 ◽  
Vol 182 (2) ◽  
pp. 1385-1398 ◽  
Author(s):  
M. Bandyopadhyay ◽  
Sandip Banerjee
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Discrete Time Delay ◽  
Predator Model

Discrete-Time Delay Dynamics of Boundedly Rational Monopoly

2012 ◽  
Author(s):  
Akio Matsumato ◽  
Ferenc Szidarovsky
Keyword(s):  
Time Delay ◽  
Discrete Time ◽  
Discrete Time Delay
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