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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 |
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. |
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