The diffusive Lotka–Volterra predator–prey system with delay

Noufaey, K.S. Al; Marchant, Timothy R.; Edwards, Maureen

doi:10.1016/j.mbs.2015.09.010

Cited by 24 publications

(13 citation statements)

References 19 publications

(22 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Reaction-diffusion phenomenon with time delays has been incorporated into many fields of biological applications. These applications have explained a number of practical applications in our everyday life by using partial differential equations (PDEs), for instance, in population ecology [15,17,20,30,31], animals [2,4,26], cell [5,19,22,25,33], chemicals [1,3,10], and heat and mass transfer [13,27]. This model can introduce instability, via a Hopf bifurcation, with the subsequent development of limit cycles.…”

Section: Introductionmentioning

confidence: 99%

“…Semi-analytical methods have been used to discuss many delay systems with reactiondiffusion phenomenon, such as delay logistic equations [6,7], predator-prey model [2], viral infection system [5], pellet systems [24], the Belousov-Zhabotinsky (BZ) reaction [9], the Brusselator model [3], the reversible Selkov model [1], the equation of Nicholsons blowflies [8], and the limited food model [4]. The outcomes for all papers that used this method revealed an excellent agreement between semi-analytical ODEs outcomes and the numerical solutions pertain to PDEs equations.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

Alfifi

2020

Adv Differ Equ

View full text Add to dashboard Cite

In this work, the semi-analytical solution is studied for the diffusive logistic equation with both mixed instantaneous and delayed density. The domain of reaction-diffusion in one dimension is shown. Delay partial differential equation is approximated with a delay ordinary differential equation system by using the Galerkin technique method. Steady-state solutions and stability analysis as well as bifurcation diagrams are derived. The effect of diffusion parameter and delay values is comprehensively studied; as a result, both parameters can destabilize or stabilize the model. We obtained that the decrease in values of the Hopf bifurcations for growth rate is associated with an increase in delay values, whereas the diffusion parameter is increased. Furthermore, comparisons between the numerical simulations and semi-analytical results present a good agreement for all examples and figures of the Hopf bifurcations. Examples of limit cycle and phase-plane map are plotted to confirm the benefits and accuracy of semi-analytical solutions result. For periodic solutions, an asymptotic method is studied after the Hopf bifurcation point for both one-and two-term semi-analytical systems.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

Alfifi

2020

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…Semi-analytical solutions have been applied to solve several complications with reaction-diffusion systems, such as predator-prey model [8], pellet systems [18], BZ reactions [6], the Brusselator system [2], mixed quadratic-cubic autocatalytic reactions [7], and logistic equations [3,4]. All these models yielded accurate solutions compared to full numerical outcomes.…”

Section: Introductionmentioning

confidence: 99%

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

Alfifi¹

2019

J. Nonlinear Sci. Appl.

View full text Add to dashboard Cite

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.

show abstract

“…For a long period of time, many significant nonlinear phenomena have been modelled and described via ordinary or partial differential equations (ODEs or PDEs). For example, these equations have been used to model population ecology [1][2][3][4], animals [5,6], health [7,8], chemicals [9,10], and business economics [11][12][13][14]. A business cycle model is utilized to explain the working of economic laws and can also be utilized to predict investment status, yield, costs, and other important factors in the business economic model.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Solutions for the Diffusive Kaldor–Kalecki Business Cycle Model with a Time Delay for Gross Product and Capital Stock

Alfifi

2021

Complexity

View full text Add to dashboard Cite

This paper discusses the stability and Hopf bifurcation analysis of the diffusive Kaldor–Kalecki model with a delay included in both gross product and capital stock functions. The reaction-diffusion domain is considered, and the Galerkin analytical method is used to derive the system of ordinary differential equations. The methodology used to determine the Hopf bifurcation points is discussed in detail. Furthermore, full diagrams of the Hopf bifurcation regions considered in the stability analysis are shown, and some numerical simulations of the limit cycle are used to confirm the theoretical outcomes. The delay investment parameter and diffusion coefficient can have great impacts on the Hopf bifurcations and stability of the business cycle model. The investment parameters for the gross product and capital stock as well as the adjustment coefficient of the production market are also studied. These parameters can cause instability in, and the stabilization of, the business cycle model. In addition, we point out that, as the delay investment parameter increases, the Hopf bifurcation points for the diffusion coefficient values decrease considerably. When the delay investment parameter has a very small value, the solution of the business cycle model tends to become steady.

show abstract

The diffusive Lotka–Volterra predator–prey system with delay

Cited by 24 publications

References 19 publications

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

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

Semi-Analytical Solutions for the Diffusive Kaldor–Kalecki Business Cycle Model with a Time Delay for Gross Product and Capital Stock

Contact Info

Product

Resources

About