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December  2014, 19(10): 3219-3244. doi: 10.3934/dcdsb.2014.19.3219

## Invading the ideal free distribution

 1 Department of Mathematics, Ohio State University, Columbus, OH 43210, United States 2 Department of Mathematics, Cleveland State University, Cleveland, OH 44115, United States

Received  August 2013 Revised  September 2013 Published  October 2014

Recently, the ideal free dispersal strategy has been proven to be evolutionarily stable in the spatially discrete as well as continuous setting. That is, at equilibrium a species adopting the strategy is immune against invasion by any species carrying a different dispersal strategy, other conditions being held equal. In this paper, we consider a two-species competition model where one of the species adopts an ideal free dispersal strategy, but is penalized by a weak Allee effect. We will show rigorously in this case that the ideal free disperser is invasible by a range of non-ideal free strategies, illustrating the trade-off between the advantage of being an ideal free disperser and the setback caused by the weak Allee effect. Moreover, an integral criterion is given to determine the stability/instability of one of the semi-trivial steady states, which is always linearly neutrally stable due to the degeneracy caused by the weak Allee effect.
Citation: King-Yeung Lam, Daniel Munther. Invading the ideal free distribution. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3219-3244. doi: 10.3934/dcdsb.2014.19.3219
##### References:
 [1] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130. doi: 10.1080/17513758.2010.529169. [2] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271. doi: 10.1080/17513750701450227. [3] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb., 137A (2007), 497-518. doi: 10.1017/S0308210506000047. [4] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math Bios. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [5] X. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Cont. Dyn. Sys., 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841. [6] E. N. Dancer, Positivity of maps and applications, in Topological nonlinear analysis, Nonlinear Differential Equations Appl., (eds. Matzeu and Vignoli), Birkhauser, Boston, 15 1995, 303-340. [7] C. P. Doncaster, et al., Balanced dispersal between spatially varying local populations: an alternative to the source-sink model, The American Naturalist, 150 (1997), 425-445. [8] H. Dreisig, Ideal free distributions of nectar foraging bumblebees, Oikos, 72 (1995), 161-172. doi: 10.2307/3546218. [9] S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Theoretical development, Acta Biotheor., 19 (1970), 16-36. [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, 2nd Ed., Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [11] T. Grand, Foraging site selection by juvenile coho salmon: Ideal free distribution with unequal competitors, Animal Behavior, 53 (1997), 185-196. [12] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991. [13] S.-B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2. [14] M. Kennedy and R. D. Gray, Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution, Oikos, 68 (1993), 158-166. doi: 10.2307/3545322. [15] L. Korobenko and E. Braverman, On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations, J. Math. Biol. (to appear). doi: 10.1007/s00285-013-0729-8. [16] K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in poulation dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830. doi: 10.1137/100819758. [17] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. [18] Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655. doi: 10.3934/dcds.2010.27.643. [19] H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645-673. [20] M. A. McPeek and R. D. Holt, The evolution fo dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1997), 1010-1027. [21] M. Milinski, An evolutionarily stable feeding strategy in sticklebacks, Zeitschrift für Tierpsychologie, 51 (1979), 36-40. doi: 10.1111/j.1439-0310.1979.tb00669.x. [22] D. W. Morris, J. E. Diffendorfer and P. Lundberg, Dispersal among habitats varying in fitness: Reciprocating migration through ideal habitat selection, Oikos, 107 (2004), 559-575. [23] D. Munther, The ideal free strategy with weak Allee effect, J. Differential Equations, 254 (2013), 1728-1740. doi: 10.1016/j.jde.2012.11.010. [24] D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000. [25] J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829. doi: 10.1007/s00285-006-0373-7. [26] H. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs 41. American Mathematical Society, Providence, Rhode Island, U.S.A., 1995.

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##### References:
 [1] I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130. doi: 10.1080/17513758.2010.529169. [2] R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padrón, The ideal free distribution as an evolutionarily stable strategy, J. Biol. Dyn., 1 (2007), 249-271. doi: 10.1080/17513750701450227. [3] R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species, Proc. Roy. Soc. Edinb., 137A (2007), 497-518. doi: 10.1017/S0308210506000047. [4] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math Bios. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17. [5] X. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Cont. Dyn. Sys., 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841. [6] E. N. Dancer, Positivity of maps and applications, in Topological nonlinear analysis, Nonlinear Differential Equations Appl., (eds. Matzeu and Vignoli), Birkhauser, Boston, 15 1995, 303-340. [7] C. P. Doncaster, et al., Balanced dispersal between spatially varying local populations: an alternative to the source-sink model, The American Naturalist, 150 (1997), 425-445. [8] H. Dreisig, Ideal free distributions of nectar foraging bumblebees, Oikos, 72 (1995), 161-172. doi: 10.2307/3546218. [9] S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat selection in birds, Theoretical development, Acta Biotheor., 19 (1970), 16-36. [10] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order, 2nd Ed., Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0. [11] T. Grand, Foraging site selection by juvenile coho salmon: Ideal free distribution with unequal competitors, Animal Behavior, 53 (1997), 185-196. [12] P. Hess, Periodic Parabolic Boundary Value Problems and Positivity, Longman Scientific & Technical, Harlow, UK, 1991. [13] S.-B. Hsu, H. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2. [14] M. Kennedy and R. D. Gray, Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution, Oikos, 68 (1993), 158-166. doi: 10.2307/3545322. [15] L. Korobenko and E. Braverman, On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations, J. Math. Biol. (to appear). doi: 10.1007/s00285-013-0729-8. [16] K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in poulation dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830. doi: 10.1137/100819758. [17] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157. [18] Y. Lou, W.-M. Ni and L. Su, An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655. doi: 10.3934/dcds.2010.27.643. [19] H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645-673. [20] M. A. McPeek and R. D. Holt, The evolution fo dispersal in spatially and temporally varying environments, The American Naturalist, 140 (1997), 1010-1027. [21] M. Milinski, An evolutionarily stable feeding strategy in sticklebacks, Zeitschrift für Tierpsychologie, 51 (1979), 36-40. doi: 10.1111/j.1439-0310.1979.tb00669.x. [22] D. W. Morris, J. E. Diffendorfer and P. Lundberg, Dispersal among habitats varying in fitness: Reciprocating migration through ideal habitat selection, Oikos, 107 (2004), 559-575. [23] D. Munther, The ideal free strategy with weak Allee effect, J. Differential Equations, 254 (2013), 1728-1740. doi: 10.1016/j.jde.2012.11.010. [24] D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000. [25] J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829. doi: 10.1007/s00285-006-0373-7. [26] H. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs 41. American Mathematical Society, Providence, Rhode Island, U.S.A., 1995.
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