On the dependence of population
size upon random dispersal rate

Song Liang; Yuan Lou

doi:10.3934/dcdsb.2012.17.2771

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

On the dependence of population size upon random dispersal rate

Song Liang ^1, and Yuan Lou ^2,

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

Abstract
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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.

Keywords: Dispersal rate, reaction-diffusion., invasion of species, population size.

Mathematics Subject Classification: Primary: 34D23; Secondary: 92D2.

Citation: Song Liang, Yuan Lou. On the dependence of population size upon random dispersal rate. Discrete & 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. Google Scholar
[2]	P. Amarasekare, Effect of non-random dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282-293. Google Scholar
[3]	I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). Google Scholar
[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. Google Scholar
[5]	A. Bezugly and Y. Lou, Reaction-diffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 983-1004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497-518. Google Scholar
[9]	R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. Google Scholar
[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. Google Scholar
[14]	W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173-211. Google Scholar
[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. Google Scholar
[16]	A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar
[17]	R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. Google Scholar
[18]	A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311-314. Google Scholar
[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. Google Scholar
[20]	K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110. Google Scholar
[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. Google Scholar
[22]	K.-Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). Google Scholar
[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. Google Scholar
[24]	K.-Y. Lam and W.-M. Ni, Dynamics of the diffusive Lotka-Volterra competition system,, in preparation., (). Google Scholar
[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., (). Google Scholar
[26]	S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113-132. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	A. W. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867-881. Google Scholar
[33]	Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. Google Scholar
[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. Google Scholar
[35]	Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. Google Scholar
[36]	J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003. Google Scholar
[37]	C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 1304-1314. Google Scholar
[38]	M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843-849. Google Scholar
[39]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. Google Scholar
[40]	W.-M. Ni, "The Mathematics of Diffusions," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

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. Google Scholar
[2]	P. Amarasekare, Effect of non-random dispersal strategies on spatial coexistence mechanisms, Journal of Animal Ecology, 79 (2010), 282-293. Google Scholar
[3]	I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, J. Biol. Dyn., (). Google Scholar
[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. Google Scholar
[5]	A. Bezugly and Y. Lou, Reaction-diffusion models with large advection coefficients, Applicable Analysis, 89 (2010), 983-1004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistenceof competing species, Proc. Roy. Soc. Edinb. Sect. A, 137 (2007), 497-518. Google Scholar
[9]	R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math. Bios. Eng., 7 (2010), 17-36. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	J. Clobert, E. Danchin, A. A. Dhondt and J. D. Nichols, eds., "Dispersal," Oxford University Press, Oxford, 2001. Google Scholar
[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. Google Scholar
[14]	W. Ding and S. Lenhart, Optimal harvesting of a spatially explicit fishery model, Natural Resource Modeling J., 22 (2009), 173-211. Google Scholar
[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. Google Scholar
[16]	A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. Google Scholar
[17]	R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. Google Scholar
[18]	A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 33 (1983), 311-314. Google Scholar
[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. Google Scholar
[20]	K. Kurata and J. P. Shi, Optimal spatial harvesting strategy and symmetry-breaking, Appl. Math. Optim., 58 (2008), 89-110. Google Scholar
[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. Google Scholar
[22]	K.-Y. Lam, Limiting profiles of semilinear elliptic equationswith large advection in population dynamics. II,, preprint., (). Google Scholar
[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. Google Scholar
[24]	K.-Y. Lam and W.-M. Ni, Dynamics of the diffusive Lotka-Volterra competition system,, in preparation., (). Google Scholar
[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., (). Google Scholar
[26]	S. Lenhart, S. Stojanovic and V. Protopopescu, A minimax problem for semilinear nonlocal competitive systems, Applied Math. Optim., 28 (1993), 113-132. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	A. W. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	F. Li and N. K. Yip, Long time behavior of some epidemic models, Dis. Cont. Dyn. Syst. Ser. B, 16 (2011), 867-881. Google Scholar
[33]	Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. Google Scholar
[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. Google Scholar
[35]	Y. Lou and T. Nagylaki, A semilinear parabolic system For migration and selection in population genetics, J. Diff. Eqs., 181 (2002), 388-418. Google Scholar
[36]	J. D. Murray, "Mathematical Biology. II. Spatial Models and Biomedical Applications," Third edition, Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, New York, 2003. Google Scholar
[37]	C. Neuhauser, Mathematical challenges in spatial ecology, Notices Amer. Math. Soc., 48 (2001), 1304-1314. Google Scholar
[38]	M. Neubert, Marine reserves and optimal harvesting, Ecol. Letters, 6 (2003), 843-849. Google Scholar
[39]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. Google Scholar
[40]	W.-M. Ni, "The Mathematics of Diffusions," CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, (2011), to appear. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations," Corrected reprint of the 1967 original, Springer-Verlag, New York, 1984. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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On the dependence of population size upon random dispersal rate

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