American Institute of Mathematical Sciences

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July  2013, 18(5): 1275-1290. doi: 10.3934/dcdsb.2013.18.1275

Finite-time quenching of competing species with constrained boundary evaporation

Shu Dai 1, , Dong Li 2, and  Kun Zhao 3,

1. 

CGG, Houston, TX 77072, United States
2. 

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
3. 

Department of Mathematics, Tulane University, New Orleans, LA 70118, United States

Received  July 2012 Revised  February 2013 Published  March 2013

We consider a class of SKT type reaction-cross diffusion models with vanishing random diffusion coefficients. For homogeneous Dirichlet boundary conditions we prove non-existence of global-in-time non-trivial non-negative smooth solutions. Some numerical results are also presented, suggesting the possibility of finite-time extinction.
Keywords: Reaction-cross diffusion equations, non-existence, Dirichlet boundary condition, global smooth solution..
Mathematics Subject Classification: Primary: 35K51, 35B60; Secondary: 92D2.
Citation: Shu Dai, Dong Li, Kun Zhao. Finite-time quenching of competing species with constrained boundary evaporation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1275-1290. doi: 10.3934/dcdsb.2013.18.1275
References:
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Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 1193. doi: 10.3934/dcds.2003.9.1193. Google Scholar

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Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 719. doi: 10.3934/dcds.2004.10.719. Google Scholar

[6]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system,, Math. Z., 194 (1987), 375. doi: 10.1007/BF01162244. Google Scholar

[7]

P. Fife, Asymptotic states for equations of reaction and diffusion,, Bull. Amer. Math. Soc., 84 (1978), 693. Google Scholar

[8]

G. Galiano, M. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model,, Numer. Math., 93 (2003), 655. doi: 10.1007/s002110200406. Google Scholar

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W. Gurney and R. Nisbet, The regulation of inhomogeneous populations,, J. Theoretical Biology, 52 (1975), 441. doi: 10.1016/0022-5193(75)90011-9. Google Scholar

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W. Gurney and R. Nisbet, A note on non-linear population transport,, J. Theoretical Biology, 56 (1976), 249. doi: 10.1016/S0022-5193(76)80056-2. Google Scholar

[11]

G. Hardin, The competitive exclusion principle,, Science, 131 (1960), 1292. doi: 10.1126/science.131.3409.1292. Google Scholar

[12]

J. Jackson and L. Segel, Dissipative structure: An explanation and an ecological example,, J. Theoretical Biology, 37 (1972), 545. doi: 10.1016/0022-5193(72)90090-2. Google Scholar

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E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoretical Biology, 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[14]

J. Kim, Smooth solutions to a quasilinear system of diffusion equations for a certain population model,, Nonlinear Analysis, 8 (1984), 1121. doi: 10.1016/0362-546X(84)90115-9. Google Scholar

[15]

D. Le, Global existence for a class of strongly coupled parabolic systems,, Ann. Mat. Pura Appl., 185 (2006), 133. doi: 10.1007/s10231-004-0131-7. Google Scholar

[16]

D. Le and T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. Amer. Math. Soc., 133 (2005), 1985. doi: 10.1090/S0002-9939-05-07867-6. Google Scholar

[17]

D. Le, L. Nguyen and T. Nguyen, Shigesada-Kawasaki-Teramoto model on higher dimensional domains,, Electron. J. Differential Equations, (2003). Google Scholar

[18]

S. Levin, Dispersion and Population Interactions,, American Naturalist, 108 (1974), 207. Google Scholar

[19]

S. Levin, Some mathematical questions in biology - VII,, Lectures on Mathematics in the Life Sciences, 8 (1976). Google Scholar

[20]

S. Levin, Studies in mathematical biology. Part II. Populations and communities,, MAA Studies in Mathematics, 16 (1978). Google Scholar

[21]

Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains,, Discrete Contin. Dyn. Syst., 12 (2005), 185. Google Scholar

[22]

Y. Lou, S. Martinez and W. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 6 (2000), 175. Google Scholar

[23]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79. doi: 10.1006/jdeq.1996.0157. Google Scholar

[24]

Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193. doi: 10.3934/dcds.1998.4.193. Google Scholar

[25]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035. Google Scholar

[26]

