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:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology,, University of Chicago Press, (1931). doi: 10.5962/bhl.title.7313. Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336. doi: 10.1006/jdeq.1998.3440. Google Scholar

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biological Dynamics, 6 (2012), 117. doi: 10.1080/17513758.2010.529169. Google Scholar

[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. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[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. Google Scholar

[6]

M. 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. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Differential Equations, 5 (1980), 999. doi: 10.1080/03605308008820162. Google Scholar

[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. doi: 10.3934/cpaa.2012.11.1699. Google Scholar

[48]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[49]

R. D. Holt, Predation, apparent competition and the structure of prey communities,, Theoretical Population Biology, 12 (1977), 197. doi: 10.1016/0040-5809(77)90042-9. Google Scholar

[50]

R. D. Holt and G. A. Polis, A theoretical gramework for intraguild predation,, The American Naturalist, 149 (1997), 745. Google Scholar

[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. doi: 10.1007/s002850100106. Google Scholar

[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. doi: 10.1090/S0002-9939-00-05808-1. Google Scholar

[53]

V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar

[54]

C.-Y. Kao, Y. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains,, Math. Biosci. Eng., 5 (2008), 315. doi: 10.3934/mbe.2008.5.315. Google Scholar

[55]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal,, Discrete and Continuous Dynamical Systems, 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar

[56]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms,, J. Math. Anal. Appl., 323 (2006), 1387. doi: 10.1016/j.jmaa.2005.11.065. Google Scholar

[57]

P. Kareiva and G. Odell, Swarms of predators exhibit 'preytaxis' if individual predators use area-restricted search,, American Naturalist, 130 (1987), 233. doi: 10.1086/284707. Google Scholar

[58]

S. Kirkland, C.-K. Li and S. J. Schreiber, On the evolution of dispersal in patchy environments,, SIAM Journal on Applied Mathematics, 66 (2006), 1366. doi: 10.1137/050628933. Google Scholar

[59]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,, J. Differential Equations, 250 (2011), 161. doi: 10.1016/j.jde.2010.08.028. Google Scholar

[60]

K.-Y. Lam and Y. Lou, Evolution of conditional dispersal: Evolutionarily stable strategies in spatial models,, J. Math. Biology, (2013). doi: 10.1007/s00285-013-0650-1. Google Scholar

[61]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics,, Discrete Contin. Dyn. Syst., 28 (2010), 1051. doi: 10.3934/dcds.2010.28.1051. Google Scholar

[62]

A. C. Lazer, Some remarks on periodic solutions of parabolic differential equations,, Dynamical systems II, (1982), 227. Google Scholar

[63]

D. Le and T. T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. of AMS, 133 (2005), 1985. doi: 10.1090/S0002-9939-05-07867-6. Google Scholar

[64]

D. Le and T. T. Nguyen, Persistence for a class of triangular cross diffusion parabolic systems,, Adv. Nonlinear Stud., 5 (2005), 493. Google Scholar

[65]

D. Le and T. T. Nguyen, Global attractors and uniform persistence for cross diffusion parabolic systems,, Dynam. Systems Appl., 16 (2007), 361. Google Scholar

[66]

J. M. Lee, T. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis,, Bull. Math. Biol., 70 (2008), 654. doi: 10.1007/s11538-007-9271-4. Google Scholar

[67]

J. M. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems,, J. Biol. Dyn., 3 (2009), 551. doi: 10.1080/17513750802716112. Google Scholar

[68]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species,, J. Differential Equations, 223 (2006), 400. doi: 10.1016/j.jde.2005.05.010. Google Scholar

[69]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics,, Japan J. Indust. Appl. Math., 23 (2006), 275. doi: 10.1007/BF03167595. Google Scholar

[70]

R. May and W. Leonard, Nonlinear aspects of competition between three species,, Special issue on mathematics and the social and biological sciences. SIAM J. Appl. Math., 29 (1975), 243. doi: 10.1137/0129022. Google Scholar

[71]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments,, American Naturalist, 140 (1992), 1010. doi: 10.1086/285453. Google Scholar

[72]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21. doi: 10.2307/1999300. Google Scholar

[73]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models,, An extended version of the Japanese edition, (1980). Google Scholar

[74]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence,, Math. Biosci., 210 (2007), 34. doi: 10.1016/j.mbs.2007.05.007. Google Scholar

[75]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities,, Dissertation, (2011). Google Scholar

[76]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (): 459. doi: 10.1007/BF01453979. Google Scholar

[77]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[78]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, Journal of Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[79]

H. L. Smith, Monotone Dynamical Systems,, An introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs, (1995). Google Scholar

[80]

M. Turelli, Re-Examination of stability in randomly varying versus deterministic environments with comments on stochastic theory of limiting similarity,, Theoretical Population Biology, 13 (1978), 244. doi: 10.1016/0040-5809(78)90045-X. Google Scholar

show all references

References:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology,, University of Chicago Press, (1931). doi: 10.5962/bhl.title.7313. Google Scholar

[2]

H. Amann and J. López-Gómez, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146 (1998), 336. doi: 10.1006/jdeq.1998.3440. Google Scholar

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biological Dynamics, 6 (2012), 117. doi: 10.1080/17513758.2010.529169. Google Scholar

[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. doi: 10.1016/j.jmaa.2006.09.007. Google Scholar

[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. Google Scholar

[6]

M. 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. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Differential Equations, 5 (1980), 999. doi: 10.1080/03605308008820162. Google Scholar

[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. doi: 10.3934/cpaa.2012.11.1699. Google Scholar

[48]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[49]

R. D. Holt, Predation, apparent competition and the structure of prey communities,, Theoretical Population Biology, 12 (1977), 197. doi: 10.1016/0040-5809(77)90042-9. Google Scholar

