# American Institute of Mathematical Sciences

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 and 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), 517-534. doi: 10.1023/B:JODY.0000009746.52357.28. [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), 167-212. [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), 1551-1570. doi: 10.1016/j.jde.2008.01.004. [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), 503-524. doi: 10.1016/S1468-1218(02)00077-9. [5] W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277. [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), 542-562. doi: 10.1016/j.jmaa.2014.11.027. [7] W. Feng and W. Ruan, Coexistence, Permanence, and stability in a three species competition model, Acta. Math. Appl. Sinica (English Ser.), 12 (1996), 443-446. doi: 10.1007/BF02029074. [8] Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations, Discrete Continuous Dynamical Systems - B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79. [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), 141-160. doi: 10.3934/cpaa.2011.10.141. [10] X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model, Nonlinear Analysis: Real World Applications, 10 (2009), 2207-2224. doi: 10.1016/j.nonrwa.2008.04.005. [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), 265-290. doi: 10.3934/dcds.2010.26.265. [12] J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics, Nonlinear Analysis: Theory, Methods & Applications, 65 (2006), 301-320. doi: 10.1016/j.na.2005.05.014. [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, Methods & Applications, 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. [14] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis: Theory, Methods & Applications, 44 (2001), 239-246. doi: 10.1016/S0362-546X(99)00261-8. [15] 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. [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), 1-72. [17] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105. [18] A. W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering (Mathematics and Its Applications), 1989 Edition, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-015-3937-1. [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), 171-196. doi: 10.3934/dcdsb.2011.15.171. [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), 393-414. doi: 10.3934/dcdsb.2010.13.393. [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), 505-518. doi: 10.3934/dcdsb.2015.20.505. [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), 591-602. doi: 10.1002/num.1690110605. [23] X. Lu and W. Feng, Dynamics and numerical simulations of food-chain populations, Applied Mathematics and Computations, 65 (1994), 335-344. doi: 10.1016/0096-3003(94)90186-4. [24] S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, Journal of Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [25] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [26] C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations, SIAM J. Sci. Comput., 25 (2003), 164-185. doi: 10.1137/S1064827502409912. [27] D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems, Advances in Mathematics, 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0. [28] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rat. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. [29] A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI, 1994. [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), 153-200. doi: 10.1016/j.jde.2007.03.025. [31] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533. doi: 10.1023/A:1016690424892. [32] D. Xu and X. Q. Zhao, Bistable waves in an epidemic model, Journal of Dynamics and Differential Equations, 16 (2004), 679-707, and Erratum, Journal of Dynamics and Differential Equations, 17 (2005), 219-247. doi: 10.1007/s10884-004-6113-z.

