# American Institute of Mathematical Sciences

July  2017, 16(4): 1103-1120. doi: 10.3934/cpaa.2017053

## Traveling waves in a three species competition-cooperation system

 1 Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, NC 28403, USA 2 Department of Mathematics, California State University Northridge, CA 91330, USA

Received  March 2014 Revised  February 2017 Published  April 2017

This paper studies the traveling wave solutions to a three species competition cooperation system, which is derived from a spatially averaged and temporally delayed Lotka Volterra system. The existence of the traveling waves is investigated via a new type of monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain two species Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions.

Citation: Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1103-1120. doi: 10.3934/cpaa.2017053
##### References:
 [1] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Zeitschrift fur Angewandte Mathematik und Physik, 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [2] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. Poincare H., Anal. non Lineaire, 9 (1992), 497-572. [3] A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Diff. Eqs., 244 (2008), 1551-1570.  doi: 10.1016/j.jde.2008.01.004. [4] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. [5] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system, Nonlinear Analysis: Real World Applications, 4 (2003), 503-524.  doi: 10.1016/S1468-1218(02)00077-9. [6] X. Hou and W. Feng, Traveling waves and their stability in a coupled reaction diffusion system, Communications on Pure and Applied Analysis, 10 (2011), 141-160.  doi: 10.3934/cpaa.2011.10.141. [7] Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model Ⅰ: Singular Perturbations, Discrete and Continuous Dynamical Systems-Series B, 3 (2003), 79-95.  doi: 10.3934/dcdsb.2003.3.79. [8] W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0. [9] L. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Analysis: Real World Applications, 12 (2011), 3691-3700.  doi: 10.1016/j.nonrwa.2011.07.002. [10] H. Ikeda, Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion, Hiroshima Math. J., 32 (2002), 87-124. [11] J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics, Nonlinear Analysis, 65 (2006), 301-320.  doi: 10.1016/j.na.2005.05.014. [12] J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Analysis, Theory, Methods & Applications, 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G. [13] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis, 44 (2001), 239-246.  doi: 10.1016/S0362-546X(99)00261-8. [14] Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion, Nonlinear Analysis, Theory, methods & Applications, 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [15] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system revisited, Discrete and Continuous Dynamical Systems -Series B, 15 (2011), 171-196.  doi: 10.3934/dcdsb.2011.15.171. [16] B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowestwave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008. [17] P. Miller, Stability of non-monotone waves in a three-species reaction-diffusion model, Proceedings of the Royal Society of Edinburgh Section A Mathematics, Cambridge University Press, 129 (1999), 125-152.  doi: 10.1017/S0308210500027499. [18] D. Sattinger, On the stability of traveling waves, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0. [19] I. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. , vol 140, Amer. Math. Soc. , Providence, RI. 1994. [20] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynamics and Diff. Eq., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.

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##### References:
 [1] P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Travelling fronts for the KPP equation with spatio-temporal delay, Zeitschrift fur Angewandte Mathematik und Physik, 53 (2002), 103-122.  doi: 10.1007/s00033-002-8145-8. [2] H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. Poincare H., Anal. non Lineaire, 9 (1992), 497-572. [3] A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Diff. Eqs., 244 (2008), 1551-1570.  doi: 10.1016/j.jde.2008.01.004. [4] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. [5] N. Fei and J. Carr, Existence of travelling waves with their minimal speed for a diffusing Lotka-Volterra system, Nonlinear Analysis: Real World Applications, 4 (2003), 503-524.  doi: 10.1016/S1468-1218(02)00077-9. [6] X. Hou and W. Feng, Traveling waves and their stability in a coupled reaction diffusion system, Communications on Pure and Applied Analysis, 10 (2011), 141-160.  doi: 10.3934/cpaa.2011.10.141. [7] Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model Ⅰ: Singular Perturbations, Discrete and Continuous Dynamical Systems-Series B, 3 (2003), 79-95.  doi: 10.3934/dcdsb.2003.3.79. [8] W. Huang, Problem on minimum wave speed for a Lotka-Volterra reaction-diffusion competition model, J. Dynam. Differential Equations, 22 (2010), 285-297.  doi: 10.1007/s10884-010-9159-0. [9] L. Hung, Traveling wave solutions of competitive-cooperative Lotka-Volterra systems of three species, Nonlinear Analysis: Real World Applications, 12 (2011), 3691-3700.  doi: 10.1016/j.nonrwa.2011.07.002. [10] H. Ikeda, Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion, Hiroshima Math. J., 32 (2002), 87-124. [11] J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics, Nonlinear Analysis, 65 (2006), 301-320.  doi: 10.1016/j.na.2005.05.014. [12] J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system, Nonlinear Analysis, Theory, Methods & Applications, 27 (1996), 579-587.  doi: 10.1016/0362-546X(95)00221-G. [13] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis, 44 (2001), 239-246.  doi: 10.1016/S0362-546X(99)00261-8. [14] Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion, Nonlinear Analysis, Theory, methods & Applications, 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [15] A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system revisited, Discrete and Continuous Dynamical Systems -Series B, 15 (2011), 171-196.  doi: 10.3934/dcdsb.2011.15.171. [16] B. Li, H. Weinberger and M. Lewis, Spreading speeds as slowestwave speeds for cooperative systems, Math. Biosci., 196 (2005), 82-98.  doi: 10.1016/j.mbs.2005.03.008. [17] P. Miller, Stability of non-monotone waves in a three-species reaction-diffusion model, Proceedings of the Royal Society of Edinburgh Section A Mathematics, Cambridge University Press, 129 (1999), 125-152.  doi: 10.1017/S0308210500027499. [18] D. Sattinger, On the stability of traveling waves, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0. [19] I. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. , vol 140, Amer. Math. Soc. , Providence, RI. 1994. [20] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynamics and Diff. Eq., 13 (2001), 651-687.  doi: 10.1023/A:1016690424892.
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