# American Institue of Mathematical Sciences

2017, 22(2): 647-668. doi: 10.3934/dcdsb.2017031

## The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China 2 Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK 3 Department of Mathematical and Life Sciences, Hiroshima University, Kagamiyama 1-3-1, Higashi-hiroshima 739-0046, Japan and JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

1Corresponding Author

Received  January 2016 Revised  June 2016 Published  December 2016

A delayed reaction-diffusion Schnakenberg system with Neumann boundary conditions is considered in the context of long range biological self-organisation dynamics incorporating gene expression delays. We perform a detailed stability and Hopf bifurcation analysis and derive conditions for determining the direction of bifurcation and the stability of the bifurcating periodic solution. The delay-diffusion driven instability of the unique spatially homogeneous steady state solution and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated, with limited simulations to support our theoretical analysis. These studies analytically demonstrate that the modelling of gene expression time delays in Turing systems can eliminate or disrupt the formation of a stationary heterogeneous pattern in the Schnakenberg system.

Citation: Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
##### References:
 [1] B. Alberts, A. Johnson, P. Walter, J. Lewis, M. Raff and K. Roberts, Molecular Biology of The Cell 5th ed. Garland Science, New York, 2002. [2] E. Beretta, Y. Kuang, Geometry stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [3] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer, New York, 1992. [4] K. L. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8. [5] Y. Chen, A. F. Schier, Lefty proteins are long-range inhibitors of squint-mediated nodal signaling, Curr. Biol., 12 (2002), 2124-2128. [6] S. Chen, J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonl. Anal. RWA, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004. [7] S. Chen, J. Shi, J. Wei, A note on Hopf bifurcation in delayed diffusive Lotka-Volterra predator-prey system, Comput. Math. Appl., 62 (2011), 2240-2245. doi: 10.1016/j.camwa.2011.07.011. [8] S. Chen, J. Shi, J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commu. Pure. Appl. Anal., 12 (2013), 481-501. doi: 10.3934/cpaa.2013.12.481. [9] S. Chen, J. Shi, J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38. doi: 10.1007/s00332-012-9138-1. [10] C. F. Drew, C. M. Lin, T. X. Jiang, G. Blunt, C. Mou, C. M. Chuong and D. J. Headon, The Edar subfamily in feather placode formation, Developmental Biology, 305 (2007), 232-245. [11] T. Faria, L. Magalhses, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Diff. Equa, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144. [12] T. Faria, L. Magalhses, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Diff. Equa., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145. [13] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. [14] T. Faria, On the study of singularities for a planar system with two delays, Dyn. Contin. Discrete Implus. Syst., 10 (2003), 357-371. [15] E. A. Gaffney, N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Trans. Amer. Math. Soc, 12 (1972), 30-39. doi: 10.1007/s11538-006-9066-z. [16] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [17] K. P. Hadeler, S. Ruan, Interaction of diffusion and delay, Dyn. Con. Dis. Sys. Series B., 8 (2007), 95-105. doi: 10.3934/dcdsb.2007.8.95. [18] M. P. Harris, S. Williamson, J. F. Fallon, H. Meinhardt, R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci., 102 (2005), 11734-11739. [19] M. P. Harris, S. Williamson, J. F. Fallon, H. Meinhardt, R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci. USA, 102 (2005), 11734-11739. [20] H. Juan, H. Hamada, Roles of nodal-lefty regulatory loops in embryonic patterning of vertebrates, Genes Cells., 6 (2001), 923-930. [21] J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol., 13 (2003), 1398-1408. [22] T. Miura, K. Shiota, Extracellular matrix environment influences chondrogenic pattern formation in limb bud micromass culture: Experimental verification of theoretical models, Anat. Rec., 258 (2000), 100-107. [23] T. Miura, K. Shiota, TGFβ2 acts as an activator molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture, Dev. Dyn., 217 (2000), 241-249. [24] T. Miura, K. Shiota, G. Morriss-Kay, P. K. Maini, Mixed-mode pattern in doublefoot mutant mouse limb-Turing reaction-diffusion model on a growing domain during limb development, J. Theor. Biol., 240 (2006), 562-573. doi: 10.1016/j.jtbi.2005.10.016. [25] Y. Morita, Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions, Japan J. Appl. Math., 1 (1984), 39-65. doi: 10.1007/BF03167861. [26] C. Mou, B. Jackson, P. Schneider, P. A. Overbeek, D. J. Headon, Generation of the primary hair follicle pattern, Proceedings Of The National Academy Of Sciences Of The United States Of America, 103 (2006), 9075-9080. [27] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. [28] L. A. F. de Oliveira, Instability of homogeneous periodic solutions of parabolic-delay equations, J. Diff. Equa., 109 (1994), 42-76. doi: 10.1006/jdeq.1994.1044. [29] S. Ruan, Absolute stability, conditional stability an bifurcation in Kolmogorov-type predator-prey systems with discrete delay, Quart. Appl. Math., 59 (2001), 159-176. [30] S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 61 (1998), 15-32. doi: 10.1093/imamat/61.1.15. [31] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Con. Dis. Impul. Sys. Series A: Math. Anal., 10 (2003), 863-874. [32] R. Sakuma, Y. Ohnishi, C. Meno, H. Fujii, H. Juan, J. Takeuchi, T. Ogura, E. Li, K. Miyazono, H. Hamada, Inhibition of nodal signalling by lefty mediated through interaction with common receptors and efficient diffusion, Genes Cells., 7 (2002), 401-412. [33] L. A. Segel, J. L. Jackson, Dissipative structure. an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559. [34] L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr. Biol. , 13 (2003), R7-R9,401-412. [35] J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0. [36] S. Seirin Lee, E. A. Gaffney, Aberrant behaviours of reaction diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-010-9533-4. [37] S. Seirin Lee, E. A. Gaffney, R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-011-9634-8. [38] S. Seirin Lee, E. A. Gaffney, N. A. M. Monk, The influence of gene expression time delays on Gierer-Meihardt pattern formation systems, Bull. Math. Biol., 72 (2010), 2139-2160. doi: 10.1007/s11538-010-9532-5. [39] S. Sick, S. Reinker, J. Timmer, T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450. doi: 10.1126/science.1130088. [40] A. Sorkin, M. von Zastrow, Signal transduction and endocytosis: Close encounters of many kinds, Nature Reviews Molecular Cell Biol., 3 (2002), 600-614. [41] C. N. Tennyson, H. J. Klamut, R. G. Worton, The human dystrophin gene requires 16 hr to be transcribed and is contranscriptionally spliced, Nat. Gen., 9 (1995), 184-190. [42] A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. [43] J. Xia, Z. Liu, R. Yuan, S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type Ⅱ functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200. doi: 10.1137/080728512.