[50]

R. D. Holt and G. A. Polis, A theoretical gramework for intraguild predation,, The American Naturalist, 149 (1997), 745. Google Scholar

[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. doi: 10.1007/s002850100106. Google Scholar

[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. doi: 10.1090/S0002-9939-00-05808-1. Google Scholar

[53]

V. Hutson, S. Martínez, K. Mischaikow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483. doi: 10.1007/s00285-003-0210-1. Google Scholar

[54]

C.-Y. Kao, Y. Lou and E. Yanagida, Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains,, Math. Biosci. Eng., 5 (2008), 315. doi: 10.3934/mbe.2008.5.315. Google Scholar

[55]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal,, Discrete and Continuous Dynamical Systems, 26 (2010), 551. doi: 10.3934/dcds.2010.26.551. Google Scholar

[56]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms,, J. Math. Anal. Appl., 323 (2006), 1387. doi: 10.1016/j.jmaa.2005.11.065. Google Scholar

[57]

P. Kareiva and G. Odell, Swarms of predators exhibit 'preytaxis' if individual predators use area-restricted search,, American Naturalist, 130 (1987), 233. doi: 10.1086/284707. Google Scholar

[58]

S. Kirkland, C.-K. Li and S. J. Schreiber, On the evolution of dispersal in patchy environments,, SIAM Journal on Applied Mathematics, 66 (2006), 1366. doi: 10.1137/050628933. Google Scholar

[59]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model,, J. Differential Equations, 250 (2011), 161. doi: 10.1016/j.jde.2010.08.028. Google Scholar

[60]

K.-Y. Lam and Y. Lou, Evolution of conditional dispersal: Evolutionarily stable strategies in spatial models,, J. Math. Biology, (2013). doi: 10.1007/s00285-013-0650-1. Google Scholar

[61]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics,, Discrete Contin. Dyn. Syst., 28 (2010), 1051. doi: 10.3934/dcds.2010.28.1051. Google Scholar

[62]

A. C. Lazer, Some remarks on periodic solutions of parabolic differential equations,, Dynamical systems II, (1982), 227. Google Scholar

[63]

D. Le and T. T. Nguyen, Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,, Proc. of AMS, 133 (2005), 1985. doi: 10.1090/S0002-9939-05-07867-6. Google Scholar

[64]

D. Le and T. T. Nguyen, Persistence for a class of triangular cross diffusion parabolic systems,, Adv. Nonlinear Stud., 5 (2005), 493. Google Scholar

[65]

D. Le and T. T. Nguyen, Global attractors and uniform persistence for cross diffusion parabolic systems,, Dynam. Systems Appl., 16 (2007), 361. Google Scholar

[66]

J. M. Lee, T. Hillen and M. A. Lewis, Continuous traveling waves for prey-taxis,, Bull. Math. Biol., 70 (2008), 654. doi: 10.1007/s11538-007-9271-4. Google Scholar

[67]

J. M. Lee, T. Hillen and M. A. Lewis, Pattern formation in prey-taxis systems,, J. Biol. Dyn., 3 (2009), 551. doi: 10.1080/17513750802716112. Google Scholar

[68]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species,, J. Differential Equations, 223 (2006), 400. doi: 10.1016/j.jde.2005.05.010. Google Scholar

[69]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics,, Japan J. Indust. Appl. Math., 23 (2006), 275. doi: 10.1007/BF03167595. Google Scholar

[70]

R. May and W. Leonard, Nonlinear aspects of competition between three species,, Special issue on mathematics and the social and biological sciences. SIAM J. Appl. Math., 29 (1975), 243. doi: 10.1137/0129022. Google Scholar

[71]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments,, American Naturalist, 140 (1992), 1010. doi: 10.1086/285453. Google Scholar

[72]

X. Mora, Semilinear parabolic problems define semiflows on $C^k$ spaces,, Trans. Amer. Math. Soc., 278 (1983), 21. doi: 10.2307/1999300. Google Scholar

[73]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models,, An extended version of the Japanese edition, (1980). Google Scholar

[74]

L. Roques and F. Hamel, Mathematical analysis of the optimal habitat configurations for species persistence,, Math. Biosci., 210 (2007), 34. doi: 10.1016/j.mbs.2007.05.007. Google Scholar

[75]

D. Ryan, Fitness Dependent Dispersal in Intraguild Predation Communities,, Dissertation, (2011). Google Scholar

[76]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions,, Math. Ann., 258 (): 459. doi: 10.1007/BF01453979. Google Scholar

[77]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species,, J. Theoret. Biol., 79 (1979), 83. doi: 10.1016/0022-5193(79)90258-3. Google Scholar

[78]

J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, Journal of Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar

[79]

H. L. Smith, Monotone Dynamical Systems,, An introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs, (1995). Google Scholar

[80]

M. Turelli, Re-Examination of stability in randomly varying versus deterministic environments with comments on stochastic theory of limiting similarity,, Theoretical Population Biology, 13 (1978), 244. doi: 10.1016/0040-5809(78)90045-X. Google Scholar

[1]

Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65

[2]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 41-67. doi: 10.3934/dcds.2008.21.41

[3]

Henri Berestycki, Luca Rossi. Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 19-61. doi: 10.3934/dcds.2009.25.19

[4]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085

[5]

Heather Finotti, Suzanne Lenhart, Tuoc Van Phan. Optimal control of advective direction in reaction-diffusion population models. Evolution Equations & Control Theory, 2012, 1 (1) : 81-107. doi: 10.3934/eect.2012.1.81

[6]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385

[7]

Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057

[8]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85

[9]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[10]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[11]

Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

[12]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

[13]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415

[14]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[15]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

[16]

Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062

[17]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124

[18]

Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85

[19]

Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861

[20]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]