Hopf bifurcation in a reaction-diffusion-advection equation with nonlocal delay effect

Abstract This paper investigates the dynamics of a general reaction-diffusion-advection equation with nonlocal delay effect and Dirichlet boundary condition. The existence and stability of positive spatially nonhomogeneous steady state solution are shown. By analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized equation, the existence of Hopf bifurcation is proved. We introduce the weighted space to overcome the hurdle from advection term. We also show that the effect of adding a term advection along environmental gradients to Hopf bifurcation values for a Logistic equation with nonlocal delay.

[1]  N. Britton.  Aggregation and the competitive exclusion principle. , 1989, Journal of Theoretical Biology.

[2]  Shangjiang Guo,et al.  Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect , 2015 .

[3]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[4]  Hong Xiang,et al.  Hopf bifurcation for a delayed predator-prey diffusion system with Dirichlet boundary condition , 2017, Appl. Math. Comput..

[5]  Yongli Song,et al.  Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity , 2019, Discrete & Continuous Dynamical Systems - B.

[6]  Yuan Yuan,et al.  Spatially nonhomogeneous equilibrium in a reaction–diffusion system with distributed delay , 2011 .

[7]  Li Ma,et al.  Stability and Bifurcation in a Delayed Reaction–Diffusion Equation with Dirichlet Boundary Condition , 2016, J. Nonlinear Sci..

[8]  Junjie Wei,et al.  Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence , 2012 .

[9]  Yuan Lou,et al.  Hopf bifurcation in a delayed reaction-diffusion-advection population model , 2017, 1706.02087.

[10]  Yongli Song,et al.  Hopf bifurcation in a reaction–diffusion equation with distributed delay and Dirichlet boundary condition , 2017 .

[11]  Wan-Tong Li,et al.  Stability of bifurcating periodic solutions in a delayed reaction–diffusion population model , 2010 .

[12]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[13]  Junjie Wei,et al.  Hopf bifurcations in a reaction-diffusion population model with delay effect , 2009 .

[14]  Jianshe Yu,et al.  Stability and bifurcations in a nonlocal delayed reaction–diffusion population model , 2016 .

[15]  Junjie Wei,et al.  Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation☆ , 2010 .

[16]  Nicholas F. Britton,et al.  Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model , 1990 .

[17]  Shawgy Hussein,et al.  Stability and Hopf bifurcation for a delay competition diffusion system , 2002 .

[18]  Junping Shi,et al.  Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect , 2012 .

[19]  Junjie Wei,et al.  Bifurcation Analysis for a Delayed Diffusive Logistic Population Model in the Advective Heterogeneous Environment , 2020, Journal of Dynamics and Differential Equations.

[20]  Wan-Tong Li,et al.  Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions , 2008 .

[21]  Margaret C. Memory.  Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion , 1989 .

[22]  Shangjiang Guo,et al.  Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect , 2016 .

[23]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[24]  Wenzhang Huang,et al.  Stability and Hopf Bifurcation for a Population Delay Model with Diffusion Effects , 1996 .

[25]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[26]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[27]  Kiyoshi Yoshida,et al.  The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology , 1982 .

[28]  Michael G. Crandall,et al.  Bifurcation, perturbation of simple eigenvalues, itand linearized stability , 1973 .

[29]  Rong Yuan,et al.  Bifurcations in a Diffusive Predator–Prey Model with Beddington–DeAngelis Functional Response and Nonselective Harvesting , 2018, J. Nonlinear Sci..

[30]  S. Levin,et al.  Diffusion and Ecological Problems: Modern Perspectives , 2013 .