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September  2012, 17(6): 2047-2072. doi: 10.3934/dcdsb.2012.17.2047

Evolution of mixed dispersal in periodic environments

1. 

Department of Mathematics, The Ohio State University, Columbus, OH 43210

2. 

Department of Mathematics, Mathematical Bioscience Institute, Ohio State University, Columbus, Ohio 43210

3. 

Department of Mathematics and Statistics, Auburn University, Auburn University, AL 36849-5310

Received  April 2011 Revised  May 2011 Published  May 2012

Random dispersal describes the movement of organisms between adjacent spatial locations. However, the movement of some organisms such as seeds of plants can occur between non-adjacent spatial locations and is thus non-local. We propose to study a mixed dispersal strategy, which is a combination of random dispersal and non-local dispersal. More specifically, we assume that a fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes non-local dispersal. We investigate how such mixed dispersal affects the invasion of a single species and also how mixed dispersal strategy will evolve in spatially heterogeneous but temporally constant environment.
Citation: Chiu-Yen Kao, Yuan Lou, Wenxian Shen. Evolution of mixed dispersal in periodic environments. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2047-2072. doi: 10.3934/dcdsb.2012.17.2047
References:
[1]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar

[2]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl. (9), 86 (2006), 271. Google Scholar

[3]

J. Clobert, E. Danchin, A. Dhondt and J. Nichols, eds., "Dispersal,", Oxford University Press, (2001). Google Scholar

[4]

C. Cortázar, J. Coville, M. Elgueta and S. Martinez, A nonlocal inhomogeneous dispersal process,, J. Differential Equations, 241 (2007), 332. doi: 10.1016/j.jde.2007.06.002. Google Scholar

[5]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems,, SIAM J. Math. Anal., 41 (2009), 2136. doi: 10.1137/090751682. Google Scholar

[6]

C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal,, J. Biological Dynamics, 6 (2012), 395. Google Scholar

[7]

R. Cousens, C. Dytham and R. Law, "Dispersal in Plants: A Population Perspective,", Oxford University Press, (2008). Google Scholar

[8]

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

[9]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[10]

J. Coville, J. Dávila and S. Martínez, 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

[11]

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

[12]

W. Fagan and F. Lutscher, Average dispersal success: Linking home range, dispersal, and metapopulation dynamics to reserve design,, Ecol. Appl., 16 (2006), 820. Google Scholar

[13]

B. Gustafsson, H.-O. Kreiss and J. Oliger, "Time-Dependent Problems and Difference Methods,", Pure and Applied Mathematics (New York), (1995). Google Scholar

[14]

I. Hanski, "Metapopulation Ecology,", Oxford Univ. Press, (1999). Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math., 840 (1981). Google Scholar

[17]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations,, Rocky Mountain Journal of Mathematics, (). Google Scholar

[18]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal,, Comm. Pure Appl. Anal., (). Google Scholar

[19]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221. doi: 10.1017/S0956792506006462. Google Scholar

[20]

V. Hutson, S. Martinez, 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

[21]

V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain Journal of Mathematics, 38 (2008), 1147. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[22]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Critical domain size,, SIAM J. Appl. Math., 71 (2011), 1241. doi: 10.1137/100788033. Google Scholar

[23]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal,, Disc. Cont. Dynam. Sys. Series A, 26 (2010), 551. Google Scholar

[24]

R. W. van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (2007), 107. doi: 10.1007/BF02459473. Google Scholar

[25]

R. W. van Kirk and M. A. Lewis, Edgepermeability and population persistence in isolated habitat patches,, Natural Resources Modeling, 12 (2009), 37. Google Scholar

[26]

C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. McCain, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology,, J. Theor. Biol., 210 (2001), 201. doi: 10.1006/jtbi.2000.2287. Google Scholar

[27]

R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods,", SIAM Publications, (1998). Google Scholar

[28]

S. A. Levin, H. C. Muller-Landau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective,, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575. doi: 10.1146/annurev.ecolsys.34.011802.132428. Google Scholar

[29]

W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[30]

G. Lindfield and J. Penny, "Numerical Methods Using MATLAB,", Prentice Hall, (2000). Google Scholar

[31]

F. Lutscher, Nonlocal dispersal and averaging in heterogeneous landscapes,, Applicable Analysis, 89 (2010), 1091. doi: 10.1080/00036811003735816. Google Scholar

[32]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Appl. Math., 65 (2005), 1305. doi: 10.1137/S0036139904440400. Google Scholar

[33]

T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. doi: 10.1016/j.tpb.2011.09.006. Google Scholar

[34]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar

[35]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[36]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitat,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[37]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997). Google Scholar

[38]

J. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. doi: 10.1093/biomet/38.1-2.196. Google Scholar

show all references

References:
[1]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar

[2]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl. (9), 86 (2006), 271. Google Scholar

[3]

J. Clobert, E. Danchin, A. Dhondt and J. Nichols, eds., "Dispersal,", Oxford University Press, (2001). Google Scholar

[4]