show all references

##### References:
 [1] B. Alberts, A. Johnson, P. Walter, J. Lewis, M. Raff and K. Roberts, Molecular Biology of The Cell 5th ed. Garland Science, New York, 2002. [2] E. Beretta, Y. Kuang, Geometry stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [3] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory Springer, New York, 1992. [4] K. L. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8. [5] Y. Chen, A. F. Schier, Lefty proteins are long-range inhibitors of squint-mediated nodal signaling, Curr. Biol., 12 (2002), 2124-2128. [6] S. Chen, J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production saturation and gene expression time delays, Nonl. Anal. RWA, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004. [7] S. Chen, J. Shi, J. Wei, A note on Hopf bifurcation in delayed diffusive Lotka-Volterra predator-prey system, Comput. Math. Appl., 62 (2011), 2240-2245. doi: 10.1016/j.camwa.2011.07.011. [8] S. Chen, J. Shi, J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commu. Pure. Appl. Anal., 12 (2013), 481-501. doi: 10.3934/cpaa.2013.12.481. [9] S. Chen, J. Shi, J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction-diffusion systems, J. Nonlinear Sci., 23 (2013), 1-38. doi: 10.1007/s00332-012-9138-1. [10] C. F. Drew, C. M. Lin, T. X. Jiang, G. Blunt, C. Mou, C. M. Chuong and D. J. Headon, The Edar subfamily in feather placode formation, Developmental Biology, 305 (2007), 232-245. [11] T. Faria, L. Magalhses, Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. Diff. Equa, 122 (1995), 181-200. doi: 10.1006/jdeq.1995.1144. [12] T. Faria, L. Magalhses, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Diff. Equa., 122 (1995), 201-224. doi: 10.1006/jdeq.1995.1145. [13] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. [14] T. Faria, On the study of singularities for a planar system with two delays, Dyn. Contin. Discrete Implus. Syst., 10 (2003), 357-371. [15] E. A. Gaffney, N. A. M. Monk, Gene expression time delays and Turing pattern formation systems, Trans. Amer. Math. Soc, 12 (1972), 30-39. doi: 10.1007/s11538-006-9066-z. [16] A. Gierer, H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [17] K. P. Hadeler, S. Ruan, Interaction of diffusion and delay, Dyn. Con. Dis. Sys. Series B., 8 (2007), 95-105. doi: 10.3934/dcdsb.2007.8.95. [18] M. P. Harris, S. Williamson, J. F. Fallon, H. Meinhardt, R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci., 102 (2005), 11734-11739. [19] M. P. Harris, S. Williamson, J. F. Fallon, H. Meinhardt, R. O. Prum, Molecular evidence for an activator-inhibitor mechanism in development of embryonic feather branching, Proc. Natl. Acad. Sci. USA, 102 (2005), 11734-11739. [20] H. Juan, H. Hamada, Roles of nodal-lefty regulatory loops in embryonic patterning of vertebrates, Genes Cells., 6 (2001), 923-930. [21] J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol., 13 (2003), 1398-1408. [22] T. Miura, K. Shiota, Extracellular matrix environment influences chondrogenic pattern formation in limb bud micromass culture: Experimental verification of theoretical models, Anat. Rec., 258 (2000), 100-107. [23] T. Miura, K. Shiota, TGFβ2 acts as an activator molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture, Dev. Dyn., 217 (2000), 241-249. [24] T. Miura, K. Shiota, G. Morriss-Kay, P. K. Maini, Mixed-mode pattern in doublefoot mutant mouse limb-Turing reaction-diffusion model on a growing domain during limb development, J. Theor. Biol., 240 (2006), 562-573. doi: 10.1016/j.jtbi.2005.10.016. [25] Y. Morita, Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions, Japan J. Appl. Math., 1 (1984), 39-65. doi: 10.1007/BF03167861. [26] C. Mou, B. Jackson, P. Schneider, P. A. Overbeek, D. J. Headon, Generation of the primary hair follicle pattern, Proceedings Of The National Academy Of Sciences Of The United States Of America, 103 (2006), 9075-9080. [27] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. [28] L. A. F. de Oliveira, Instability of homogeneous periodic solutions of parabolic-delay equations, J. Diff. Equa., 109 (1994), 42-76. doi: 10.1006/jdeq.1994.1044. [29] S. Ruan, Absolute stability, conditional stability an bifurcation in Kolmogorov-type predator-prey systems with discrete delay, Quart. Appl. Math., 59 (2001), 159-176. [30] S. Ruan, Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling, IMA J. Appl. Math., 61 (1998), 15-32. doi: 10.1093/imamat/61.1.15. [31] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Con. Dis. Impul. Sys. Series A: Math. Anal., 10 (2003), 863-874. [32] R. Sakuma, Y. Ohnishi, C. Meno, H. Fujii, H. Juan, J. Takeuchi, T. Ogura, E. Li, K. Miyazono, H. Hamada, Inhibition of nodal signalling