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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 & 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.  doi: 10.1080/17513758.2010.529169.  Google Scholar

[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.  doi: 10.1080/17513750701450227.  Google Scholar

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species,, Proc. Roy. Soc. Edinb., 137A (2007), 497.  doi: 10.1017/S0308210506000047.  Google Scholar

[4]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math Bios. Eng., 7 (2010), 17.  doi: 10.3934/mbe.2010.7.17.  Google Scholar

[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.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

E. N. Dancer, Positivity of maps and applications,, in Topological nonlinear analysis, 15 (1995), 303.   Google Scholar

[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.   Google Scholar

[8]

H. Dreisig, Ideal free distributions of nectar foraging bumblebees,, Oikos, 72 (1995), 161.  doi: 10.2307/3546218.  Google Scholar

[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.   Google Scholar

[10]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order,, 2nd Ed., (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[11]

T. Grand, Foraging site selection by juvenile coho salmon: Ideal free distribution with unequal competitors,, Animal Behavior, 53 (1997), 185.   Google Scholar

[12]

P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991).   Google Scholar

[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.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[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.  doi: 10.2307/3545322.  Google Scholar

[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.  Google Scholar

[16]

K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in poulation dynamics II,, SIAM J. Math. Anal., 44 (2012), 1808.  doi: 10.1137/100819758.  Google Scholar

[17]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[19]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645.   Google Scholar

[20]

M. A. McPeek and R. D. Holt, The evolution fo dispersal in spatially and temporally varying environments,, The American Naturalist, 140 (1997), 1010.   Google Scholar

[21]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks,, Zeitschrift für Tierpsychologie, 51 (1979), 36.  doi: 10.1111/j.1439-0310.1979.tb00669.x.  Google Scholar

[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.   Google Scholar

[23]

D. Munther, The ideal free strategy with weak Allee effect,, J. Differential Equations, 254 (2013), 1728.  doi: 10.1016/j.jde.2012.11.010.  Google Scholar

[24]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979.   Google Scholar

[25]

J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect,, J. Math. Biol., 52 (2006), 807.  doi: 10.1007/s00285-006-0373-7.  Google Scholar

[26]

H. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs 41. American Mathematical Society, (1995).   Google Scholar

show all references

References:
[1]

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

[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.  doi: 10.1080/17513750701450227.  Google Scholar

[3]

R. S. Cantrell, C. Cosner and Y. Lou, Advection mediated coexistence of competing species,, Proc. Roy. Soc. Edinb., 137A (2007), 497.  doi: 10.1017/S0308210506000047.  Google Scholar

[4]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution,, Math Bios. Eng., 7 (2010), 17.  doi: 10.3934/mbe.2010.7.17.  Google Scholar

[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.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

E. N. Dancer, Positivity of maps and applications,, in Topological nonlinear analysis, 15 (1995), 303.   Google Scholar

[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.   Google Scholar

[8]

H. Dreisig, Ideal free distributions of nectar foraging bumblebees,, Oikos, 72 (1995), 161.  doi: 10.2307/3546218.  Google Scholar

[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.   Google Scholar

[10]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equation of Second Order,, 2nd Ed., (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar

[11]

T. Grand, Foraging site selection by juvenile coho salmon: Ideal free distribution with unequal competitors,, Animal Behavior, 53 (1997), 185.   Google Scholar

[12]

P. Hess, Periodic Parabolic Boundary Value Problems and Positivity,, Longman Scientific & Technical, (1991).   Google Scholar

[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.  doi: 10.1090/S0002-9947-96-01724-2.  Google Scholar

[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.  doi: 10.2307/3545322.  Google Scholar

[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.  Google Scholar

[16]

K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in poulation dynamics II,, SIAM J. Math. Anal., 44 (2012), 1808.  doi: 10.1137/100819758.  Google Scholar

[17]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion,, J. Differential Equations, 131 (1996), 79.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[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.  doi: 10.3934/dcds.2010.27.643.  Google Scholar

[19]

H. Matano, Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems,, J. Fac. Sci. Univ. Tokyo, 30 (1984), 645.   Google Scholar

[20]

M. A. McPeek and R. D. Holt, The evolution fo dispersal in spatially and temporally varying environments,, The American Naturalist, 140 (1997), 1010.   Google Scholar

[21]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks,, Zeitschrift für Tierpsychologie, 51 (1979), 36.  doi: 10.1111/j.1439-0310.1979.tb00669.x.  Google Scholar

[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.   Google Scholar

[23]

D. Munther, The ideal free strategy with weak Allee effect,, J. Differential Equations, 254 (2013), 1728.  doi: 10.1016/j.jde.2012.11.010.  Google Scholar

[24]

D. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems,, Indiana Univ. Math. J., 21 (): 979.   Google Scholar

[25]

J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect,, J. Math. Biol., 52 (2006), 807.  doi: 10.1007/s00285-006-0373-7.  Google Scholar

[26]

H. Smith, Monotone Dynamical Systems,, Mathematical Surveys and Monographs 41. American Mathematical Society, (1995).   Google Scholar

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