# American Institute of Mathematical Sciences

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.

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##### 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.
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