Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Wang, Xinhui; Liu, Haihong

doi:10.4236/am.2012.36099

Cited by 4 publications

(3 citation statements)

References 18 publications

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“…Lotka-Volterra models were first introduced in 1925 by Lotka and Volterra in 1926 [1][2]. Researchers have significantly developed or modified the model, see [3][4][5][6][7][8] and references therein. One modification is called allelopathic effect.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of predator-prey with allelopathic effect

Fitria¹,

Suryanto²,

Trisilowati³

2015

AIP Conference Proceedings

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In this paper we study the dynamics of predator-prey model with allelopathic effect on prey. We find that the model has four equilibrium points that are extinction of prey equilibrium, the extinction of predator equilibrium, the extinction of both predator and prey, and the coexistence equilibrium. It is shown that the extinction of predator equilibrium and the extinction of both predator and prey equilibrium are unstable, while the extinction of prey equilibrium is conditionally stable. The coexistence equilibrium is unconditionally stable. The analytical results are confirmed by numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Stability analysis of predator-prey with allelopathic effect

Fitria¹,

Suryanto²,

Trisilowati³

2015

AIP Conference Proceedings

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“…Thus, many scientists and researchers have been focusing on the study of controlling the infections. There has been much interest in mathematical modeling of HIV dynamics [Yu & Zou, 2012;Tian et al, 2014;Mittler et al, 1998;Nelson et al, 2000;Revilla & Garcia-Ramos, 2003;Jiang et al, 2009;Zhu & Zou, 2008Nolan, 1997;Wagner & Hewlett, 1999;Wang et al, 2012;Prashant et al, 2010]. HIV mathematical models can provide insights into the dynamics of viral load in vivo and may play a significant role in the development of a better understanding of HIV/AIDs and drug therapies.…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

Zhao

Liu

Yan

2015

Int. J. Bifurcation Chaos

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In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.

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“…The eigenvalue obtained from equation (7) will be analyzed in the similar way to the one performed in [12] and [11]. For τ = 0, the equation (7) becomes λ 2 + p 0 λ + p 1 = 0 .…”

Section: The Mathematical Modelmentioning

confidence: 99%

Mathematical model of tuberculosis epidemic with recovery time delay

Iskandar

Chaniago

Munzir

et al. 2017

AIP Conference Proceedings

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Abstract. Tuberculosis (TB) is a contagious disease which can cause death. The disease is caused by Mycobacterium Tuberculosis which generally affects lungs and other organs such as lymph gland, intestine, kidneys, uterus, bone, and brain. The spread of TB occurs through the bacteria-contaminated air which is inhaled into the lungs. The symptoms of the TB patients are cough, chest pain, shortness of breath, appetite lose, weight lose, fever, cold, and fatigue. World Health Organization (WHO) reported that Indonesia placed the second in term of the most TB cases after India which has 23 % cases while China is reported to have 10 % cases in global. TB has become one of the greatest death threats in global. One way to countermeasure TB disease is by administering vaccination. However, a medication is needed when one has already infected. The medication can generally take 6 months of time which consists of two phases, inpatient and outpatient. Mathematical models to analyze the spread of TB have been widely developed. One of them is the SEIR type model. In this model the population is divided into four groups, which are suspectible (S), exposed (S), infected (I), recovered (R). In fact, a TB patient needs to undergo medication with a period of time in order to recover. This article discusses a model of TB spread with considering the term of recovery (time delay). The model is developed in SIR type where the population is divided into three groups, suspectible (S), infected (I), and recovered (R). Here, the vaccine is given to the susceptible group and the time delay is considered in the group undergoing the medication.

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Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

Cited by 4 publications

References 18 publications

Stability analysis of predator-prey with allelopathic effect

Stability analysis of predator-prey with allelopathic effect

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

Mathematical model of tuberculosis epidemic with recovery time delay

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