# 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 & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701
##### References:
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Bendahmane, Weak and classical solutions to predator-prey system with cross-diffusion,, Nonlinear Analysis: TMA, 73 (2010), 2489. doi: 10.1016/j.na.2010.06.021. Google Scholar [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. doi: 10.1002/cpa.3160470105. Google Scholar [8] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients,, Appl. Anal., 89 (2010), 983. doi: 10.1080/00036810903479723. Google Scholar [9] J. E. Billotti and J. P. LaSalle, Dissipative periodic processes,, Bull. Amer. Math. Soc., 77 (1971), 1082. doi: 10.1090/S0002-9904-1971-12879-3. Google Scholar [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. doi: 10.1017/S030821050001876X. Google Scholar [11] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315. doi: 10.1007/BF00167155. Google Scholar [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. doi: 10.1137/0522068. Google Scholar [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, (2003). doi: 10.1002/0470871296. Google Scholar [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. doi: 10.1080/17513750701450227. Google Scholar [15] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199. doi: 10.1016/j.mbs.2006.09.003. Google Scholar [16] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497. doi: 10.1017/S0308210506000047. Google Scholar [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. doi: 10.1016/j.jde.2008.07.024. Google Scholar [18] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17. Google Scholar [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. doi: 10.1007/s00285-011-0486-5. Google Scholar [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. Google Scholar [21] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model,, J. Math. Biol., 57 (2008), 361. doi: 10.1007/s00285-008-0166-2. Google Scholar [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. doi: 10.1512/iumj.2008.57.3204. Google Scholar [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. doi: 10.3934/dcds.2004.10.719. Google Scholar [24] C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation,, Theoretical Population Biology, 67 (2005), 101. doi: 10.1016/j.tpb.2004.09.002. Google Scholar [25] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9. Google Scholar [26] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal,, J. Biol. Dynamics, 6 (2012), 395. doi: 10.1080/17513758.2011.588341. Google Scholar [27] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Annali di Matematica, 185 (2006), 461. doi: 10.1007/s10231-005-0163-7. Google Scholar [28] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921. doi: 10.1016/j.jde.2010.07.003. Google Scholar [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. doi: 10.1137/060676854. Google Scholar [30] M. Crandall and P. 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Schneider, Line sum symmetric scaling of square nonnegative matrices,, Mathematical Programming Study, 25 (): 124. Google Scholar [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. doi: 10.1016/j.jtbi.2008.09.024. Google Scholar [37] S. D. Fretwell and H. L. Lucas, On territorial behaviour and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 16. doi: 10.1007/BF01601953. Google Scholar [38] S. D. Fretwell, Populations in A Seasonal Environment,, Princeton University Press, (1972). Google Scholar [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. doi: 10.1007/s11538-011-9662-4. Google Scholar [40] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats,, Bull. Math. Biol., 71 (2009), 1793. doi: 10.1007/s11538-009-9425-7. Google Scholar [41] J. K. Hale, Dynamical systems and stability,, J. Math. Anal. Appl., 26 (1969), 39. doi: 10.1016/0022-247X(69)90175-9. Google Scholar [42] J. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Anaysis, 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [43] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar [44] D. Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics 840),, Springer-Verlag, (1981). Google Scholar [45] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Research Notes in Mathematics Series, (1991). Google Scholar [46] P. Hess and T. 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show all references

