American Institute of Mathematical Sciences

November  2012, 17(8): 2771-2788. doi: 10.3934/dcdsb.2012.17.2771

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

Received  March 2011 Revised  August 2011 Published  July 2012

This paper concerns the dependence of the population size for a single species on its random dispersal rate and its applications on the invasion of species. The population size of a single species often depends on its random dispersal rate in non-trivial manners. Previous results show that the population size is usually not a monotone function of the random dispersal rate. We construct some examples to illustrate that the population size, as a function of the random dispersal rate, can have at least two local maxima. As an application we illustrate that the invasion of exotic species depends upon the random dispersal rate of the resident species in complicated manners. Previous results show that the total population is maximized at some intermediate random dispersal rate for several classes of local intrinsic growth rates. We find one family of local intrinsic growth rates such that the total population is maximized exactly at zero random dispersal rate. We show that the population distribution becomes flatter in average if we increase the random dispersal rate, and the environmental gradient is always steeper than the population distribution, at least in some average sense. We also discuss the dependence of the population size on movement rates in other contexts and propose some open problems.
Citation: Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2771-2788. doi: 10.3934/dcdsb.2012.17.2771
References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profile of the steady states for an SIS epidemic reaction-diffusion model, Discre. Cont. Dyn. Sys., 21 (2008), 1-20. [2] P. Amarasekare, Effect of non-random dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282-293. [3] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). [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), 379-397. [5] A. Bezugly and Y. Lou, Reaction-diffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 983-1004. [6] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. [7] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199-214. [8] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497-518. [9] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. [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), 627-658. [11] X. F. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. [12] J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. [13] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688-704. [14] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173-211. [15] 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. [16] A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [17] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. [18] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311-314. [19] W. Z. Huang, M. A. Han and K. Y. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Bios. Eng., 7 (2010), 51-66. [20] K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110. [21] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161-181. [22] K.-Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). [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), 1051-1067. [24] K.-Y. Lam and W.-M. Ni, Dynamics of the diffusive Lotka-Volterra competition system,, in preparation., (). [25] J. Langebrake, L. Riotte-Lambert, 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., (). [26] S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113-132. [27] S. Lenhart, S. Stojanovic and V. Protopopescu, A two-sided 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), 135-152. [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. [29] A. W. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619. [30] 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), 575-604. [31] F. Li, L. P Wang and Y. Wang, On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Dis. Cont. Dyn. Syst. Ser. B, 15 (2011), 669-686. [32] F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867-881. [33] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. [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), 171-205. [35] Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. [36] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003. [37] C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 1304-1314. [38] M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843-849. [39] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [40] W.-M. Ni, "The Mathematics of Diffusions," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. [41] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. [42] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction diffusion model. I, J. Diff. Eqs., 247 (2009), 1096-1119. [43] R. Peng and S. Q. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonl. Anal. TMA, 71 (2009), 239-247. [44] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [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. [46] X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility andchemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. [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), 505-531.

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References:
 [1] L. Allen, B. Bolker, Y. Lou and A. Nevai, Asymptotic profile of the steady states for an SIS epidemic reaction-diffusion model, Discre. Cont. Dyn. Sys., 21 (2008), 1-20. [2] P. Amarasekare, Effect of non-random dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282-293. [3] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). [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), 379-397. [5] A. Bezugly and Y. Lou, Reaction-diffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 983-1004. [6] R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, UK, 2003. [7] R. S. Cantrell, C. Cosner and Y. Lou, Movement towards better environments and the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199-214. [8] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497-518. [9] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. [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), 627-658. [11] X. F. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. [12] J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. [13] W. Ding, H. Finotti, S. Lenhart, Y. Lou and Q. Ye, Optimal control of growth coefficient on a steady-state population model, Nonlinear Analysis: Real World Applications, 11 (2010), 688-704. [14] W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173-211. [15] 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. [16] A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. [17] R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. [18] A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311-314. [19] W. Z. Huang, M. A. Han and K. Y. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Math. Bios. Eng., 7 (2010), 51-66. [20] K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110. [21] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Diff. Eqs., 250 (2011), 161-181. [22] K.-Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). [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), 1051-1067. [24] K.-Y. Lam and W.-M. Ni, Dynamics of the diffusive Lotka-Volterra competition system,, in preparation., (). [25] J. Langebrake, L. Riotte-Lambert, 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., (). [26] S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113-132. [27] S. Lenhart, S. Stojanovic and V. Protopopescu, A two-sided 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), 135-152. [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. [29] A. W. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619. [30] 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), 575-604. [31] F. Li, L. P Wang and Y. Wang, On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Dis. Cont. Dyn. Syst. Ser. B, 15 (2011), 669-686. [32] F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867-881. [33] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. [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), 171-205. [35] Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. [36] J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003. [37] C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 1304-1314. [38] M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843-849. [39] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [40] W.-M. Ni, "The Mathematics of Diffusions," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. [41] A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives," Second edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. [42] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction diffusion model. I, J. Diff. Eqs., 247 (2009), 1096-1119. [43] R. Peng and S. Q. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonl. Anal. TMA, 71 (2009), 239-247. [44] M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. [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. [46] X. F. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility andchemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560. [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), 505-531.
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