-
Previous Article
Swarming in domains with boundaries: Approximation and regularization by nonlinear diffusion
- DCDS-B Home
- This Issue
-
Next Article
Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center
Spatial propagation for a parabolic system with multiple species competing for single resource
School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China |
A model of $m$ species competing for a single growth-limiting resource is considered. We aim to use the dynamics of such a problem to describe the invasion and spread of $m$ species which are introduced localized in space $\mathbb{R}^N$. The existence, uniqueness and uniform boundedness of the Cauchy problem are investigated by semigroup theory and local $L^p$-estimates. The asymptotic speed of spread is achieved by uniform persistence ideas. The existence of traveling wave is obtained by upper-lower solutions and sliding techniques. Our result shows that the asymptotic speed of spread for $m$ species is characterized by the minimum wave speed of the positive traveling wave solutions associated with this system.
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion,
and nerve pulse propagation, in: J. A. Goldstein (Ed.), Partial Differential Equations and
Related Topics, in: Lecture Notes in Math., Springer-Verlag, 446 (1975), 5-49. |
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal,
Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293.
doi: 10.1142/S0218202504003234. |
[4] |
D. Daners and P. K. Medina,
Abstract Evolution Equations, Periodic Problems and Applications, Longman Scientific & Technical, 1992. |
[5] |
Y. H. Du,
Order Structure and Topological Methods in Nonlinear Partial Differential Equations, vol. 1, Maximum Principles and Applications, World Scientific, 2006.
doi: 10.1142/9789812774446. |
[6] |
A. Ducrot,
Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.
doi: 10.1016/j.jde.2016.02.023. |
[7] |
A. Ducrot and T. Giletti,
Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.
doi: 10.1007/s00285-013-0713-3. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[9] |
S. I. Hollis, R. H. Martin and M. Pierre,
Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.
doi: 10.1137/0518057. |
[10] |
Y. Hosono and B. Ilyas,
Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277-290.
|
[11] |
Y. Hosono and B. Ilyas,
Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
doi: 10.1142/S0218202595000504. |
[12] |
S.-B. Hsu, H. L. Smith and P. Waltman,
Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483.
|
[13] |
S.-B. Hsu and P. Waltman,
On a system of reaction-diffusion equations arising from competition in the unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[14] |
S.-B. Hsu and F.-B. Wang,
On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.
doi: 10.3934/cpaa.2011.10.1479. |
[15] |
W. Z. Huang,
Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765.
doi: 10.1007/s10884-004-6115-x. |
[16] |
W. Z. Huang,
Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59.
doi: 10.1016/j.jmaa.2005.04.084. |
[17] |
W. Z. Huang,
Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor, Discrete Contin. Dyn. Syst., 24 (2009), 883-896.
doi: 10.3934/dcds.2009.24.883. |
[18] |
A. Kallen,
Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal. Theory Methods Appl., 8 (1984), 851-856.
doi: 10.1016/0362-546X(84)90107-X. |
[19] |
C. R. Kennedy and R. Aris,
Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1007/BF02460793. |
[20] |
D. Le,
Global attractors and steady state solutions for a class of reaction-diffusion systems, J. Differential Equations, 147 (1998), 1-29.
doi: 10.1006/jdeq.1998.3435. |
[21] |
D. Le,
Coexistence with Chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.
|
[22] |
D. Le and H. L. Smith,
A parabolic system modeling microbial competition in an unmixed bio-reactor, J. Differential Equations, 130 (1996), 59-91.
doi: 10.1006/jdeq.1996.0132. |
[23] |
P. Magal and X.-Q. Zhao,
Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[24] |
J. D. Murray,
Mathematical Biology, Springer, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[25] |
H. Nie, J. H. Wu and Z. G. Wang,
Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. |
[26] |
H. L. Smith and H. R. Thieme,
Chemostats and epidemics: Competition for nutrients/hosts, Mathematical Biosciences & Engineering, 10 (2013), 1635-1650.
doi: 10.3934/mbe.2013.10.1635. |
[27] |
H. L. Smith and X.-Q. Zhao,
Traveling waves in a bio-reactor model, Nonlinear Anal. Real World Appl., 5 (2004), 895-909.
doi: 10.1016/j.nonrwa.2004.05.001. |
[28] |
Z. C. Wang and J. H. Wu,
Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.
doi: 10.1098/rspa.2009.0377. |
[29] |
Z. C. Wang and J. H. Wu,
Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.
doi: 10.1016/j.jmaa.2011.06.084. |
[30] |
Z. G Wang, H. Nie and J. H. Wu,
Existence and uniqueness of traveling waves for a reaction-diffusion model with general response functions, J. Math. Anal. Appl., 450 (2017), 406-426.