##### References:
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Bendahmane, Weak and classical solutions to predator-prey system with cross-diffusion,, Nonlinear Analysis: TMA, 73 (2010), 2489. doi: 10.1016/j.na.2010.06.021. Google Scholar [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. doi: 10.1002/cpa.3160470105. Google Scholar [8] A. Bezuglyy and Y. Lou, Reaction-diffusion models with large advection coefficients,, Appl. Anal., 89 (2010), 983. doi: 10.1080/00036810903479723. Google Scholar [9] J. E. Billotti and J. P. LaSalle, Dissipative periodic processes,, Bull. Amer. Math. Soc., 77 (1971), 1082. doi: 10.1090/S0002-9904-1971-12879-3. Google Scholar [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. doi: 10.1017/S030821050001876X. Google Scholar [11] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics,, J. Math. Biol., 29 (1991), 315. doi: 10.1007/BF00167155. Google Scholar [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. doi: 10.1137/0522068. Google Scholar [13] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, (2003). doi: 10.1002/0470871296. Google Scholar [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. doi: 10.1080/17513750701450227. Google Scholar [15] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion,, Math. Biosci., 204 (2006), 199. doi: 10.1016/j.mbs.2006.09.003. Google Scholar [16] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497. doi: 10.1017/S0308210506000047. Google Scholar [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. doi: 10.1016/j.jde.2008.07.024. Google Scholar [18] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution,, Math. Biosci. Eng., 7 (2010), 17. doi: 10.3934/mbe.2010.7.17. Google Scholar [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. doi: 10.1007/s00285-011-0486-5. Google Scholar [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. Google Scholar [21] X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model,, J. Math. Biol., 57 (2008), 361. doi: 10.1007/s00285-008-0166-2. Google Scholar [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. doi: 10.1512/iumj.2008.57.3204. Google Scholar [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. doi: 10.3934/dcds.2004.10.719. Google Scholar [24] C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation,, Theoretical Population Biology, 67 (2005), 101. doi: 10.1016/j.tpb.2004.09.002. Google Scholar [25] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489. doi: 10.1016/S0022-247X(02)00575-9. Google Scholar [26] C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal,, J. Biol. Dynamics, 6 (2012), 395. doi: 10.1080/17513758.2011.588341. Google Scholar [27] J. Coville, On uniqueness and monotonicity of solutions of non-local reaction diffusion equation,, Annali di Matematica, 185 (2006), 461. doi: 10.1007/s10231-005-0163-7. Google Scholar [28] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators,, J. Differential Equations, 249 (2010), 2921. doi: 10.1016/j.jde.2010.07.003. Google Scholar [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. doi: 10.1137/060676854. Google Scholar [30] M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar [31] M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. Google Scholar [32] S. Dehaene, The neural basis of the Weber-Fechner law: A logarithmic mental number line,, Trends in Cognitive Sciences, 7 (2003), 145. doi: 10.1016/S1364-6613(03)00055-X. Google Scholar [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. doi: 10.1006/jmaa.2001.7815. Google Scholar [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. doi: 10.1007/s002850050120. Google Scholar [35] B. Eaves, A. Hoffman, U. Rothblum and H. Schneider, Line sum symmetric scaling of square nonnegative matrices,, Mathematical Programming Study, 25 (): 124. Google Scholar [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. doi: 10.1016/j.jtbi.2008.09.024. Google Scholar [37] S. D. Fretwell and H. L. Lucas, On territorial behaviour and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 16. doi: 10.1007/BF01601953. Google Scholar [38] S. D. Fretwell, Populations in A Seasonal Environment,, Princeton University Press, (1972). Google Scholar [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. doi: 10.1007/s11538-011-9662-4. Google Scholar [40] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats,, Bull. Math. Biol., 71 (2009), 1793. doi: 10.1007/s11538-009-9425-7. Google Scholar [41] J. K. Hale, Dynamical systems and stability,, J. Math. Anal. Appl., 26 (1969), 39. doi: 10.1016/0022-247X(69)90175-9. Google Scholar [42] J. Hale and P. Waltman, Persistence in infinite-dimensional systems,, SIAM Journal on Mathematical Anaysis, 20 (1989), 388. doi: 10.1137/0520025. Google Scholar [43] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Popul. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar [44] D. Henry, Geometric Theory of Semilinear Parabolic Equations (Lecture Notes in Mathematics 840),, Springer-Verlag, (1981). Google Scholar [45] P. Hess, Periodic-parabolic Boundary Value Problems and Positivity,, Pitman Research Notes in Mathematics Series, (1991). Google Scholar [46] P. Hess and T. 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