Stability in Insect Host-Parasite Models

Hassell, M. P.; May, Robert M.

doi:10.2307/3133

Cited by 841 publications

(387 citation statements)

References 41 publications

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“…As prey below the critical density are effectively in a refuge, this analysis nicely demonstrates the stabilizing role of refugia that protect a fixed number of prey (e. g., Rosenzweig and MacArthur, 1963;Hassell and May, 1973;Maynard Smith, 1974;Murdoch and Oaten, 1975;Hassell, 1978;Sih, 1987;Ives and Dobson, 1987;Ruxton, 1995;Hochberg and Holt, 1995;Křivan, 1998). When the refuge is large enough, i. e., when the critical prey density below which prey are protected is high, provided predators are able to regulate the prey population (i. e., when handling times are small) then the two species coexist at an equilibrium.…”

Section: Discussionmentioning

confidence: 68%

On the Gause predator–prey model with a refuge: A fresh look at the history

Křivan

2011

Journal of Theoretical Biology

100

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Cite this article as: Vlastimil Křivan, On the gause predator-prey model with a refuge: a fresh look at the history, Journal of Theoretical Biology, doi:10.1016Biology, doi:10. /j.jtbi.2011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.On the Gause predator-prey model with a refuge: A fresh look at the history AbstractThis article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology G. F. Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a (1) trajectories converge to a limit cycle, as predicted by F. G. Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.

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Section: Discussionmentioning

confidence: 68%

On the Gause predator–prey model with a refuge: A fresh look at the history

Křivan

2011

Journal of Theoretical Biology

100

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“…A c c e p t e d m a n u s c r i p t (Beddington et al, 1978;Hassell, 1984;Hassell and May, 1973;May and Hassell, 1988;Waage and Hassell, 1982;Waage et al, 1985).…”

Section: Accepted Manuscriptmentioning

confidence: 99%

Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases

Tang

Cheke

2010

Journal of Theoretical Biology

115

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. peer-00585804, version 1 -14 Apr 2011Author manuscript, published in "Journal of Theoretical Biology 264, 2 (2010) 623" DOI : 10.1016/j.jtbi.2010.02.034 A c c e p t e d m a n u s c r i p t A c c e p t e d m a n u s c r i p tAbstract Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest-natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided.Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications.Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial.To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics.

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“…Smith [12]). Hassell and May [6] have shown that the addition of a large refuge to a model, which in the absence of prey refuge exhibits a divergent oscillation, can replace the oscillatory behavior with a stable equilibrium. Based on the above, an attempt is made in the present investigation to study a three-component predator-prey model in which one prey species takes refuge and two predator species feed on the same prey species with competition among themselves.…”

Section: Introductionmentioning

confidence: 99%

“…The study of the consequences of prey refuge on the dynamics of predator-prey interactions can be recognized as a major but rather challenging issue in applied mathematics and theoretical ecology (cf. Hassell and May [6], Hassell [7], Holling [8,9], Hoy [10], Huang et al [11], Smith [12]). Some of the empirical and theoretical works based on prey refuge have concluded that the refuge used by prey has a stabilizing effect on predator-prey interactions and also the prey species can be prevented from extinction by using this policy (cf.…”

Section: Introductionmentioning

confidence: 99%

Dynamical behaviour of a two-predator model with prey refuge

Sarwardi

Mandal²,

Ray

2013

J Biol Phys

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A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.

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Stability in Insect Host-Parasite Models

Cited by 841 publications

References 41 publications

On the Gause predator–prey model with a refuge: A fresh look at the history

On the Gause predator–prey model with a refuge: A fresh look at the history

Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases

Dynamical behaviour of a two-predator model with prey refuge

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