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On limit systems for some population models with crossdiffusion
On the dependence of population size upon random dispersal rate
1.  Department of Environmental and Global Health, College of Public Health and Health Professions and Emerging Pathogens Institute, University of Florida, Gainesville, FL 32610, United States 
2.  Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210 
References:
[1] 
L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profile of the steady states for an SIS epidemic reactiondiffusion model, Discre. Cont. Dyn. Sys., 21 (2008), 120. Google Scholar 
[2] 
P. Amarasekare, Effect of nonrandom dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282293. Google Scholar 
[3] 
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). Google Scholar 
[4] 
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379397. Google Scholar 
[5] 
A. Bezugly and Y. Lou, Reactiondiffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 9831004. Google Scholar 
[6] 
R. S. Cantrell and C. Cosner, "Spatial Ecology via ReactionDiffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. Google Scholar 
[7] 
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199214. Google Scholar 
[8] 
R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497518. Google Scholar 
[9] 
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 1736. Google Scholar 
[10] 
X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627658. Google Scholar 
[11] 
X. F. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reactiondiffusionadvection model, J. Math. Biol., 57 (2008), 361386. Google Scholar 
[12] 
J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. Google Scholar 
[13] 
W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steadystate population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688704. Google Scholar 
[14] 
W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173211. Google Scholar 
[15] 
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reactiondiffusion model, J. Math. Biol., 37 (1998), 6183. Google Scholar 
[16] 
A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New YorkMontreal, Que.London, 1969. Google Scholar 
[17] 
R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 17931817. Google Scholar 
[18] 
A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311314. Google Scholar 
[19] 
W. Z. Huang, M. A. Han and K. Y. Liu, Dynamics of an SIS reactiondiffusion epidemic model for disease transmission, Math. Bios. Eng., 7 (2010), 5166. Google Scholar 
[20] 
K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetrybreaking, Appl. Math. Optim., 58 (2008), 89110. Google Scholar 
[21] 
K.Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161181. Google Scholar 
[22] 
K.Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). Google Scholar 
[23] 
K.Y. Lam and W.M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Cont. Dyn. Sys. A, 28 (2010), 10511067. Google Scholar 
[24] 
K.Y. Lam and W.M. Ni, Dynamics of the diffusive LotkaVolterra competition system,, in preparation., (). Google Scholar 
[25] 
J. Langebrake, L. RiotteLambert, C. W. Osenberg and P. De Leenheer, Differential movement and movement bias models for marine protected areas,, accepted for publication in Journal of Mathematical Biology., (). Google Scholar 
[26] 
S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113132. Google Scholar 
[27] 
S. Lenhart, S. Stojanovic and V. Protopopescu, A twosided game for nonlocal competitive systems with control on the source terms, in "Variational and Free Boundary Problems" (eds. A. Friedman and J. Spruck), IMA Volumes in Math. and its Applications, 53, Springer, New York, (1993), 135152. Google Scholar 
[28] 
S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman and Hall/CRC, Boca Raton, FL, 2007. Google Scholar 
[29] 
A. W. Leung and S. Stojanovic, Optimal control for elliptic VolterraLotka type equations, J. Math. Anal. Appl., 173 (1993), 603619. Google Scholar 
[30] 
S. A. Levin, H. C. MullerLandau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575604. Google Scholar 
[31] 
F. Li, L. P Wang and Y. Wang, On the effects of migration and interspecific competitions in steady state of some LotkaVolterra model, Dis. Cont. Dyn. Syst. Ser. B, 15 (2011), 669686. Google Scholar 
[32] 
F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867881. Google Scholar 
[33] 
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400426. Google Scholar 
[34] 
Y. Lou, Some challenging mathematical problems in evolution of dispersal andpopulation dynamics, in "Tutor. Math. Biosci. IV" (ed. A. Friedman), Lect. Notes Mathematics, 1922, Springer, Berlin, (2008), 171205. Google Scholar 
[35] 
Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388418. Google Scholar 
[36] 
J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, SpringerVerlag, New York, 2003. Google Scholar 
[37] 
C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 13041314. Google Scholar 
[38] 
M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843849. Google Scholar 
[39] 
W.M. Ni, Diffusion, crossdiffusion, and their spikelayer steady states, Notices Amer. Math. Soc., 45 (1998), 918. Google Scholar 
[40] 
W.M. Ni, "The Mathematics of Diffusions," CBMSNSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. Google Scholar 
[41] 
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, SpringerVerlag, New York, 2001. Google Scholar 
[42] 
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction diffusion model. I, J. Diff. Eqs., 247 (2009), 10961119. Google Scholar 
[43] 
R. Peng and S. Q. Liu, Global stability of the steady states of an SIS epidemic reactiondiffusion model, Nonl. Anal. TMA, 71 (2009), 239247. Google Scholar 
[44] 
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, SpringerVerlag, New York, 1984. Google Scholar 
[45] 
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 
[46] 
X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility andchemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535560. Google Scholar 
[47] 
