May  2016, 21(3): 815-836. doi: 10.3934/dcdsb.2016.21.815

On existence of wavefront solutions in mixed monotone reaction-diffusion systems

1. 

Department of Mathematics and Statistics, UNC Wilmington, Wilmington, NC 28403

2. 

Department of Mathematics, Computer Science, and Statistics, Purdue University Calumet, Hammond, IN 46323, United States

3. 

Department of Mathematics and Statistics, University of North Carolina in Wilmington, Wilmington, NC 28403

Received  July 2015 Revised  September 2015 Published  January 2016

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.
Citation: Wei Feng, Weihua Ruan, Xin Lu. On existence of wavefront solutions in mixed monotone reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 815-836. doi: 10.3934/dcdsb.2016.21.815
References:
[1]

S. Ai, S.-N. Chow and Y. Yi, Traveling wave solutions in a tissue interaction model for skin pattern formation,, Journal of Dynamics and Differential Equations, 15 (2003), 517.  doi: 10.1023/B:JODY.0000009746.52357.28.  Google Scholar

[2]

J. C. Alexander, R. A. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew Math., 410 (1990), 167.   Google Scholar

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A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

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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.  doi: 10.1016/S1468-1218(02)00077-9.  Google Scholar

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W. Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195.  doi: 10.1080/00036819408840277.  Google Scholar

[6]

W. Feng and X. Lu, Traveling waves and competitive exclusion in models of resource competition and mating interference,, J. Math. Anal. Appl., 424 (2015), 542.  doi: 10.1016/j.jmaa.2014.11.027.  Google Scholar

[7]

W. Feng and W. Ruan, Coexistence, Permanence, and stability in a three species competition model,, Acta. Math. Appl. Sinica (English Ser.), 12 (1996), 443.  doi: 10.1007/BF02029074.  Google Scholar

[8]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations,, Discrete Continuous Dynamical Systems - B, 3 (2003), 79.  doi: 10.3934/dcdsb.2003.3.79.  Google Scholar

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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.  doi: 10.3934/cpaa.2011.10.141.  Google Scholar

[10]

X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model,, Nonlinear Analysis: Real World Applications, 10 (2009), 2207.  doi: 10.1016/j.nonrwa.2008.04.005.  Google Scholar

[11]

X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Continuous Dynamical Systems-A, 26 (2010), 265.  doi: 10.3934/dcds.2010.26.265.  Google Scholar

[12]

J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301.  doi: 10.1016/j.na.2005.05.014.  Google Scholar

[13]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579.  doi: 10.1016/0362-546X(95)00221-G.  Google Scholar

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Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239.  doi: 10.1016/S0362-546X(99)00261-8.  Google Scholar

[15]

Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion,, Nonlinear Analysis: Theory, 28 (1997), 145.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

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A. Kolmogorov, A. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter,, Bjul. Moskovskovo Gov. Iniv., 17 (1937), 1.   Google Scholar

[17]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[18]

A. W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering (Mathematics and Its Applications),, 1989 Edition, (1989).  doi: 10.1007/978-94-015-3937-1.  Google Scholar

[19]

A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited,, Discrete and Continuous Dynamical Systems - Series B, 15 (2011), 171.  doi: 10.3934/dcdsb.2011.15.171.  Google Scholar

[20]

G. Lin, W. Li and M. Ma, Traveling wave solutions in delayed reaction diffusio system with applications to multi-species models,, Discrete and Continuous Dynamical Systems - B, 13 (2010), 393.  doi: 10.3934/dcdsb.2010.13.393.  Google Scholar

[21]

X. Liu and P. Weng, Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system,, Discrete and Continuous Dynamical Systems - B, 20 (2015), 505.  doi: 10.3934/dcdsb.2015.20.505.  Google Scholar

[22]

X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591.  doi: 10.1002/num.1690110605.  Google Scholar

[23]

X. Lu and W. Feng, Dynamics and numerical simulations of food-chain populations,, Applied Mathematics and Computations, 65 (1994), 335.  doi: 10.1016/0096-3003(94)90186-4.  Google Scholar

[24]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, Journal of Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[25]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992).   Google Scholar

[26]

C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations,, SIAM J. Sci. Comput., 25 (2003), 164.  doi: 10.1137/S1064827502409912.  Google Scholar

[27]

D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems,, Advances in Mathematics, 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[28]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Rat. Mech. Anal., 73 (1980), 69.  doi: 10.1007/BF00283257.  Google Scholar

[29]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, Transl. Math. Monograhs, 140 (1994).   Google Scholar

[30]

Z.-C. Wang, W.-T. Li and S. Ruan, Existence and Stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, Journal of Differential Equations, 238 (2007), 153.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[31]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

[32]

D. Xu and X. Q. Zhao, Bistable waves in an epidemic model,, Journal of Dynamics and Differential Equations, 16 (2004), 679.  doi: 10.1007/s10884-004-6113-z.  Google Scholar

show all references

References:
[1]

S. Ai, S.-N. Chow and Y. Yi, Traveling wave solutions in a tissue interaction model for skin pattern formation,, Journal of Dynamics and Differential Equations, 15 (2003), 517.  doi: 10.1023/B:JODY.0000009746.52357.28.  Google Scholar

[2]

J. C. Alexander, R. A. Gardner and C. K. R. T. Jones, A topological invariant arising in the stability analysis of traveling waves,, J. Reine Angew Math., 410 (1990), 167.   Google Scholar

[3]

A. Boumenir and V. Nguyen, Perron theorem in monotone iteration method for traveling waves in delayed reaction-diffusion equations,, Journal of Differential Equations, 244 (2008), 1551.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

