Hopf bifurcations in a class of reaction-diffusion equations including two discrete time delays: An algorithm for determining Hopf bifurcation, and its applications

Asl, 2003 Bi, 2013, Bifurcations in delay differential equations and applications to tumor and immune system interaction models, SIAM J Appl Dyn Syst, 12, 1847, 10.1137/120887898 Chow, 1982 Cooke, 1986, On zeroes of some transcendental equationsm, Funkcialaj Ekvacioj, 29, 77 D‘Huys, 2013, Synchronisation and scaling properties of chaotic networks with multiple delays, EPL (Europhysics Letters), 103(1), 10013, 10.1209/0295-5075/103/10013 Galach, 2003, Dynamics of the tumor-immune system competition-the effect of time delay, Int J Appl Math Comput Sci, 13, 395 Hadeler, 2007, Interaction of diffusion and delay, Discrete Contin Dyn Syst - SerB, 8 (1), 95 Hassard, 1981 Iri, 2014, Dynamics of turing pattern under time-delayed feedback, J Phys Soc Jpn, 83, 024001, 10.7566/JPSJ.83.024001 Karaoğlu, 2014, Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays, Chaos Solitons Fractals, 68, 159, 10.1016/j.chaos.2014.07.011 Kayan, 2017, An algorithm for Hopf bifurcation analysis of a delayed reaction-diffusion model, Nonlinear Dyn, 89, 345, 10.1007/s11071-017-3458-5 Kirschner, 1998, Modeling immunotherapy of the tumor-immune interaction, Nonlinear Dyn, 37(3), 235 Kuang, 1993 Kuznetsov, 1998 Lengyel, 1991, Modeling of turing structures in the chlorite-iodide-malonic acid-starch reaction system, Science, 251(4994), 650, 10.1126/science.251.4994.650 Lin, 2012, Stability analysis of delay differential equations with two discrete delays, Can Appl Math Q, 20 (4), 519 Merdan, 2015, Hopf bifurcations in Lengyel-Epstein reaction-diffusion model with discrete time delay, Nonlinear Dyn, 79, 1757, 10.1007/s11071-014-1772-8 Merdan, 2016, Delay effects on the dynamics of the Lengyel-Epstein reaction-diffusion model, 125 Murray, 2003 Ruan, 2001, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart Appl Math, 59, 159, 10.1090/qam/1811101 Seydel, 2010 Shi J.. Partial differential equations and mathematical biology2004; http://www.resnet.wm.edu/~Jxshix/math490/lecture-chap1.pdf. Strogatz, 1994 Svetlana, 2014, Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system, Phil Trans R Soc A, 372, 20140014, 10.1098/rsta.2014.0014 Townley, 2000, Existence and learning of oscillations in recurrent neural networks, IEEE Trans Neural Netw, 11, 205, 10.1109/72.822523 Volpert, 2009, Reaction-diffusion waves in biologys, Phys Life Rev, 6(4), 267, 10.1016/j.plrev.2009.10.002 Wang, 2016, Dynamical analysis for a model of asset prices with two delays, Physica A, 447, 297, 10.1016/j.physa.2015.12.054 Wei, 1999, Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255, 10.1016/S0167-2789(99)00009-3 Wu, 1996 Yafia, 2007, Hopf bifurcation in differential equations with delay for tumor-immune system competition model, SIAM J Appl Math, 67(6), 1693, 10.1137/060657947 Yanchuk, 2014, Pattern formation in systems with multiple delayed feedbacks, Phys Rev Lett, 112(17), 174103, 10.1103/PhysRevLett.112.174103 Yanchuk, 2015, Dynamical systems with multiple long-delayed feedbacks: multiscale analysis and spatiotemporal equivalence, Phys Rev E Stat Nonlin Soft Matter Phys, 92(4), 042903, 10.1103/PhysRevE.92.042903