X. F. Wang and Y. P. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., LX (2002), 505531. Google Scholar 
show all references
References:
[1] 
L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profile of the steady states for an SIS epidemic reactiondiffusion model, Discre. Cont. Dyn. Sys., 21 (2008), 120. Google Scholar 
[2] 
P. Amarasekare, Effect of nonrandom dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282293. Google Scholar 
[3] 
I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). Google Scholar 
[4] 
F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379397. Google Scholar 
[5] 
A. Bezugly and Y. Lou, Reactiondiffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 9831004. Google Scholar 
[6] 
R. S. Cantrell and C. Cosner, "Spatial Ecology via ReactionDiffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. Google Scholar 
[7] 
R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199214. Google Scholar 
[8] 
R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497518. Google Scholar 
[9] 
R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 1736. Google Scholar 
[10] 
X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunction of elliptic operator with large convection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627658. Google Scholar 
[11] 
X. F. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reactiondiffusionadvection model, J. Math. Biol., 57 (2008), 361386. Google Scholar 
[12] 
J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. Google Scholar 
[13] 
W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steadystate population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688704. Google Scholar 
[14] 
W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173211. Google Scholar 
[15] 
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reactiondiffusion model, J. Math. Biol., 37 (1998), 6183. Google Scholar 
[16] 
A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New YorkMontreal, Que.London, 1969. Google Scholar 
[17] 
R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 17931817. Google Scholar 
[18] 
A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311314. Google Scholar 
[19] 
W. Z. Huang, M. A. Han and K. Y. Liu, Dynamics of an SIS reactiondiffusion epidemic model for disease transmission, Math. Bios. Eng., 7 (2010), 5166. Google Scholar 
[20] 
K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetrybreaking, Appl. Math. Optim., 58 (2008), 89110. Google Scholar 
[21] 
K.Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161181. Google Scholar 
[22] 
K.Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). Google Scholar 
[23] 
K.Y. Lam and W.M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Cont. Dyn. Sys. A, 28 (2010), 10511067. Google Scholar 
[24] 
K.Y. Lam and W.M. Ni, Dynamics of the diffusive LotkaVolterra competition system,, in preparation., (). Google Scholar 
[25] 
J. Langebrake, L. RiotteLambert, C. W. Osenberg and P. De Leenheer, Differential movement and movement bias models for marine protected areas,, accepted for publication in Journal of Mathematical Biology., (). Google Scholar 
[26] 
S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113132. Google Scholar 
[27] 
S. Lenhart, S. Stojanovic and V. Protopopescu, A twosided game for nonlocal competitive systems with control on the source terms, in "Variational and Free Boundary Problems" (eds. A. Friedman and J. Spruck), IMA Volumes in Math. and its Applications, 53, Springer, New York, (1993), 135152. Google Scholar 
[28] 
S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman and Hall/CRC, Boca Raton, FL, 2007. Google Scholar 
[29] 
A. W. Leung and S. Stojanovic, Optimal control for elliptic VolterraLotka type equations, J. Math. Anal. Appl., 173 (1993), 603619. Google Scholar 
[30] 
S. A. Levin, H. C. MullerLandau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575604. Google Scholar 
[31] 
F. Li, L. P Wang and Y. Wang, On the effects of migration and interspecific competitions in steady state of some LotkaVolterra model, Dis. Cont. Dyn. Syst. Ser. B, 15 (2011), 669686. Google Scholar 
[32] 
F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867881. Google Scholar 
[33] 
Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400426. Google Scholar 
[34] 
Y. Lou, Some challenging mathematical problems in evolution of dispersal andpopulation dynamics, in "Tutor. Math. Biosci. IV" (ed. A. Friedman), Lect. Notes Mathematics, 1922, Springer, Berlin, (2008), 171205. Google Scholar 
[35] 
Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388418. Google Scholar 
[36] 
J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, SpringerVerlag, New York, 2003. Google Scholar 
[37] 
C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 13041314. Google Scholar 
[38] 
M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843849. Google Scholar 
[39] 
W.M. Ni, Diffusion, crossdiffusion, and their spikelayer steady states, Notices Amer. Math. Soc., 45 (1998), 918. Google Scholar 
[40] 
W.M. Ni, "The Mathematics of Diffusions," CBMSNSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. Google Scholar 
[41] 
A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, SpringerVerlag, New York, 2001. Google Scholar 
[42] 
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction diffusion model. I, J. Diff. Eqs., 247 (2009), 10961119. Google Scholar 
[43] 
R. Peng and S. Q. Liu, Global stability of the steady states of an SIS epidemic reactiondiffusion model, Nonl. Anal. TMA, 71 (2009), 239247. Google Scholar 
[44] 
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, SpringerVerlag, New York, 1984. Google Scholar 
[45] 
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 
[46] 
X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility andchemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535560. Google Scholar 
[47] 
X. F. Wang and Y. P. Wu, Qualitative analysis on a chemotactic diffusion model for two species competing for a limited resource, Quart. Appl. Math., LX (2002), 505531. Google Scholar 
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