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

Advances in Difference Equations

article

Alfifi, H. Y.¹ 

[1] Department of Basic Sciences, College of Education, Imam Abdulrahman Bin Faisal University

关键词: Semi-analytical solution;    Logistic equation;    Stability analysis;    Hopf bifurcation;    Limit cycle;    Reaction–diffusion;    Delay equation;   

DOI  :  10.1186/s13662-020-02613-0

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY