Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation

Alfifi, Hassan Yahya; Marchant, Timothy R.; Nelson, Mark

doi:10.1093/imamat/hxs060

Cited by 20 publications

(24 citation statements)

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“…The Runge-Kutta fourth-order technique [5,18] and a Crank-Nicholson finite-difference scheme [4,6] are introduced to the ODE and PDE numerical solutions and the spatial and temporal discretizations used in all the examples and figures are ∆x = 0.05 and ∆t = 5 × 10 −3 , respectively. The proportion of inaccuracy, which is considered to be the difference between the two-term semi-analytical solutions and the numerical solutions is defined in this paper as the unconditional value of the difference divided by the precise value times 100.…”

Section: Governing Equationsmentioning

confidence: 99%

“…In our everyday life, delayed reaction-diffusion models have revealed a number of oscillatory phenomena with the use of continuous well-stirred tank reactor (CSTR). CSTRs have obtained good results with experimental oscillatory systems in the theoretical research, for instance in logistical equations [4], the viral infection model [14,20,27], Nicholson's blowflies equation [5], and the limited food model [1].…”

Section: Introductionmentioning

confidence: 99%

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Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

Alfifi¹

2019

J. Nonlinear Sci. Appl.

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In the one-dimensional reaction-diffusion domain of this study, semi-analytical solutions are used for a delayed viral infection system with logistic growth. Through an ordinary differential equations system, the Galerkin technique is believed to estimate the prevailing partial differential equations. In addition, Hopf bifurcation maps are constructed. The effect of diffusion coefficient stricture and delay on the model is comprehensively investigated, and the outcomes demonstrate that diffusion and delay can stabilize or destabilize the system. We found that, as the delay parameter values rise, the values of the Hopf bifurcations for growth and the rates of viral death are augmented, whereas the rate of production is decreased. For the growth, production, and death rates strictures, there is determination of an asymptotically unstable region and a stable region. Illustrations of the unstable and stable limit cycles, as well as the Hopf bifurcation points, are found to prove the formerly revealed outcomes in the Hopf bifurcation map. The results of the semi-analytical solutions and numerical assessments revealed that the semi-analytical solutions are highly effective.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

Alfifi¹

2019

J. Nonlinear Sci. Appl.

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“…There is a difference of 18% between the two-term semi-analytical and numerical solutions, up to f = 15. Semi-analytical solutions for a 2-D geometry generally have slightly larger errors than those in a 1-D geometry, see [28,29,8]. Figure 4 shows the steady-state reactant concentrations u, v and w versus x, for the 2-D case.…”

Section: Steady-state Solutionsmentioning

confidence: 99%

Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

2016

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The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail. Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

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“…Above these curves limit cycles can occur while under the curves only stable solutions occur and there are no Hopf bifurcation points. In general, small values of the delay parameters leads to a stable solution while large delays, which implies feedback from the distant past, leads to instabilities [1,2]. The Hopf curve of Hopf bifurcations can be described by the linear equations τ 2 = −0.994τ 1 + 3.11 and τ 2 = −τ 1 + 3.23 for one and two-term semi-analytical solutions respectively.…”

Section: Local Stability Bifurcation Diagrams and Oscillatory Solutionsmentioning

confidence: 99%

The diffusive Lotka–Volterra predator–prey system with delay

Noufaey

Marchant

Edwards

2015

Mathematical Biosciences

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Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semianalytical and numerical solutions of the governing equations. The diffusive Lotka-Volterra predator-prey system with delay Abstract. Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.

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Semi-analytical solutions for the 1- and 2-D diffusive Nicholson's blowflies equation

Cited by 20 publications

References 0 publications

Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

The diffusive Lotka–Volterra predator–prey system with delay

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