C. Cortázar, J. Coville, M. Elgueta and S. Martinez, A nonlocal inhomogeneous dispersal process,, J. Differential Equations, 241 (2007), 332. doi: 10.1016/j.jde.2007.06.002. Google Scholar

[5]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Existence and asymptotic behavior of solutions to some inhomogeneous nonlocal diffusion problems,, SIAM J. Math. Anal., 41 (2009), 2136. doi: 10.1137/090751682. Google Scholar

[6]

C. Cosner, J. Dávila and S. Martínez, Evolutionary stability of ideal free nonlocal dispersal,, J. Biological Dynamics, 6 (2012), 395. Google Scholar

[7]

R. Cousens, C. Dytham and R. Law, "Dispersal in Plants: A Population Perspective,", Oxford University Press, (2008). Google Scholar

[8]

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

[9]

J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080. doi: 10.1016/j.jde.2007.11.002. Google Scholar

[10]

J. Coville, J. Dávila and S. Martínez, 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

[11]

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

[12]

W. Fagan and F. Lutscher, Average dispersal success: Linking home range, dispersal, and metapopulation dynamics to reserve design,, Ecol. Appl., 16 (2006), 820. Google Scholar

[13]

B. Gustafsson, H.-O. Kreiss and J. Oliger, "Time-Dependent Problems and Difference Methods,", Pure and Applied Mathematics (New York), (1995). Google Scholar

[14]

I. Hanski, "Metapopulation Ecology,", Oxford Univ. Press, (1999). Google Scholar

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theor. Pop. Biol., 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8. Google Scholar

[16]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Math., 840 (1981). Google Scholar

[17]

G. Hetzer, W. Shen and A. Zhang, Effects of spatial variations and dispersal strategies on principal eigenvalues of dispersal operators and spreading speeds of monostable equations,, Rocky Mountain Journal of Mathematics, (). Google Scholar

[18]

G. Hetzer, T. Nguyen and W. Shen, Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal,, Comm. Pure Appl. Anal., (). Google Scholar

[19]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221. doi: 10.1017/S0956792506006462. Google Scholar

[20]

V. Hutson, S. Martinez, 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

[21]

V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain Journal of Mathematics, 38 (2008), 1147. doi: 10.1216/RMJ-2008-38-4-1147. Google Scholar

[22]

Y. Jin and M. A. Lewis, Seasonal influences on population spread and persistence in streams: Critical domain size,, SIAM J. Appl. Math., 71 (2011), 1241. doi: 10.1137/100788033. Google Scholar

[23]

C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal,, Disc. Cont. Dynam. Sys. Series A, 26 (2010), 551. Google Scholar

[24]

R. W. van Kirk and M. A. Lewis, Integrodifference models for persistence in fragmented habitats,, Bulletin of Mathematical Biology, 59 (2007), 107. doi: 10.1007/BF02459473. Google Scholar

[25]

R. W. van Kirk and M. A. Lewis, Edgepermeability and population persistence in isolated habitat patches,, Natural Resources Modeling, 12 (2009), 37. Google Scholar

[26]

C. T. Lee, M. F. Hoopes, J. Diehl, W. Gilliland, G. Huxel, E. V. Leaver, K. McCain, J. Umbanhowar and A. Mogilner, Non-local concepts and models in biology,, J. Theor. Biol., 210 (2001), 201. doi: 10.1006/jtbi.2000.2287. Google Scholar

[27]

R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods,", SIAM Publications, (1998). Google Scholar

[28]

S. A. Levin, H. C. Muller-Landau, R. Nathan and J. Chave, The ecology and evolution of seed dispersal: A theoretical perspective,, Annu. Rev. Eco. Evol. Syst., 34 (2003), 575. doi: 10.1146/annurev.ecolsys.34.011802.132428. Google Scholar

[29]

W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real World Appl., 11 (2010), 2302. doi: 10.1016/j.nonrwa.2009.07.005. Google Scholar

[30]

G. Lindfield and J. Penny, "Numerical Methods Using MATLAB,", Prentice Hall, (2000). Google Scholar

[31]

F. Lutscher, Nonlocal dispersal and averaging in heterogeneous landscapes,, Applicable Analysis, 89 (2010), 1091. doi: 10.1080/00036811003735816. Google Scholar

[32]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Appl. Math., 65 (2005), 1305. doi: 10.1137/S0036139904440400. Google Scholar

[33]

T. Nagylaki, Clines with partial panmixia,, Theor. Popul. Biol., 81 (2012), 45. doi: 10.1016/j.tpb.2011.09.006. Google Scholar

[34]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar

[35]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 249 (2010), 747. doi: 10.1016/j.jde.2010.04.012. Google Scholar

[36]

W. Shen and A. Zhang, Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitat,, Proc. Amer. Math. Soc., 140 (2012), 1681. doi: 10.1090/S0002-9939-2011-11011-6. Google Scholar

[37]

N. Shigesada and K. Kawasaki, "Biological Invasions: Theory and Practice,", Oxford University Press, (1997). Google Scholar

[38]

J. Skellam, Random dispersal in theoretical populations,, Biometrika, 38 (1951), 196. doi: 10.1093/biomet/38.1-2.196. Google Scholar

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