M. Morisita, Habitat preference and evaluation of environment of an animal. Experimental studies on the population density of an ant-lion, Glenuroides japonicus M'L. (I),, Physiol. Ecol. Japan, 5 (1952), 1. Google Scholar

[27]

A. Okubo, "Ecology and Diffusion,", Tokyo: Tsukiji Shokan, (1975). Google Scholar

[28]

M. Pozio and A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system,, Nonlinear Anal., 14 (1990), 657. doi: 10.1016/0362-546X(90)90043-G. Google Scholar

[29]

R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics,, J. Differential Equations, 118 (1995), 219. doi: 10.1006/jdeq.1995.1073. Google Scholar

[30]

G. Rosen, Effects of diffusion on the stability of the equilibrium in multi-species ecological systems,, Bull. Math. Biol., 39 (1977), 373. Google Scholar

[31]

W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558. doi: 10.1006/jmaa.1996.0039. Google Scholar

[32]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049. doi: 10.3934/dcds.2003.9.1049. Google Scholar

[33]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theo. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[34]

S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems,, J. Differential Equations, 185 (2002), 281. doi: 10.1006/jdeq.2002.4169. Google Scholar

[35]

P. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933. doi: 10.1090/S0002-9939-07-08978-2. Google Scholar

[36]

P. Tuoc, On global existence of solutions to a cross-diffusion system,, J. Math. Anal. Appl., 343 (2008), 826. doi: 10.1016/j.jmaa.2008.01.089. Google Scholar

[37]

A. Turing, The Chemical Basis of Morphogenesis,, Phil. Transact. Royal Soc. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar

[38]

Y. Wu, Qualitative studies of solutions for some cross-diffusion systems,, China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, (1997), 177. Google Scholar

[39]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics,, Nonlinear Analysis, 21 (1993), 603. doi: 10.1016/0362-546X(93)90004-C. Google Scholar

show all references

References:
[1]

D. Blatt and H. Comins, Prey-predator models in spatially heterogeneous environments,, J. Theoretical Biology, 48 (1974), 75. doi: 10.1016/0022-5193(74)90180-5. Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion,, SIAM J. Math. Anal., 36 (2004), 301. doi: 10.1137/S0036141003427798. Google Scholar

[3]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion,, J. Differential Equations, 224 (2006), 39. doi: 10.1016/j.jde.2005.08.002. Google Scholar

[4]

Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,, Discrete Contin. Dyn. Syst., 9 (2003), 1193. doi: 10.3934/dcds.2003.9.1193. Google Scholar

[5]

Y. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,, Discrete Contin. Dyn. Syst., 10 (2004), 719. doi: 10.3934/dcds.2004.10.719. Google Scholar

[6]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system,, Math. Z., 194 (1987), 375. doi: 10.1007/BF01162244. Google Scholar

[7]

P. Fife, Asymptotic states for equations of reaction and diffusion,, Bull. Amer. Math. Soc., 84 (1978), 693. Google Scholar

[8]

G. Galiano, M. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model,, Numer. Math., 93 (2003), 655. doi: 10.1007/s002110200406. Google Scholar

[9]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations,, J. Theoretical Biology, 52 (1975), 441. doi: 10.1016/0022-5193(75)90011-9. Google Scholar

[10]

W. Gurney and R. Nisbet, A note on non-linear population transport,, J. Theoretical Biology, 56 (1976), 249. doi: 10.1016/S0022-5193(76)80056-2. Google Scholar

[11]

G. Hardin, The competitive exclusion principle,, Science, 131 (1960), 1292. doi: 10.1126/science.131.3409.1292. Google Scholar

[12]

J. Jackson and L. Segel, Dissipative structure: An explanation and an ecological example,, J. Theoretical Biology, 37 (1972), 545. doi: 10.1016/0022-5193(72)90090-2. Google Scholar

[13]

E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoretical Biology, 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[14]

J. Kim, Smooth solutions to a quasilinear system of diffusion equations for a certain population model,, Nonlinear Analysis, 8 (1984), 1121. doi: 10.1016/0362-546X(84)90115-9. Google Scholar

[15]

D. Le, Global existence for a class of strongly coupled parabolic systems,, Ann. Mat. Pura Appl., 185 (2006), 133. doi: 10.1007/s10231-004-0131-7. Google Scholar

