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

Traveling waves in a three species competition-cooperation system

Xiaojie Hou 1, and  Yi Li 2,

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.

Keywords: Traveling wave, spatio-temporal delay, Lotka Volterra, competition cooperation, existence.
Mathematics Subject Classification: Primary: 35B40, 35B50, 35B51, 35K40, 35K57; Secondary: 35Q92.
Citation: Xiaojie Hou, Yi Li. Traveling waves in a three species competition-cooperation system. Communications on Pure & 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.  Google Scholar

[2]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. Poincare H., Anal. non Lineaire, 9 (1992), 497-572.   Google Scholar

[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.  Google Scholar

[4]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Sattinger, On the stability of traveling waves, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

H. Berestycki and L. Nirenberg, Travelling fronts in cylinders, Ann. Inst. Poincare H., Anal. non Lineaire, 9 (1992), 497-572.   Google Scholar

[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.  Google Scholar

[4]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Sattinger, On the stability of traveling waves, Adv. in Math., 22 (1976), 312-355.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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