# American Institute of Mathematical Sciences

March  2015, 20(2): 505-518. doi: 10.3934/dcdsb.2015.20.505

## Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system

 1 School of Mathematics, South China Normal University, Guangzhou, Guangdong 510631, China 2 School of Mathematics, South China Normal University, Guangzhou 510631

Received  July 2013 Revised  October 2014 Published  January 2015

This paper is concerned with a three dimensional diffusive Lotka-Volterra system which is combined with cooperative-competitive interactions between the three species. By using the method of super-sub solutions and comparison principle with cross iteration, some results on the asymptotic spreading speed of the system are established under certain assumptions on the parameters appearing in the system.
Citation: Yubin Liu, Peixuan Weng. Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 505-518. doi: 10.3934/dcdsb.2015.20.505
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (ed. J. A. Goldstein), Leture Notes in Mathematics, Springer, Berlin, 446 (1975), 5-49. [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [3] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [4] Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bulletin of Math. Biol., 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008. [5] X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. [6] A. M. Huang, P. X. Weng and Y. H. Huang, Stability of a three-dimensional diffusive Lotka-Volterra system of type-K with delays, Appl. Anal., 92 (2013), 2357-2374. doi: 10.1080/00036811.2012.738360. [7] A. M. Huang and P. X. Weng , Traveling wavefronts for a Lotka-Volterra system of type-$K$ with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129. doi: 10.1016/j.nonrwa.2012.09.002. [8] L. C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl., 12 (2011), 3691-3700. doi: 10.1016/j.nonrwa.2011.07.002. [9] L. C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251. doi: 10.1007/s13160-012-0056-2. [10] S. B. Hsu and X.-Q. Zhao, Speading speed and traveling wave for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [11] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [12] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164. doi: 10.1016/0362-546X(95)00142-I. [13] A. W. Leung , X. J. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924. doi: 10.1016/j.jmaa.2007.05.066. [14] A. W. Leung , X. J. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 171-196. doi: 10.3934/dcdsb.2011.15.171. [15] M. A. Lewis, B. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144. [16] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Commun. Pure Appl. Math., 60 (2007), 1-40. Erratum: 61 (2008), 137-138. doi: 10.1002/cpa.20154. [17] G. Lin and W. T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689. doi: 10.1017/S0956792512000198. [18] G. Lin, Asymptotic spreading fastened by inter-specific coupled nonlinearities: a cooperative system, Phys. D, 241 (2012), 705-710. doi: 10.1016/j.physd.2011.12.007. [19] R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6. [20] R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-312. doi: 10.1016/0025-5564(89)90027-8. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [23] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [24] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [25] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94. [26] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model, J. Diff. Eqns., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [27] Q. R. Wang and K. Zhou, Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity, J. Comput. Appl. Math., 233 (2010), 2549-2562. doi: 10.1016/j.cam.2009.11.002. [28] M. X. Wang, Nonlinear Parabolic Equations, Science Press, Beijing, 1993. [29] H. F. Weinberger, Asymptotic behavior of a model in population genetics, in Nonlinear Partial Equations and applications (ed. J. M. Chadam), Lecture Notes in Math., 648, Springer-Verlag. New York, 1978, 47-96. [30] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [31] H. F. Weinberger, M. A. Lewis and B. T. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [32] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Maths., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [33] P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Diff. Eqns., 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020. [34] Q. X. Ye and Z. Y. Li, Theory of Reaction Diffusion Equations, Science Press, Beijing, 1994.