[4]

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.  doi: 10.1016/S1468-1218(02)00077-9.  Google Scholar

[5]

W. Feng, Permanence effect in a three-species food chain model,, Applicable Analysis, 54 (1994), 195.  doi: 10.1080/00036819408840277.  Google Scholar

[6]

W. Feng and X. Lu, Traveling waves and competitive exclusion in models of resource competition and mating interference,, J. Math. Anal. Appl., 424 (2015), 542.  doi: 10.1016/j.jmaa.2014.11.027.  Google Scholar

[7]

W. Feng and W. Ruan, Coexistence, Permanence, and stability in a three species competition model,, Acta. Math. Appl. Sinica (English Ser.), 12 (1996), 443.  doi: 10.1007/BF02029074.  Google Scholar

[8]

Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations,, Discrete Continuous Dynamical Systems - B, 3 (2003), 79.  doi: 10.3934/dcdsb.2003.3.79.  Google Scholar

[9]

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.  doi: 10.3934/cpaa.2011.10.141.  Google Scholar

[10]

X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model,, Nonlinear Analysis: Real World Applications, 10 (2009), 2207.  doi: 10.1016/j.nonrwa.2008.04.005.  Google Scholar

[11]

X. Hou, Y. Li and K. R. Meyer, Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities,, Discrete and Continuous Dynamical Systems-A, 26 (2010), 265.  doi: 10.3934/dcds.2010.26.265.  Google Scholar

[12]

J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics,, Nonlinear Analysis: Theory, 65 (2006), 301.  doi: 10.1016/j.na.2005.05.014.  Google Scholar

[13]

J. I. Kanel and L. Zhou, Existence of wave front solutions and estimates of wave speed for a competition-diffusion system,, Nonlinear Analysis: Theory, 27 (1996), 579.  doi: 10.1016/0362-546X(95)00221-G.  Google Scholar

[14]

Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system,, Nonlinear Analysis: Theory, 44 (2001), 239.  doi: 10.1016/S0362-546X(99)00261-8.  Google Scholar

[15]

Y. Kan-on, Fisher wave fronts for the lotka-volterra competition model with diffusion,, Nonlinear Analysis: Theory, 28 (1997), 145.  doi: 10.1016/0362-546X(95)00142-I.  Google Scholar

[16]

A. Kolmogorov, A. Petrovskii and N. Piskunov, A study of the equation of diffusion with increase in the quantity of matter,, Bjul. Moskovskovo Gov. Iniv., 17 (1937), 1.   Google Scholar

[17]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 36 (1998), 389.  doi: 10.1007/s002850050105.  Google Scholar

[18]

A. W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering (Mathematics and Its Applications),, 1989 Edition, (1989).  doi: 10.1007/978-94-015-3937-1.  Google Scholar

[19]

A. W. Leung, X. Hou and W. Feng, Traveling wave solutions for Lotka-Volterra system re-visited,, Discrete and Continuous Dynamical Systems - Series B, 15 (2011), 171.  doi: 10.3934/dcdsb.2011.15.171.  Google Scholar

[20]

G. Lin, W. Li and M. Ma, Traveling wave solutions in delayed reaction diffusio system with applications to multi-species models,, Discrete and Continuous Dynamical Systems - B, 13 (2010), 393.  doi: 10.3934/dcdsb.2010.13.393.  Google Scholar

[21]

X. Liu and P. Weng, Asymptotic spreading of a three dimensional Lotka-Volterra cooperative-competitive system,, Discrete and Continuous Dynamical Systems - B, 20 (2015), 505.  doi: 10.3934/dcdsb.2015.20.505.  Google Scholar

[22]

X. Lu, Monotone method and convergence acceleration for finite-difference solutions of parabolic problems with time delays,, Numer. Meth. Part. Diff. Eqn.s, 11 (1995), 591.  doi: 10.1002/num.1690110605.  Google Scholar

[23]

X. Lu and W. Feng, Dynamics and numerical simulations of food-chain populations,, Applied Mathematics and Computations, 65 (1994), 335.  doi: 10.1016/0096-3003(94)90186-4.  Google Scholar

[24]

S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations,, Journal of Differential Equations, 237 (2007), 259.  doi: 10.1016/j.jde.2007.03.014.  Google Scholar

[25]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations,, Plenum Press, (1992).   Google Scholar

[26]

C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations,, SIAM J. Sci. Comput., 25 (2003), 164.  doi: 10.1137/S1064827502409912.  Google Scholar

[27]

D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems,, Advances in Mathematics, 22 (1976), 312.  doi: 10.1016/0001-8708(76)90098-0.  Google Scholar

[28]

M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion,, Arch. Rat. Mech. Anal., 73 (1980), 69.  doi: 10.1007/BF00283257.  Google Scholar

[29]

A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems,, Transl. Math. Monograhs, 140 (1994).   Google Scholar

[30]

Z.-C. Wang, W.-T. Li and S. Ruan, Existence and Stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay,, Journal of Differential Equations, 238 (2007), 153.  doi: 10.1016/j.jde.2007.03.025.  Google Scholar

[31]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay,, Journal of Dynamics and Differential Equations, 13 (2001), 651.  doi: 10.1023/A:1016690424892.  Google Scholar

[32]

D. Xu and X. Q. Zhao, Bistable waves in an epidemic model,, Journal of Dynamics and Differential Equations, 16 (2004), 679.  doi: 10.1007/s10884-004-6113-z.  Google Scholar

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