American Institute of Mathematical Sciences

May  2014, 34(5): 1701-1745. doi: 10.3934/dcds.2014.34.1701

Reaction-diffusion-advection models for the effects and evolution of dispersal

 1 Department of Mathematics, University of Miami, Coral Gables, FL 33124, United States

Received  June 2013 Revised  August 2013 Published  October 2013

This review describes reaction-advection-diffusion models for the ecological effects and evolution of dispersal, and mathematical methods for analyzing those models. The topics covered include models for a single species, models for ecological interactions between species, and models for the evolution of dispersal strategies. The models are all set on bounded domains. The mathematical methods include spectral theory, specifically the theory of principal eigenvalues for elliptic operators, maximum principles and comparison theorems, bifurcation theory, and persistence theory.
Citation: Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701
References:
 [1] W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, 1931. doi: 10.5962/bhl.title.7313. [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440. [3] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biological Dynamics, 6 (2012) , 117-130. doi: 10.1080/17513758.2010.529169. [4] P. Bates and G. Zhao, Existence, uniqueness, and stability of the stationary solution to a nonlocal equation arising in population dispersal, J. Math. Anal. Appl, 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [5] F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canadian Appllied Mathematics Quarterly, 3 (1995), 379-397. [6] M. Bendahmane, Weak and classical solutions to predator-prey system with cross-diffusion, Nonlinear Analysis: TMA, 73 (2010), 2489-2503. doi: 10.1016/j.na.2010.06.021. [7] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105. [8] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010), 983-1004. doi: 10.1080/00036810903479723. [9] J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 1971, 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. [10] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X. [11] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155. [12] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [14] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. of Biological Dynamics, 1 (2007), 249-271. doi: 10.1080/17513750701450227. [15] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003. [16] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047. [17] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024. [18] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [19] R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965. doi: 10.1007/s00285-011-0486-5. [20] R. S Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal free dispersal in spatial population models with nonlocal dispersal, Canadian Applied Math. Quarterly, 20 (2012), 15-38. [21] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2. [22] X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204. [23] Y. S. 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. [24] C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation, Theoretical Population Biology, 67 (2005),101-108. doi: 10.1016/j.tpb.2004.09.002. [25] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population? J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9. [26] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dynamics, 6 (2012), 395-405. doi: 10.1080/17513758.2011.588341. [27] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [28] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [29] J. Coville, J. Dávila and S. Martnez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854. [30] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [31] M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [32] S. Dehaene, The neural basis of the Weber-Fechner law: A logarithmic mental number line, Trends in Cognitive Sciences, 7 (2003), 145-147. doi: 10.1016/S1364-6613(03)00055-X. [33] M. Delgado and A. Suárez, On the structure of the positive solutions of the logistic equation with nonlinear diffusion, J. Math. Anal. Appl., 268 (2002), 200-216. doi: 10.1006/jmaa.2001.7815. [34] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120. [35] B. Eaves, A. Hoffman, U. Rothblum and H. Schneider, Line sum symmetric scaling of square nonnegative matrices, Mathematical Programming Study, 25 (l985), 124-141. [36] S. Flaxman and Y. Lou, Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators, J. Theoretical Biology, 256 (2009), 187-200. doi: 10.1016/j.jtbi.2008.09.024. [37] S. D. Fretwell and H. L. Lucas, On territorial behaviour and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 16-36. doi: 10.1007/BF01601953. [38] S. D. Fretwell, Populations in A Seasonal Environment, Princeton University Press, 1972. [39] R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biology, 74 (2012), 257-299. doi: 10.1007/s11538-011-9662-4. [40] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7. [41] J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59. doi: 10.1016/0022-247X(69)90175-9. [42] J. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Anaysis, 20 (1989), 388-395. doi: 10.1137/0520025. [43] A. Hastings, Can spatial variation alone lead to selection for dispersal? Theor. Popul. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8. [44] D. Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics 840), Springer-Verlag, Berlin-New York, 1981. [45] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [46] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030. doi: 10.1080/03605308008820162. [47] G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal, Comm. Pure and Applied Analysis, 11 (2012), 1699-1722. doi: 10.3934/cpaa.2012.11.1699. [48] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [49] R. D. Holt, Predation, apparent competition and the structure of prey communities, Theoretical Population Biology, 12 (1977), 197-229. doi: 10.1016/0040-5809(77)90042-9. [50] R. D. Holt and G. A. Polis, A theoretical gramework for intraguild predation, The American Naturalist, 149 (1997), 745-769. [51] V. Hutson, K. Mischaikow and P. Polácik, The evolution of dispersal rates in a heterogenous time-periodic environment, J. Math. Biology, 43 (2001), 501-533. doi: 10.1007/s002850100106. [52] V. Hutson, W. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679(electronic). doi: 10.1090/S0002-9939-00-05808-1. [53] V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. 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References:
 [1] W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, 1931. doi: 10.5962/bhl.title.7313. [2] H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374. doi: 10.1006/jdeq.1998.3440. [3] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biological Dynamics, 6 (2012) , 117-130. doi: 10.1080/17513758.2010.529169. [4] P. Bates and G. Zhao, Existence, uniqueness, and stability of the stationary solution to a nonlocal equation arising in population dispersal, J. Math. Anal. Appl, 332 (2007), 428-440. doi: 10.1016/j.jmaa.2006.09.007. [5] F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environments, Canadian Appllied Mathematics Quarterly, 3 (1995), 379-397. [6] M. Bendahmane, Weak and classical solutions to predator-prey system with cross-diffusion, Nonlinear Analysis: TMA, 73 (2010), 2489-2503. doi: 10.1016/j.na.2010.06.021. [7] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92. doi: 10.1002/cpa.3160470105. [8] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients, Appl. Anal., 89 (2010), 983-1004. doi: 10.1080/00036810903479723. [9] J. E. Billotti and J. P. LaSalle, Dissipative periodic processes, Bull. Amer. Math. Soc., 77 1971, 1082-1088. doi: 10.1090/S0002-9904-1971-12879-3. [10] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318. doi: 10.1017/S030821050001876X. [11] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155. [12] R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064. doi: 10.1137/0522068. [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. [14] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. of Biological Dynamics, 1 (2007), 249-271. doi: 10.1080/17513750701450227. [15] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003. [16] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047. [17] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024. [18] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [19] R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol., 65 (2012), 943-965. doi: 10.1007/s00285-011-0486-5. [20] R. S Cantrell, C. Cosner, Y. Lou and D. Ryan, Evolutionary stability of ideal free dispersal in spatial population models with nonlocal dispersal, Canadian Applied Math. Quarterly, 20 (2012), 15-38. [21] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2. [22] X. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204. [23] Y. S. 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. [24] C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation, Theoretical Population Biology, 67 (2005),101-108. doi: 10.1016/j.tpb.2004.09.002. [25] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population? J. Math. Anal. Appl., 277 (2003), 489-503. doi: 10.1016/S0022-247X(02)00575-9. [26] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal, J. Biol. Dynamics, 6 (2012), 395-405. doi: 10.1080/17513758.2011.588341. [27] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation, Annali di Matematica, 185 (2006), 461-485. doi: 10.1007/s10231-005-0163-7. [28] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921-2953. doi: 10.1016/j.jde.2010.07.003. [29] J. Coville, J. Dávila and S. Martnez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity, SIAM J. Math. Anal., 39 (2008), 1693-1709. doi: 10.1137/060676854. [30] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [31] M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. [32] S. Dehaene, The neural basis of the Weber-Fechner law: A logarithmic mental number line, Trends in Cognitive Sciences, 7 (2003), 145-147. doi: 10.1016/S1364-6613(03)00055-X. [33] M. Delgado and A. Suárez, On the structure of the positive solutions of the logistic equation with nonlinear diffusion, J. Math. Anal. Appl., 268 (2002), 200-216. doi: 10.1006/jmaa.2001.7815. [34] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120. [35] B. Eaves, A. Hoffman, U. Rothblum and H. Schneider, Line sum symmetric scaling of square nonnegative matrices, Mathematical Programming Study, 25 (l985), 124-141. [36] S. Flaxman and Y. Lou, Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators, J. Theoretical Biology, 256 (2009), 187-200. doi: 10.1016/j.jtbi.2008.09.024. [37] S. D. Fretwell and H. L. Lucas, On territorial behaviour and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 16-36. doi: 10.1007/BF01601953. [38] S. D. Fretwell, Populations in A Seasonal Environment, Princeton University Press, 1972. [39] R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biology, 74 (2012), 257-299. doi: 10.1007/s11538-011-9662-4. [40] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7. [41] J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl., 26 (1969), 39-59. doi: 10.1016/0022-247X(69)90175-9. [42] J. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Anaysis, 20 (1989), 388-395. doi: 10.1137/0520025. [43] A. 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