[16]

D. Le and T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. Amer. Math. Soc., 133 (2005), 1985. doi: 10.1090/S0002-9939-05-07867-6. Google Scholar

[17]

D. Le, L. Nguyen and T. Nguyen, Shigesada-Kawasaki-Teramoto model on higher dimensional domains,, Electron. J. Differential Equations, (2003). Google Scholar

[18]

S. Levin, Dispersion and Population Interactions,, American Naturalist, 108 (1974), 207. Google Scholar

[19]

S. Levin, Some mathematical questions in biology - VII,, Lectures on Mathematics in the Life Sciences, 8 (1976). Google Scholar

[20]

S. Levin, Studies in mathematical biology. Part II. Populations and communities,, MAA Studies in Mathematics, 16 (1978). Google Scholar

[21]

Y. Li and C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains,, Discrete Contin. Dyn. Syst., 12 (2005), 185. Google Scholar

[22]

Y. Lou, S. Martinez and W. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion,, Discrete Contin. Dyn. Syst., 6 (2000), 175. Google Scholar

[23]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79. doi: 10.1006/jdeq.1996.0157. Google Scholar

[24]

Y. Lou, W. Ni and Y. Wu, On the global existence of a cross-diffusion system,, Discrete Contin. Dyn. Syst., 4 (1998), 193. doi: 10.3934/dcds.1998.4.193. Google Scholar

[25]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations,, J. Math. Biol., 9 (1980), 49. doi: 10.1007/BF00276035. Google Scholar

[26]

M. Morisita, Habitat preference and evaluation of environment of an animal. Experimental studies on the population density of an ant-lion, Glenuroides japonicus M'L. (I),, Physiol. Ecol. Japan, 5 (1952), 1. Google Scholar

[27]

A. Okubo, "Ecology and Diffusion,", Tokyo: Tsukiji Shokan, (1975). Google Scholar

[28]

M. Pozio and A. Tesei, Global existence of solutions for a strongly coupled quasilinear parabolic system,, Nonlinear Anal., 14 (1990), 657. doi: 10.1016/0362-546X(90)90043-G. Google Scholar

[29]

R. Redlinger, Existence of the global attractor for a strongly coupled parabolic system arising in population dynamics,, J. Differential Equations, 118 (1995), 219. doi: 10.1006/jdeq.1995.1073. Google Scholar

[30]

G. Rosen, Effects of diffusion on the stability of the equilibrium in multi-species ecological systems,, Bull. Math. Biol., 39 (1977), 373. Google Scholar

[31]

W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients,, J. Math. Anal. Appl., 197 (1996), 558. doi: 10.1006/jmaa.1996.0039. Google Scholar

[32]

K. Ryu and I. Ahn, Positive steady-states for two interacting species models with linear self-cross diffusions,, Discrete Contin. Dyn. Syst., 9 (2003), 1049. doi: 10.3934/dcds.2003.9.1049. Google Scholar

[33]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theo. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[34]

S. Shim, Uniform boundedness and convergence of solutions to cross-diffusion systems,, J. Differential Equations, 185 (2002), 281. doi: 10.1006/jdeq.2002.4169. Google Scholar

[35]

P. Tuoc, Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions,, Proc. Amer. Math. Soc., 135 (2007), 3933. doi: 10.1090/S0002-9939-07-08978-2. Google Scholar

[36]

P. Tuoc, On global existence of solutions to a cross-diffusion system,, J. Math. Anal. Appl., 343 (2008), 826. doi: 10.1016/j.jmaa.2008.01.089. Google Scholar

[37]

A. Turing, The Chemical Basis of Morphogenesis,, Phil. Transact. Royal Soc. B, 237 (1952), 37. doi: 10.1098/rstb.1952.0012. Google Scholar

[38]

Y. Wu, Qualitative studies of solutions for some cross-diffusion systems,, China-Japan Symposium on Reaction-Diffusion Equations and their Applications and Computational Aspects (Shanghai, (1997), 177. Google Scholar

[39]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics,, Nonlinear Analysis, 21 (1993), 603. doi: 10.1016/0362-546X(93)90004-C. Google Scholar

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