doi: 10.1016/j.jmaa.2017.01.017. |
[31] |
X.-S. Wang, H. Y. Wang and J. H. Wu,
Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. A, 32 (2012), 3303-3324.
doi: 10.3934/dcds.2012.32.3303. |
[32] |
J. H. Wu,
Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[33] |
J. H. Wu,
Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.
doi: 10.1016/S0362-546X(98)00250-8. |
[34] |
J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687; J. H. Wu and X. F. Zou, Erratum to "Traveling wave fronts of reaction-diffusion systems with delays", J. Dynam. Differential Equations, 20 (2008), 531-533.
doi: 10.1007/s10884-007-9090-1. |
[35] |
D. S. Xu and X.-Q. Zhao,
Asymptotic speed of spread and traveling waves for a nonlocal epidemic model, Discrete Contin. Dyn. Syst. B, 5 (2005), 1043-1056.
doi: 10.3934/dcdsb.2005.5.1043. |
[36] |
Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu,
Introduction to Reaction-Diffusion Equations, (in Chinese), Science Press, Beijing, 2011. |
[37] |
E. Zeidler,
Nonlinear Functional Analysis and its Applications, I, Fixed-Point Theorems, Springer-verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[38] |
Z. Q. Zhao,
Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal., 73 (2010), 1481-1490.
doi: 10.1016/j.na.2010.04.008. |
show all references
References:
[1] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion,
and nerve pulse propagation, in: J. A. Goldstein (Ed.), Partial Differential Equations and
Related Topics, in: Lecture Notes in Math., Springer-Verlag, 446 (1975), 5-49. |
[2] |
D. G. Aronson and H. F. Weinberger,
Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
J. M. Arrieta, J. W. Cholewa, T. Dlotko and A. Rodríguez-Bernal,
Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293.
doi: 10.1142/S0218202504003234. |
[4] |
D. Daners and P. K. Medina,
Abstract Evolution Equations, Periodic Problems and Applications, Longman Scientific & Technical, 1992. |
[5] |
Y. H. Du,
Order Structure and Topological Methods in Nonlinear Partial Differential Equations, vol. 1, Maximum Principles and Applications, World Scientific, 2006.
doi: 10.1142/9789812774446. |
[6] |
A. Ducrot,
Spatial propagation for a two component reaction-diffusion system arising in population dynamics, J. Differential Equations, 260 (2016), 8316-8357.
doi: 10.1016/j.jde.2016.02.023. |
[7] |
A. Ducrot and T. Giletti,
Convergence to a pulsating travelling wave for an epidemic reaction-diffusion system with non-diffusive susceptible population, J. Math. Biol., 69 (2014), 533-552.
doi: 10.1007/s00285-013-0713-3. |
[8] |
D. Gilbarg and N. S. Trudinger,
Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. |
[9] |
S. I. Hollis, R. H. Martin and M. Pierre,
Global existence and boundedness in reaction-diffusion systems, SIAM J. Math. Anal., 18 (1987), 744-761.
doi: 10.1137/0518057. |
[10] |
Y. Hosono and B. Ilyas,
Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277-290.
|
[11] |
Y. Hosono and B. Ilyas,
Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
doi: 10.1142/S0218202595000504. |
[12] |
S.-B. Hsu, H. L. Smith and P. Waltman,
Dynamics of competition in the unstirred chemostat, Canad. Appl. Math. Quart., 2 (1994), 461-483.
|
[13] |
S.-B. Hsu and P. Waltman,
On a system of reaction-diffusion equations arising from competition in the unstirred chemostat, SIAM J. Appl. Math., 53 (1993), 1026-1044.
doi: 10.1137/0153051. |
[14] |
S.-B. Hsu and F.-B. Wang,
On a mathematical model arising from competition of phytoplankton species for a single nutrient with internal storage: steady state analysis, Commun. Pure Appl. Anal., 10 (2011), 1479-1501.
doi: 10.3934/cpaa.2011.10.1479. |
[15] |
W. Z. Huang,
Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765.
doi: 10.1007/s10884-004-6115-x. |
[16] |
W. Z. Huang,
Uniqueness of traveling wave solutions for a biological reaction-diffusion equation, J. Math. Anal. Appl., 316 (2006), 42-59.
doi: 10.1016/j.jmaa.2005.04.084. |
[17] |
W. Z. Huang,
Co-existence of traveling waves for a model of microbial growth and competition in a flow reactor, Discrete Contin. Dyn. Syst., 24 (2009), 883-896.
doi: 10.3934/dcds.2009.24.883. |
[18] |
A. Kallen,
Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal. Theory Methods Appl., 8 (1984), 851-856.