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics (ed. J. A. Goldstein), Leture Notes in Mathematics, Springer, Berlin, 446 (1975), 5-49. [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. [3] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73. doi: 10.1016/0022-0396(79)90080-9. [4] Y. Hosono, The minimal speed of traveling fronts for a diffusive Lotka-Volterra competition model, Bulletin of Math. Biol., 60 (1998), 435-448. doi: 10.1006/bulm.1997.0008. [5] X. J. Hou and A. W. Leung, Traveling wave solutions for a competitive reaction-diffusion system and their asymptotics, Nonlinear Anal. Real World Appl., 9 (2008), 2196-2213. doi: 10.1016/j.nonrwa.2007.07.007. [6] A. M. Huang, P. X. Weng and Y. H. Huang, Stability of a three-dimensional diffusive Lotka-Volterra system of type-K with delays, Appl. Anal., 92 (2013), 2357-2374. doi: 10.1080/00036811.2012.738360. [7] A. M. Huang and P. X. Weng , Traveling wavefronts for a Lotka-Volterra system of type-$K$ with delays, Nonlinear Anal. Real World Appl., 14 (2013), 1114-1129. doi: 10.1016/j.nonrwa.2012.09.002. [8] L. C. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Anal. Real World Appl., 12 (2011), 3691-3700. doi: 10.1016/j.nonrwa.2011.07.002. [9] L. C. Hung, Exact traveling wave solutions for diffusive Lotka-Volterra systems of two competing species, Jpn. J. Ind. Appl. Math., 29 (2012), 237-251. doi: 10.1007/s13160-012-0056-2. [10] S. B. Hsu and X.-Q. Zhao, Speading speed and traveling wave for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016. [11] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363. doi: 10.1137/S0036141093244556. [12] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164. doi: 10.1016/0362-546X(95)00142-I. [13] A. W. Leung , X. J. Hou and Y. Li, Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities, J. Math. Anal. Appl., 338 (2008), 902-924. doi: 10.1016/j.jmaa.2007.05.066. [14] A. W. Leung , X. J. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 171-196. doi: 10.3934/dcdsb.2011.15.171. [15] M. A. Lewis, B. T. Li and H. F. Weinberger, Spreading speed and linear determinacy for two-species competition models, J. Math. Biol., 45 (2002), 219-233. doi: 10.1007/s002850200144. [16] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with application, Commun. Pure Appl. Math., 60 (2007), 1-40. Erratum: 61 (2008), 137-138. doi: 10.1002/cpa.20154. [17] G. Lin and W. T. Li, Asymptotic spreading of competition diffusion systems: The role of interspecific competitions, European J. Appl. Math., 23 (2012), 669-689. doi: 10.1017/S0956792512000198. [18] G. Lin, Asymptotic spreading fastened by inter-specific coupled nonlinearities: a cooperative system, Phys. D, 241 (2012), 705-710. doi: 10.1016/j.physd.2011.12.007. [19] R. Lui, Biological growth and spread modeled by systems of recursions. I. Mathematical theory, Math. Biosci., 93 (1989), 269-295. doi: 10.1016/0025-5564(89)90026-6. [20] R. Lui, Biological growth and spread modeled by systems of recursions. II. Biological theory, Math. Biosci., 93 (1989), 297-312. doi: 10.1016/0025-5564(89)90027-8. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [23] K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615. doi: 10.2307/2000859. [24] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. [25] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equatons and asymptotic speeds for the spread of populatins, J. Reine Angew. Math., 306 (1979), 94-121. doi: 10.1515/crll.1979.306.94. [26] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion model, J. Diff. Eqns., 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X. [27] Q. R. Wang and K. Zhou, Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity, J. Comput. Appl. Math., 233 (2010), 2549-2562. doi: 10.1016/j.cam.2009.11.002. [28] M. X. Wang, Nonlinear Parabolic Equations, Science Press, Beijing, 1993. [29] H. F. Weinberger, Asymptotic behavior of a model in population genetics, in Nonlinear Partial Equations and applications (ed. J. M. Chadam), Lecture Notes in Math., 648, Springer-Verlag. New York, 1978, 47-96. [30] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028. [31] H. F. Weinberger, M. A. Lewis and B. T. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [32] P. X. Weng, H. X. Huang and J. H. Wu, Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction, IMA J. Appl. Maths., 68 (2003), 409-439. doi: 10.1093/imamat/68.4.409. [33] P. X. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Diff. Eqns., 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020. [34] Q. X. Ye and Z. Y. Li, Theory of Reaction Diffusion Equations, Science Press, Beijing, 1994.

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