Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Tang, Sanyi; Liang, Juhua

doi:10.1016/j.na.2012.08.013

Cited by 67 publications

(37 citation statements)

References 26 publications

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“…Numerous studies have focused on the glucoseinsulin regulatory system via a mathematical model of delay differential equations. Recently, Huang et al proposed two novel mathematical models with impulsive injections of insulin or its analogues for type 1 and type 2 diabetes mellitus [13], and similar impulsive differential equations have been widely used in integrated pest management [28,29,41,42]. In their paper, Huang et al assumed that the constant glucose infusion rate G in is described by a continuous process and insulin is injected once the blood glucose level reaches a threshold or at a fixed time.…”

Section: Resultsmentioning

confidence: 99%

“…In this case the models (1.2) and (1.3) can be rewritten as a Filippov system, a model which has been applied widely in many fields of science and engineering. Furthermore, the theory of Filippov systems is being recognized as not only richer than the corresponding theory of continuous systems, but also as representing a more natural framework for the mathematical modelling of real-world phenomena [18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Ag(t)(c + Mi(t)/(n + I(t))mentioning

confidence: 99%

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Modelling the regulatory system for diabetes mellitus with a threshold window

Tang

Cheke

2015

Communications in Nonlinear Science and Numerical Simulation

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Please cite this article as: Yang, J., Tang, S., Cheke, R.A., Modelling the regulatory system for diabetes mellitus with a threshold window, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http:// dx.doi.org/10.1016/j.cnsns. 2014.08.012 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 proof before it is published in its final 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.Modelling the regulatory system for diabetes mellitus with a threshold window AbstractPiecewise (or non-smooth) glucose-insulin models with threshold windows for type 1 and type 2 diabetes mellitus are proposed and analyzed with a view to improving understanding of the glucose-insulin regulatory system. For glucose-insulin models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between regular equilibria and a pseudo-equilibrium are studied. Furthermore, the sufficient and necessary conditions for the global stability of regular equilibria and the pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Sliding bifurcations related to boundary node bifurcations were investigated with theoretical and numerical techniques, and insulin clinical therapies are discussed. For glucose-insulin models with a threshold window, the effects of glucose thresholds or the widths of threshold windows on the durations of insulin therapy and glucose infusion were addressed. The duration of the effects of an insulin injection is sensitive to the variation of thresholds. Our results indicate that blood glucose level can be maintained within a normal range using piecewise glucose-insulin models with a single threshold or a threshold window. Moreover, our findings suggest that it is critical to individualize insulin therapy for each patient separately, based on initial blood glucose levels.

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Section: Resultsmentioning

confidence: 99%

Section: Ag(t)(c + Mi(t)/(n + I(t))mentioning

confidence: 99%

Modelling the regulatory system for diabetes mellitus with a threshold window

Tang

Cheke

2015

Communications in Nonlinear Science and Numerical Simulation

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“…A common assumption in such models is that the human control activities occur instantaneously, but this is seldom the case with interventions or control strategies usually lasting for a given period. Recently, a threshold policy (TP) has been proposed to describe density-dependent and persistent interventions, which are implemented when the case numbers exceeds a certain value and are suspended when they fall below a critical level [23][24][25][26][27][28][29]. Therefore, our main purpose is to extend the existing models on WNV as a non-smooth system by considering density-dependent and non-instantaneous control measures, based on the threshold policy idea, to examine whether a threshold policy could be used to control the transmission dynamics of WNV more effectively than reliance on existing impulsive differential equations.…”

Section: A C C E P T E D Mmentioning

confidence: 99%

A threshold policy to interrupt transmission of West Nile Virus to birds

Zhou

Xiao

Cheke

2016

Applied Mathematical Modelling

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Highlights• Propose a Filippov model of West Nile Virus with density-dependent culling strategy.• Solutions approach either endemic equilibrium for subsystems or a pseudo-equilibrium.• Results indicate that a previously chosen level of infected birds can be maintained.• Strengthening mosquito culling is beneficial to curbing the spread of WNV. AbstractThis paper proposes a model of West Nile Virus (WNV) with a Filippov-type control strategy of culling mosquitoes implemented once the number of infected birds exceeds a threshold level. The long-term dynamical behaviour of the proposed non-smooth system is investigated. It is shown that as the threshold value varies, model solutions ultimately approach either one of two endemic equilibria for two subsystems or a pseudo-equilibrium on the switching surface, which is a novel steady state. The results indicate that a previously chosen level of infected birds can be maintained when the threshold policy and other parameters are chosen properly. Numerical studies show that under the threshold policy, strengthening mosquito culling together with protecting bird population is beneficial to curbing the spread of WNV.

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“…Recently the classical Lotka-Volterra model was extended by using a piecewise saturating function to replace the linear consumption rate for considering the observed experimental results theoretically [17,24]. For simplicity, denote H(Z) = x − ET with Z = (x, y) T ∈ R 2 + , where ET describes the critical prey population threshold, and the parameter ε can be defined as follows ε = 0, H(Z) = x − ET < 0, 1, H(Z) = x − ET > 0.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, if the density of the prey population falls below the threshold ET , that is, H(Z) < 0, then ε = 0, which indicates that the prey may avoid the predator via a habitat shift by moving to the refuge and the density of the predator will decrease [2]; if the density of the prey population increases and exceeds the threshold ET , that is, H(Z) > 0, then ε = 1, which indicates that the prey population may re-appear and once again become accessible to predators [2,16,17,24]. According to the above definition, the extended classical Lotka-Volterra model with a piecewise saturating function can be defined as the following Filippov system        dx(t) dt = rx(t) − εbx(t)y(t) 1 + bhx(t) , dy(t) dt = εkbx(t)y(t) 1 + bhx(t) − δy(t),…”

Section: Introductionmentioning

confidence: 99%

Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

Tan¹,

Qin²,

Liu³

et al. 2016

J. Nonlinear Sci. Appl.

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This paper studies a Filippov predator-prey system, where chemical control strategies are proposed and analyzed. Initially, the exact sliding segment and its domains are addressed. Then the existence and stability of the regular, virtual, pseudo-equilibria and tangent points are discussed. It shows that two regular equilibria and a pseudo-equilibrium can coexist. By employing theoretical and numerical techniques several kinds of bifurcations are investigated, such as sliding bifurcations related to the boundary node (focus) bifurcations, touching bifurcations, sliding crossing bifurcation and buckling bifurcations (or sliding switching). Furthermore, it makes comparison of the obtained results with previous studies for the Filippov predator-prey system without control strategies. Some biological implications of our results with respect to pest control are also given.

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Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Cited by 67 publications

References 26 publications

Modelling the regulatory system for diabetes mellitus with a threshold window

Modelling the regulatory system for diabetes mellitus with a threshold window

A threshold policy to interrupt transmission of West Nile Virus to birds

Sliding Bifurcation Analysis and Global Dynamics for a Filippov Predator-prey System

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