by lefty mediated through interaction with common receptors and efficient diffusion, Genes Cells., 7 (2002), 401-412. [33] L. A. Segel, J. L. Jackson, Dissipative structure. an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545-559. [34] L. Solnica-Krezel, Vertebrate development: Taming the nodal waves, Curr. Biol. , 13 (2003), R7-R9,401-412. [35] J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0. [36] S. Seirin Lee, E. A. Gaffney, Aberrant behaviours of reaction diffusion self-organization models on growing domains in the presence of gene expression time delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-010-9533-4. [37] S. Seirin Lee, E. A. Gaffney, R. E. Baker, The dynamics of Turing patterns for morphogen-regulated growing domains with cellular response delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-011-9634-8. [38] S. Seirin Lee, E. A. Gaffney, N. A. M. Monk, The influence of gene expression time delays on Gierer-Meihardt pattern formation systems, Bull. Math. Biol., 72 (2010), 2139-2160. doi: 10.1007/s11538-010-9532-5. [39] S. Sick, S. Reinker, J. Timmer, T. Schlake, WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314 (2006), 1447-1450. doi: 10.1126/science.1130088. [40] A. Sorkin, M. von Zastrow, Signal transduction and endocytosis: Close encounters of many kinds, Nature Reviews Molecular Cell Biol., 3 (2002), 600-614. [41] C. N. Tennyson, H. J. Klamut, R. G. Worton, The human dystrophin gene requires 16 hr to be transcribed and is contranscriptionally spliced, Nat. Gen., 9 (1995), 184-190. [42] A. M. Turing, The chemical basis of morphoegenesis, Phil. Tans. R. Soc. London, Ser. B, 237 (1952), 37-72. [43] J. Xia, Z. Liu, R. Yuan, S. Ruan, The effects of harvesting and time delay on predator-prey systems with Holling type Ⅱ functional response, SIAM J. Appl. Math., 70 (2009), 1178-1200. doi: 10.1137/080728512.
A: The parameter space of $(a, b)$ which satisfies Theorem 2.4. In the white region, we will have a stable periodic solution for the FDEs. B: A bifurcation diagram for $\tau$ for the FDEs. This has been plotted using the value of $u(t)$ at which $v(t)=v_*$. The parameter values $(a, b)=(0.1, 0.9)$ have been chosen and the system exhibits the first Hopf bifurcation at approximately $\tau_0=0.2171$. For $\tau\in[0, \tau_0)$, $E_*=(1.0, 0.9)$ is always asymptotically stable, bifurcating first at $\tau_0$ then exhibiting subsequent bifurcations with frequency doubling prior to chaotic dynamics
A homogeneous periodic solution for $u$ with $0< \varepsilon<1$. The parameter values for the simulations are given by $a=0.1, b=0.9, \varepsilon=0.1$. A: A plot of $u$ for $d=0.5$, when equation (43) is satisfied. B: A plot of $u$ for $d=5\times 10^{-5}$, when equation (43) is not satisfied
 [1] Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975 [2] Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784 [3] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [4] Reinhard Racke. Instability of coupled systems with delay. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753 [5] Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the drift-diffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587-597. doi: 10.3934/eect.2017029 [6] Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403-422. doi: 10.3934/eect.2013.2.403 [7] Karl Peter Hadeler, Shigui Ruan. Interaction of diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 95-105. doi: 10.3934/dcdsb.2007.8.95 [8] Monika Joanna Piotrowska, Urszula Foryś, Marek Bodnar, Jan Poleszczuk. A simple model of carcinogenic mutations with time delay and diffusion. Mathematical Biosciences & Engineering, 2013, 10 (3) : 861-872. doi: 10.3934/mbe.2013.10.861 [9] Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21 [10] Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385 [11] Hong-Ming Yin. A free boundary problem arising from a stress-driven diffusion in polymers. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 191-202. doi: 10.3934/dcds.1996.2.191 [12] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [13] Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042 [14] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [15] Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717 [16] Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053 [17] Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092 [18] Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 [19] Peter E. Kloeden, Thomas Lorenz. Pullback attractors of reaction-diffusion inclusions with space-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1909-1964. doi: 10.3934/dcdsb.2017114 [20] Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321

2016 Impact Factor: 0.994