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:
[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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[37]

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

[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.

[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.

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