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April  2019, 24(4): 1785-1814. doi: 10.3934/dcdsb.2018237

Spatial propagation for a parabolic system with multiple species competing for single resource

Zhiguo Wang , Hua Nie , and  Jianhua Wu

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

* Corresponding author

Received  November 2017 Revised  April 2018 Published  August 2018

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.

Keywords: Competition model, spreading speed, traveling wave, reaction-diffusion system, Cauchy problem.
Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92D25.
Citation: Zhiguo Wang, Hua Nie, Jianhua Wu. Spatial propagation for a parabolic system with multiple species competing for single resource. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1785-1814. doi: 10.3934/dcdsb.2018237
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Longman Scientific & Technical, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. Le, Coexistence with Chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. D. Murray, Mathematical Biology, Springer, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[25]

H. Nie, J. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, Longman Scientific & Technical, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. Le, Coexistence with Chemotaxis, SIAM J. Math. Anal., 32 (2000), 504-521.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. D. Murray, Mathematical Biology, Springer, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[25]

H. Nie, J. H. Wu and Z. G. Wang, Dynamics on the Unstirred Chemostat Models, Science Press, Beijing, 2017. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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