# American Institute of Mathematical Sciences

December  2011, 15(3): 669-686. doi: 10.3934/dcdsb.2011.15.669

## On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model

 1 Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, United States 2 Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241 3 Institute of Mathematics, Hangzhou Dianzi Universitye, Xiasha Hangzhou Zhejiang 310018

Received  January 2010 Revised  March 2010 Published  February 2011

Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we investigate in a shadow system the effects of migration and interspecific competition coefficients on the existence of positive solutions. Our study shows that for any given migration, if the interspecific competition coefficient of the invader is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of a two-species Lotka-Volterra competition-diffusion model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.
Citation: Fang Li, Liping Wang, Yang Wang. On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 669-686. doi: 10.3934/dcdsb.2011.15.669
##### References:
 [1] J. M. Arrieta, A. N. Carvalho and A. Rodriguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Diff. Eqs., 168 (2000), 33-59.  Google Scholar [2] R. S. Cantrell and C. Cosner, The effect of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.  Google Scholar [3] R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252. doi: 10.1137/0153014.  Google Scholar [4] R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145. doi: 10.1007/s002850050122.  Google Scholar [5] R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.  Google Scholar [6] R. S. Cantrell, C. Cosner and V. Huston, Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559.  Google Scholar [7] R. S. Cantrell, C. Cosner and V. Huston, Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101.  Google Scholar [8] R. S. Cantrell, C. Cosner and Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynam. Differential Equations, 16 (2004), 973-1010. doi: 10.1007/s10884-004-7831-y.  Google Scholar [9] A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q.  Google Scholar [10] Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion, Disc. Cont. Dyn. Syst. A, 9 (2003), 1193-1200. doi: 10.3934/dcds.2003.9.1193.  Google Scholar [11] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.  Google Scholar [12] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion equations, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.  Google Scholar [13] Y. Du, Effects of a degeneracy in the competition model, Part I. Classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.  Google Scholar [14] Y. Du, Effects of a degeneracy in the competition model, Part II. Perturbation and dynamical behavior, J. Differential Equations, 181 (2002), 133-164. doi: 10.1006/jdeq.2001.4075.  Google Scholar [15] Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations, 193 (2003), 147-179. doi: 10.1016/S0022-0396(03)00056-1.  Google Scholar [16] J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh A, 127 (1997), 281-336.  Google Scholar [17] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic system, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8.  Google Scholar [18] A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428. doi: 10.2307/1940296.  Google Scholar [19] E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29. doi: 10.2307/1939378.  Google Scholar [20] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in Dynamical Systems and applications, World Scientific Series Applicable Analysis, 4, World Scientific, River Edge, NJ, (1995), 343-358.  Google Scholar [21] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157.  Google Scholar [22] V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161. doi: 10.1016/j.jde.2004.06.003.  Google Scholar [23] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near the degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189.  Google Scholar [24] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vicker, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.  Google Scholar [25] V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533. doi: 10.1007/s002850100106.  Google Scholar [26] M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.  Google Scholar [27] M. Iida, M. Tatsuya, H, Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 223-252. doi: 10.1007/BF03167402.  Google Scholar [28] J. Jiang, X. Liang and X. Zhao, Saddle point behavior for monotone semiflows and reaction-diffusion models, J. Differential Equations, 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.  Google Scholar [29] S. Kirkland, C. K. Li and S. J. Schreiber, On the evolution of dispersal in patchy environments, SIAM J. Appl. Math., 66 (2006), 1366-1382. doi: 10.1137/050628933.  Google Scholar [30] J. López-Gómez, Coexistence and meta-coexistence for competing species, Houston J. Math., (2003), 483-536.  Google Scholar [31] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426.  Google Scholar [32] Y. Lou, personal, communication., ().   Google Scholar [33] Y. Lou, S. Martinez and W. M. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion, Dis. Cont. Dyn. Sys., 6 (2000), 175-190.  Google Scholar [34] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131.  Google Scholar [35] Y. Lou and W. M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Diff. Eqs., 154 (1999), 157-190.  Google Scholar [36] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Dis. Cont. Dyn. Sys., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar [37] Y. Lou, S. Martinez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Diff. Eqs., 230 (2006), 720-742.  Google Scholar [38] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in non-convex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.  Google Scholar [39] M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar [40] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035.  Google Scholar [41] M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  Google Scholar [42] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications, $3^{nd}$," Interdisciplinary Applied Mathematics, Springer, New York, 2003.  Google Scholar [43] H. Ninomiya, Separatrices of competition-diffusion equations, J. Math. Kyoto Univ., 35 (1995), 539-567.  Google Scholar [44] A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives, $2^{nd}$," Interdisciplinary Applied Mathematics, 14, Springer, New York, 2001.  Google Scholar [45] S. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Popul. Biol., 21 (1982), 92-113. doi: 10.1016/0040-5809(82)90008-9.  Google Scholar [46] A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invisibility, Bull. Math. Biol., 66 (2004), 975-1008. doi: 10.1016/j.bulm.2003.10.010.  Google Scholar [47] N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997. Google Scholar [48] 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.  Google Scholar [49] I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Eqs., 61 (1986), 208-249.  Google Scholar