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), 517-534. doi: 10.1023/B:JODY.0000009746.52357.28. [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), 167-212. [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), 1551-1570. doi: 10.1016/j.jde.2008.01.004. [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), 503-524. doi: 10.1016/S1468-1218(02)00077-9. [5] W. Feng, Permanence effect in a three-species food chain model, Applicable Analysis, 54 (1994), 195-209. doi: 10.1080/00036819408840277. [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), 542-562. doi: 10.1016/j.jmaa.2014.11.027. [7] W. Feng and W. Ruan, Coexistence, Permanence, and stability in a three species competition model, Acta. Math. Appl. Sinica (English Ser.), 12 (1996), 443-446. doi: 10.1007/BF02029074. [8] Y. Hosono, Travelling waves for a diffusive Lotka-Volterra competition model I: Singular Perturbations, Discrete Continuous Dynamical Systems - B, 3 (2003), 79-95. doi: 10.3934/dcdsb.2003.3.79. [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), 141-160. doi: 10.3934/cpaa.2011.10.141. [10] X. Hou, W. Feng and X. Lu, A mathematical analysis of a pubilc goods games model, Nonlinear Analysis: Real World Applications, 10 (2009), 2207-2224. doi: 10.1016/j.nonrwa.2008.04.005. [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), 265-290. doi: 10.3934/dcds.2010.26.265. [12] J. I. Kanel, On the wave front of a competition-diffusion system in popalation dynamics, Nonlinear Analysis: Theory, Methods & Applications, 65 (2006), 301-320. doi: 10.1016/j.na.2005.05.014. [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, Methods & Applications, 27 (1996), 579-587. doi: 10.1016/0362-546X(95)00221-G. [14] Y. Kan-on, Note on propagation speed of travelling waves for a weakly coupled parabolic system, Nonlinear Analysis: Theory, Methods & Applications, 44 (2001), 239-246. doi: 10.1016/S0362-546X(99)00261-8. [15] 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. [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), 1-72. [17] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105. [18] A. W. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering (Mathematics and Its Applications), 1989 Edition, Kluwer Academic Publishers, Dordrecht, 1989. doi: 10.1007/978-94-015-3937-1. [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), 171-196. doi: 10.3934/dcdsb.2011.15.171. [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), 393-414. doi: 10.3934/dcdsb.2010.13.393. [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), 505-518. doi: 10.3934/dcdsb.2015.20.505. [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), 591-602. doi: 10.1002/num.1690110605. [23] X. Lu and W. Feng, Dynamics and numerical simulations of food-chain populations, Applied Mathematics and Computations, 65 (1994), 335-344. doi: 10.1016/0096-3003(94)90186-4. [24] S. W. Ma, Traveling waves for non-local delayed diffusion equations via auxiliary equations, Journal of Differential Equations, 237 (2007), 259-277. doi: 10.1016/j.jde.2007.03.014. [25] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [26] C. V. Pao and X. Lu, Block monotone iterative methods for numerical solutions of nonlinear parabolic equations, SIAM J. Sci. Comput., 25 (2003), 164-185. doi: 10.1137/S1064827502409912. [27] D. Sattinger, On the stability of traveling waves of nonlinear parabolic systems, Advances in Mathematics, 22 (1976), 312-355. doi: 10.1016/0001-8708(76)90098-0. [28] M. M. Tang and P. C. Fife, Propagating fronts for competing species equations with diffusion, Arch. Rat. Mech. Anal., 73 (1980), 69-77. doi: 10.1007/BF00283257. [29] A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monograhs, 140, Amer. Math. Soc., Providence, RI, 1994. [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), 153-200. doi: 10.1016/j.jde.2007.03.025. [31] J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, Journal of Dynamics and Differential Equations, 13 (2001), 651-687, and Erratum to traveling wave fronts of reaction-diffusion systems with delays, Journal of Dynamics and Differential Equations, 20 (2008), 531-533. doi: 10.1023/A:1016690424892. [32] D. Xu and X. Q. Zhao, Bistable waves in an epidemic model, Journal of Dynamics and Differential Equations, 16 (2004), 679-707, and Erratum, Journal of Dynamics and Differential Equations, 17 (2005), 219-247. doi: 10.1007/s10884-004-6113-z.
 [1] Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515 [2] Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure and Applied Analysis, 2021, 20 (2) : 623-650. doi: 10.3934/cpaa.2020283 [3] Rui Li, Yuan Lou. Some monotone properties for solutions to a reaction-diffusion model. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4445-4455. doi: 10.3934/dcdsb.2019126 [4] Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407 [5] A. Dall'Acqua. Positive solutions for a class of reaction-diffusion systems. Communications on Pure and Applied Analysis, 2003, 2 (1) : 65-76. doi: 10.3934/cpaa.2003.2.65 [6] Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 [7] Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Regular solutions and global attractors for reaction-diffusion systems without uniqueness. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1891-1906. doi: 10.3934/cpaa.2014.13.1891 [8] Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405 [9] Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083 [10] Zhoude Shao. Existence and continuity of strong solutions of partly dissipative reaction diffusion systems. Conference Publications, 2011, 2011 (Special) : 1319-1328. doi: 10.3934/proc.2011.2011.1319 [11] Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249 [12] Alexey Cheskidov, Songsong Lu. The existence and the structure of uniform global attractors for nonautonomous Reaction-Diffusion systems without uniqueness. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 55-66. doi: 10.3934/dcdss.2009.2.55 [13] Laurent Desvillettes, Klemens Fellner. Entropy methods for reaction-diffusion systems. Conference Publications, 2007, 2007 (Special) : 304-312. doi: 10.3934/proc.2007.2007.304 [14] Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 [15] Aníbal Rodríguez-Bernal, Alejandro Vidal-López. A note on the existence of global solutions for reaction-diffusion equations with almost-monotonic nonlinearities. Communications on Pure and Applied Analysis, 2014, 13 (2) : 635-644. doi: 10.3934/cpaa.2014.13.635 [16] Hideo Deguchi. A reaction-diffusion system arising in game theory: existence of solutions and spatial dominance. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3891-3901. doi: 10.3934/dcdsb.2017200 [17] Joaquin Riviera, Yi Li. Existence of traveling wave solutions for a nonlocal reaction-diffusion model of influenza a drift. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 157-174. doi: 10.3934/dcdsb.2010.13.157 [18] Klemens Fellner, Evangelos Latos, Takashi Suzuki. Global classical solutions for mass-conserving, (super)-quadratic reaction-diffusion systems in three and higher space dimensions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3441-3462. doi: 10.3934/dcdsb.2016106 [19] Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242 [20] Shi-Liang Wu, Yu-Juan Sun, San-Yang Liu. Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 921-946. doi: 10.3934/dcds.2013.33.921

2020 Impact Factor: 1.327