Stability, Persistence, and Extinction in a Predator-Prey System With Discrete and Continuous Time Delays

Freedman, H. I.; Rao, V. Sree Hari; Lakshmi, K. Jaya

doi:10.1142/9789812798893_0017

Cited by 6 publications

(9 citation statements)

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“…If either (H3) or (H4) holds, following the arguments in [14], we see that the stability of E = (K, 0, 0) cannot be preserved. Thus, we have the following theorem.…”

Section: )mentioning

confidence: 68%

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Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Ferreira¹,

Rao

2013

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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A third-order system of ordinary differential equations, modelling two predators competing for a single prey species, is analysed in this paper. A delay term modelling the delayed logistic growth of the prey is included. Fixed points of the system are identified, and a linearized stability analysis is carried out. For some parameter regime, there exists a continuum of equilibria and these equilibria may undergo a zip bifurcation. The main results presented herein are that this zip bifurcation is 'unsustainable' for certain ranges of values of the time-delay parameter. Finally, spatial diffusion is incorporated in the delay differential equation model, and it is shown that the zip bifurcation remains unsustainable.

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“…If either (H3) or (H4) holds, following the arguments in [14], we see that the stability of E = (K, 0, 0) cannot be preserved. Thus, we have the following theorem.…”

Section: )mentioning

confidence: 68%

“…If the derivative is positive (negative), then clearly stabilization (destabilization) cannot take place at that value ofτ (see [14]). That is, we analyse dδ(τ )/dτ where δ(τ ) = 0.…”

Section: Equilibria and Local Stability Analysismentioning

confidence: 99%

Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Ferreira¹,

Rao

2013

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Due to these restrictions, delay-independent stability results become less applicable. Moreover, these results are of little interest given that increasing time delays always destabilises the system [11,[13][14][15]. Hence it is worthwhile discussing results which depend on the delay (that is, delay-dependent stability results) for which the equilibrium is stable.…”

Section: Discussionmentioning

confidence: 99%

Stability analysis of resource-consumer dynamic models

Rao

Rao²

2006

ANZIAM J.

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A nutrient-consumer model involving a distributed delay in material recycling and a discrete delay in growth response has been analysed. Various easily verifiable sets of sufficient conditions for global asymptotic stability of the positive equilibrium solution of the model equations have been obtained and the length of the delay in each case has been estimated.2000 Mathematics subject classification: 34K20, 34K15, 92A15.

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“…Equations of the type (8) are well studied in literature (e.g., [8,10,11]). Letting a 0 = n 2 − (m 1 + n 1 ) 2 , a 1 = m 2 − 2ln − l 1 2 and a 2 = l 2 − 2m, a straight forward application of Theorem 4.1 of [8] yields the Theorem 3.2.…”

Section: Delay Only In Incubationmentioning

confidence: 99%

A dynamic model for infectious diseases: The role of vaccination and treatment

Rao¹,

Kumar

2015

Chaos, Solitons & Fractals

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This article is dedicated to their research supervisor Prof. V. Sree Hari Rao for his continued inspiration. a b s t r a c tUnderstanding dynamics of an infectious disease helps in designing appropriate strategies for containing its spread in a population. Recent mathematical models are aimed at studying dynamics of some specific types of infectious diseases. In this paper we propose a new model for infectious diseases spread having susceptible, infected, and recovered populations and study its dynamics in presence of incubation delays and relapse of the disease. The influence of treatment and vaccination efforts on the spread of infection in presence of time delays are studied. Sufficient conditions for local stability of the equilibria and change of stability are derived in various cases. The problem of global stability is studied for an important special case of the model. Simulations carried out in this study brought out the importance of treatment rate in controlling the disease spread. It is observed that incubation delays have influence on the system even under enhanced vaccination. The present study has clearly brought out the fact that treatment rate even in presence of time delays would contain the disease as compared to popular belief that eradication can only be done through vaccination.

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Stability, Persistence, and Extinction in a Predator-Prey System With Discrete and Continuous Time Delays

Cited by 6 publications

References 0 publications

Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Stability analysis of resource-consumer dynamic models

A dynamic model for infectious diseases: The role of vaccination and treatment

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