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doi: 10.3934/dcdsb.2021295
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## Propagation of stochastic travelling waves of cooperative systems with noise

 1 College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China 2 School of Civil and Hydraulic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

* Corresponding author: Jianhua Huang

Received  August 2021 Revised  October 2021 Early access December 2021

Fund Project: Supported by NSF of China (No.11771449, 12031020, 61841302) and NSF of Hunan Province, China (2020JJ4102)

We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.

Citation: Hao Wen, Jianhua Huang, Yuhong Li. Propagation of stochastic travelling waves of cooperative systems with noise. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021295
##### References:
 [1] X. Bao, W. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 18 (2019), 361-396.  doi: 10.3934/cpaa.2019019. [2] N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, San Diego: Academic Press, 1986. [3] F. Cao and L. Gao, Transition fronts of KPP-type lattice random equations, Electron. J. Differential Equations, 2019 (2019), Paper No. 129, 20 pp. [4] E. C. M. Crooks, On the Vol'pert theory of travelling-wave solutions for parabolic systems, Nonlinear Anal.-Theory Methods Appl., 26 (1996), 1621-1642.  doi: 10.1016/0362-546X(95)00038-W. [5] D. A. Dawson, I. Iscoe and E. A. Perkins, Super-Brownian motion: Path properties and hitting probabilities, Probab. Theory Related Fields, 83 (1989), 135-205.  doi: 10.1007/BF00333147. [6] K. D. Elworthy and H. Z. Zhao, The propagation of travelling waves for stochastic generalized KPP equations, Math. Comput. Modelling, 20 (1994), 131-166.  doi: 10.1016/0895-7177(94)90162-7. [7] R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x. [8] X. Hou and Y. Li, Traveling waves in a three species competition-cooperation system, Commun. Pure Appl. Anal., 16 (2017), 1103-1119.  doi: 10.3934/cpaa.2017053. [9] Z. Huang and Z. Liu, Stochastic traveling wave solution to stochastic generalized KPP equation, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 143-173.  doi: 10.1007/s00030-014-0279-9. [10] S. Kliem, Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support, Stochastic Process. Appl., 127 (2017), 385-418.  doi: 10.1016/j.spa.2016.06.012. [11] A. Kolmogorov, I. Petrovsky and N. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105–130. doi: 10.1016/B978-0-08-092523-3.50014-9. [12] P. Kotelenez, Comparison methods for a class of function valued stochastic partial differential equations, Probab. Theor. Relat. Fields, 93 (1992), 1-19.  doi: 10.1007/BF01195385. [13] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776.  doi: 10.1088/0951-7715/24/6/004. [14] C. Müeller, L. Mytnik and J. Quastel, Effect of noise on front propagation in reaction-diffusion equations of KPP type, Invent. Math., 184 (2011), 405-453.  doi: 10.1007/s00222-010-0292-5. [15] C. Müeller, L. Mytnik and L. Ryzhik, The speed of a random front for stochastic reaction-diffusion equations with strong noise, Commun. Math. Phys., 384 (2021), 699-732.  doi: 10.1007/s00220-021-04084-0. [16] C. Müeller and R. B. Sowers, Random travelling waves for the KPP equation with noise, J. Funct. Anal., 128 (1995), 439-498.  doi: 10.1006/jfan.1995.1038. [17] C. Müeller and R. Tribe, A phase transition for a stochastic PDE related to the contact process, Probab. Theory Related Fields, 100 (1994), 131-156.  doi: 10.1007/BF01199262. [18] J. D. Murray, Mathematical Biology, Springer-Verlag, 1993. doi: 10.1007/b98869. [19] B. Øksendal, G. Våge and H. Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1363-1381.  doi: 10.1017/S030821050000072X. [20] B. Øksendal, G. Våge and H. Z. Zhao, Two properties of stochastic KPP equations: Ergodicity and pathwise property, Nonlinearity, 14 (2001), 639-662.  doi: 10.1088/0951-7715/14/3/311. [21] L. A. Segel, Mathematical Models in Molecular and Cellular Biology, Cambridge University Press, 1980. [22] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canad. J. Math., 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8. [23] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [24] R. Tribe, A travelling wave solution to the Kolmogorov equation with noise, Stochastics Stochastics Rep., 56 (1996), 317-340.  doi: 10.1080/17442509608834047.