doi: 10.1016/0362-546X(84)90107-X. |
[19] |
C. R. Kennedy and R. Aris,
Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429.
doi: 10.1007/BF02460793. |
[20] |
D. Le,
Global attractors and steady state solutions for a class of reaction-diffusion systems, J. Differential Equations, 147 (1998), 1-29.
doi: 10.1006/jdeq.1998.3435. |
[21] |
D. Le,
Coexistence with Chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.
|
[22] |
D. Le and H. L. Smith,
A parabolic system modeling microbial competition in an unmixed bio-reactor, J. Differential Equations, 130 (1996), 59-91.
doi: 10.1006/jdeq.1996.0132. |
[23] |
P. Magal and X.-Q. Zhao,
Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[24] |
J. D. Murray,
Mathematical Biology, Springer, Berlin, 1989.
doi: 10.1007/978-3-662-08539-4. |
[25] |
H. Nie, J. H. Wu and Z. G. Wang,
Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. |
[26] |
H. L. Smith and H. R. Thieme,
Chemostats and epidemics: Competition for nutrients/hosts, Mathematical Biosciences & Engineering, 10 (2013), 1635-1650.
doi: 10.3934/mbe.2013.10.1635. |
[27] |
H. L. Smith and X.-Q. Zhao,
Traveling waves in a bio-reactor model, Nonlinear Anal. Real World Appl., 5 (2004), 895-909.
doi: 10.1016/j.nonrwa.2004.05.001. |
[28] |
Z. C. Wang and J. H. Wu,
Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237-261.
doi: 10.1098/rspa.2009.0377. |
[29] |
Z. C. Wang and J. H. Wu,
Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.
doi: 10.1016/j.jmaa.2011.06.084. |
[30] |
Z. G Wang, H. Nie and J. H. Wu,
Existence and uniqueness of traveling waves for a reaction-diffusion model with general response functions, J. Math. Anal. Appl., 450 (2017), 406-426.
doi: 10.1016/j.jmaa.2017.01.017. |
[31] |
X.-S. Wang, H. Y. Wang and J. H. Wu,
Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Contin. Dyn. Syst. A, 32 (2012), 3303-3324.
doi: 10.3934/dcds.2012.32.3303. |
[32] |
J. H. Wu,
Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[33] |
J. H. Wu,
Global bifurcation of coexistence state for the competition model in the chemostat, Nonlinear Anal., 39 (2000), 817-835.
doi: 10.1016/S0362-546X(98)00250-8. |
[34] |
J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687; J. H. Wu and X. F. Zou, Erratum to "Traveling wave fronts of reaction-diffusion systems with delays", J. Dynam. Differential Equations, 20 (2008), 531-533.
doi: 10.1007/s10884-007-9090-1. |
[35] |
D. S. Xu and X.-Q. Zhao,
Asymptotic speed of spread and traveling waves for a nonlocal epidemic model, Discrete Contin. Dyn. Syst. B, 5 (2005), 1043-1056.
doi: 10.3934/dcdsb.2005.5.1043. |
[36] |
Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu,
Introduction to Reaction-Diffusion Equations, (in Chinese), Science Press, Beijing, 2011. |
[37] |
E. Zeidler,
Nonlinear Functional Analysis and its Applications, I, Fixed-Point Theorems, Springer-verlag, New York, 1986.
doi: 10.1007/978-1-4612-4838-5. |
[38] |
Z. Q. Zhao,
Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal., 73 (2010), 1481-1490.
doi: 10.1016/j.na.2010.04.008. |
[1] |
Zhenguo Bai, Tingting Zhao. Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4063-4085. doi: 10.3934/dcdsb.2018126 |
[2] |
Manjun Ma, Xiao-Qiang Zhao. Monostable waves and spreading speed for a reaction-diffusion model with seasonal succession. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 591-606. doi: 10.3934/dcdsb.2016.21.591 |
[3] |
Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 |
[4] |
Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057 |
[5] |
Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057 |
[6] |
Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441 |
[7] |
Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083 |
[8] |
Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275 |
[9] |
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 |
[10] |
Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067 |
[11] |
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 |
[12] |
Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 |
[13] |
Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213 |
[14] |
Tianran Zhang. Traveling waves for a reaction-diffusion model with a cyclic structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1859-1870. doi: 10.3934/dcdsb.2020006 |
[15] |
Hans F. Weinberger, Kohkichi Kawasaki, Nanako Shigesada. Spreading speeds for a partially cooperative 2-species reaction-diffusion model. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1087-1098. doi: 10.3934/dcds.2009.23.1087 |
[16] |
José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure and Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 |
[17] |
Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks and Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369 |
[18] |
Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709 |
[19] |
Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure and Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319 |
[20] |
Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]