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

Finite-time quenching of competing species with constrained boundary evaporation

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.
Citation: Shu Dai, Dong Li, Kun Zhao. Finite-time quenching of competing species with constrained boundary evaporation. Discrete and 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), 75-83. 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), 301-322. 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), 39-59. 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), 1193-1200. 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), 719-730. 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), 375-396. doi: 10.1007/BF01162244.

[7]

P. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-726.

[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), 655-673. doi: 10.1007/s002110200406.

[9]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, J. Theoretical Biology, 52 (1975), 441-457. 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), 249-251. doi: 10.1016/S0022-5193(76)80056-2.

[11]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. 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), 545-559. 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), 399-415. 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), 1121-1144. 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), 133-154. 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), 1985-1992. 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), 12pp.

[18]

S. Levin, Dispersion and Population Interactions, American Naturalist, 108 (1974), 207-228.

[19]

S. Levin, Some mathematical questions in biology - VII, Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, R. I.

[20]

S. Levin, Studies in mathematical biology. Part II. Populations and communities, MAA Studies in Mathematics, 16 (1978), Mathematical Association of America, Washington, D.C.

[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), 185-192.

[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), 175-190.

[23]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. 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), 193-203. 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), 49-64. 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), 1-16, (In Japanese with English summary).

[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), 657-689. 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), 219-252. 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), 373-383.

[31]

W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578. 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), 1049-1061. 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), 83-99. 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), 281-305. 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), 3933-3941. 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), 826-834. doi: 10.1016/j.jmaa.2008.01.089.

[37]

A. Turing, The Chemical Basis of Morphogenesis, Phil. Transact. Royal Soc. B, 237 (1952), 37-72. 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, 1994), 177-187, World Sci. Publ., River Edge, NJ, (1997).

[39]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis, 21 (1993), 603-630. 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), 75-83. 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), 301-322. 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), 39-59. 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), 1193-1200. 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), 719-730. 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), 375-396. doi: 10.1007/BF01162244.

[7]

P. Fife, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc., 84 (1978), 693-726.

[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), 655-673. doi: 10.1007/s002110200406.

[9]

W. Gurney and R. Nisbet, The regulation of inhomogeneous populations, J. Theoretical Biology, 52 (1975), 441-457. 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), 249-251. doi: 10.1016/S0022-5193(76)80056-2.

[11]

G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. 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), 545-559. 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), 399-415. 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), 1121-1144. 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), 133-154. 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), 1985-1992. 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), 12pp.

[18]

S. Levin, Dispersion and Population Interactions, American Naturalist, 108 (1974), 207-228.

[19]

S. Levin, Some mathematical questions in biology - VII, Lectures on Mathematics in the Life Sciences, 8 (1976), American Mathematical Society, Providence, R. I.

[20]

S. Levin, Studies in mathematical biology. Part II. Populations and communities, MAA Studies in Mathematics, 16 (1978), Mathematical Association of America, Washington, D.C.

[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), 185-192.

[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), 175-190.

[23]

Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. 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), 193-203. 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), 49-64. 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), 1-16, (In Japanese with English summary).

[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), 657-689. 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), 219-252. 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), 373-383.

[31]

W. Ruan, Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197 (1996), 558-578. 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), 1049-1061. 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), 83-99. 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), 281-305. 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), 3933-3941. 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), 826-834. doi: 10.1016/j.jmaa.2008.01.089.

[37]

A. Turing, The Chemical Basis of Morphogenesis, Phil. Transact. Royal Soc. B, 237 (1952), 37-72. 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, 1994), 177-187, World Sci. Publ., River Edge, NJ, (1997).

[39]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis, 21 (1993), 603-630. doi: 10.1016/0362-546X(93)90004-C.

[1]

Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707

[2]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[3]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[4]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083

[5]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure and Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048

[6]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209

[7]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193

[8]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211

[9]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

[10]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

[11]

Zhenghuan Gao, Peihe Wang. Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1201-1223. doi: 10.3934/dcds.2021152

[12]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935

[13]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63

[14]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769

[15]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[16]

Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108

[17]

Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005

[18]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[19]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185

[20]

Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (121)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]