show all references

##### References:
 [1] X. Bao, W. Shen and Z. Shen, Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems, Commun. Pure Appl. Anal., 18 (2019), 361-396.  doi: 10.3934/cpaa.2019019. [2] N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, San Diego: Academic Press, 1986. [3] F. Cao and L. Gao, Transition fronts of KPP-type lattice random equations, Electron. J. Differential Equations, 2019 (2019), Paper No. 129, 20 pp. [4] E. C. M. Crooks, On the Vol'pert theory of travelling-wave solutions for parabolic systems, Nonlinear Anal.-Theory Methods Appl., 26 (1996), 1621-1642.  doi: 10.1016/0362-546X(95)00038-W. [5] D. A. Dawson, I. Iscoe and E. A. Perkins, Super-Brownian motion: Path properties and hitting probabilities, Probab. Theory Related Fields, 83 (1989), 135-205.  doi: 10.1007/BF00333147. [6] K. D. Elworthy and H. Z. Zhao, The propagation of travelling waves for stochastic generalized KPP equations, Math. Comput. Modelling, 20 (1994), 131-166.  doi: 10.1016/0895-7177(94)90162-7. [7] R. A. Fisher, The wave of advance of advantageous genes, Ann. Hum. Genet., 7 (1937), 355-369.  doi: 10.1111/j.1469-1809.1937.tb02153.x. [8] X. Hou and Y. Li, Traveling waves in a three species competition-cooperation system, Commun. Pure Appl. Anal., 16 (2017), 1103-1119.  doi: 10.3934/cpaa.2017053. [9] Z. Huang and Z. Liu, Stochastic traveling wave solution to stochastic generalized KPP equation, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 143-173.  doi: 10.1007/s00030-014-0279-9. [10] S. Kliem, Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support, Stochastic Process. Appl., 127 (2017), 385-418.  doi: 10.1016/j.spa.2016.06.012. [11] A. Kolmogorov, I. Petrovsky and N. Piskunov, Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem, Dynamics of Curved Fronts, (1988), 105–130. doi: 10.1016/B978-0-08-092523-3.50014-9. [12] P. Kotelenez, Comparison methods for a class of function valued stochastic partial differential equations, Probab. Theor. Relat. Fields, 93 (1992), 1-19.  doi: 10.1007/BF01195385. [13] B. Li and L. Zhang, Travelling wave solutions in delayed cooperative systems, Nonlinearity, 24 (2011), 1759-1776.  doi: 10.1088/0951-7715/24/6/004. [14] C. Müeller, L. Mytnik and J. Quastel, Effect of noise on front propagation in reaction-diffusion equations of KPP type, Invent. Math., 184 (2011), 405-453.  doi: 10.1007/s00222-010-0292-5. [15] C. Müeller, L. Mytnik and L. Ryzhik, The speed of a random front for stochastic reaction-diffusion equations with strong noise, Commun. Math. Phys., 384 (2021), 699-732.  doi: 10.1007/s00220-021-04084-0. [16] C. Müeller and R. B. Sowers, Random travelling waves for the KPP equation with noise, J. Funct. Anal., 128 (1995), 439-498.  doi: 10.1006/jfan.1995.1038. [17] C. Müeller and R. Tribe, A phase transition for a stochastic PDE related to the contact process, Probab. Theory Related Fields, 100 (1994), 131-156.  doi: 10.1007/BF01199262. [18] J. D. Murray, Mathematical Biology, Springer-Verlag, 1993. doi: 10.1007/b98869. [19] B. Øksendal, G. Våge and H. Z. Zhao, Asymptotic properties of the solutions to stochastic KPP equations, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 1363-1381.  doi: 10.1017/S030821050000072X. [20] B. Øksendal, G. Våge and H. Z. Zhao, Two properties of stochastic KPP equations: Ergodicity and pathwise property, Nonlinearity, 14 (2001), 639-662.  doi: 10.1088/0951-7715/14/3/311. [21] L. A. Segel, Mathematical Models in Molecular and Cellular Biology, Cambridge University Press, 1980. [22] T. Shiga, Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Canad. J. Math., 46 (1994), 415-437.  doi: 10.4153/CJM-1994-022-8. [23] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [24] R. Tribe, A travelling wave solution to the Kolmogorov equation with noise, Stochastics Stochastics Rep., 56 (1996), 317-340.  doi: 10.1080/17442509608834047.
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