show all references

##### References:
 [1] J. M. Arrieta, A. N. Carvalho and A. Rodriguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Diff. Eqs., 168 (2000), 33-59.  Google Scholar [2] R. S. Cantrell and C. Cosner, The effect of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155.  Google Scholar [3] R. S. Cantrell and C. Cosner, Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252. doi: 10.1137/0153014.  Google Scholar [4] R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145. doi: 10.1007/s002850050122.  Google Scholar [5] R. S. Cantrell and C. Cosner, "Spatial Ecology Via Reaction-Diffusion Equations," Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.  Google Scholar [6] R. S. Cantrell, C. Cosner and V. Huston, Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559.  Google Scholar [7] R. S. Cantrell, C. Cosner and V. Huston, Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101.  Google Scholar [8] R. S. Cantrell, C. Cosner and Y. Lou, Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynam. Differential Equations, 16 (2004), 973-1010. doi: 10.1007/s10884-004-7831-y.  Google Scholar [9] A. N. Carvalho and J. K. Hale, Large diffusion with dispersion, Nonlinear Anal., 17 (1991), 1139-1151. doi: 10.1016/0362-546X(91)90233-Q.  Google Scholar [10] Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion, Disc. Cont. Dyn. Syst. A, 9 (2003), 1193-1200. doi: 10.3934/dcds.2003.9.1193.  Google Scholar [11] E. Conway, D. Hoff and J. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16. doi: 10.1137/0135001.  Google Scholar [12] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion equations, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.  Google Scholar [13] Y. Du, Effects of a degeneracy in the competition model, Part I. Classical and generalized steady-state solutions, J. Differential Equations, 181 (2002), 92-132. doi: 10.1006/jdeq.2001.4074.  Google Scholar [14] Y. Du, Effects of a degeneracy in the competition model, Part II. Perturbation and dynamical behavior, J. Differential Equations, 181 (2002), 133-164. doi: 10.1006/jdeq.2001.4075.  Google Scholar [15] Y. Du, Realization of prescribed patterns in the competition model, J. Differential Equations, 193 (2003), 147-179. doi: 10.1016/S0022-0396(03)00056-1.  Google Scholar [16] J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh A, 127 (1997), 281-336.  Google Scholar [17] J. K. Hale, Large diffusivity and asymptotic behavior in parabolic system, J. Math. Anal. Appl., 118 (1986), 455-466. doi: 10.1016/0022-247X(86)90273-8.  Google Scholar [18] A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428. doi: 10.2307/1940296.  Google Scholar [19] E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit, Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29. doi: 10.2307/1939378.  Google Scholar [20] V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in Dynamical Systems and applications, World Scientific Series Applicable Analysis, 4, World Scientific, River Edge, NJ, (1995), 343-358.  Google Scholar [21] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157.  Google Scholar [22] V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161. doi: 10.1016/j.jde.2004.06.003.  Google Scholar [23] V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik, Competing species near the degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491. doi: 10.1137/S0036141002402189.  Google Scholar [24] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vicker, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517. doi: 10.1007/s00285-003-0210-1.  Google Scholar [25] V. Hutson, K. Mischaikow and P. Poláčik, The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533. doi: 10.1007/s002850100106.  Google Scholar [26] M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641. doi: 10.1007/s00285-006-0013-2.  Google Scholar [27] M. Iida, M. Tatsuya, H, Ninomiya and E. Yanagida, Diffusion-induced extinction of a superior species in a competition system, Japan J. Indust. Appl. Math., 15 (1998), 223-252. doi: 10.1007/BF03167402.  Google Scholar [28] J. Jiang, X. Liang and X. Zhao, Saddle point behavior for monotone semiflows and reaction-diffusion models, J. Differential Equations, 203 (2004), 313-330. doi: 10.1016/j.jde.2004.05.002.  Google Scholar [29] S. Kirkland, C. K. Li and S. J. Schreiber, On the evolution of dispersal in patchy environments, SIAM J. Appl. Math., 66 (2006), 1366-1382. doi: 10.1137/050628933.  Google Scholar [30] J. López-Gómez, Coexistence and meta-coexistence for competing species, Houston J. Math., (2003), 483-536.  Google Scholar [31] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426.  Google Scholar [32] Y. Lou, personal, communication., ().   Google Scholar [33] Y. Lou, S. Martinez and W. M. Ni, On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion, Dis. Cont. Dyn. Sys., 6 (2000), 175-190.  Google Scholar [34] Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131.  Google Scholar [35] Y. Lou and W. M. Ni, Diffusion vs. cross-diffusion: An elliptic approach, J. Diff. Eqs., 154 (1999), 157-190.  Google Scholar [36] Y. Lou, W. M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Dis. Cont. Dyn. Sys., 10 (2004), 435-458. doi: 10.3934/dcds.2004.10.435.  Google Scholar [37] Y. Lou, S. Martinez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Diff. Eqs., 230 (2006), 720-742.  Google Scholar [38] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in non-convex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079. doi: 10.2977/prims/1195182020.  Google Scholar [39] M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635.  Google Scholar [40] M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035.  Google Scholar [41] M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449.  Google Scholar [42] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications, $3^{nd}$," Interdisciplinary Applied Mathematics, Springer, New York, 2003.  Google Scholar [43] H. Ninomiya, Separatrices of competition-diffusion equations, J. Math. Kyoto Univ., 35 (1995), 539-567.  Google Scholar [44] A. Okubo and S. Levin, "Diffusion and Ecological Problems: Modern Perspectives, $2^{nd}$," Interdisciplinary Applied Mathematics, 14, Springer, New York, 2001.  Google Scholar [45] S. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Popul. Biol., 21 (1982), 92-113. doi: 10.1016/0040-5809(82)90008-9.  Google Scholar [46] A. B. Potapov and M. A. Lewis, Climate and competition: The effect of moving range boundaries on habitat invisibility, Bull. Math. Biol., 66 (2004), 975-1008. doi: 10.1016/j.bulm.2003.10.010.  Google Scholar [47] N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice," Oxford Series in Ecology and Evolution, Oxford University Press, Oxford, New York, Tokyo, 1997. Google Scholar [48] 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.  Google Scholar [49] I. Takagi, Point-condensation for a reaction-diffusion system, J. Diff. Eqs., 61 (1986), 208-249.  Google Scholar
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