• Previous Article
    On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems
  • DCDS-B Home
  • This Issue
  • Next Article
    Persistence and extinction of diffusing populations with two sexes and short reproductive season
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

[1]

Bo Duan, Zhengce Zhang. A two-species weak competition system of reaction-diffusion-advection with double free boundaries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 801-829. doi: 10.3934/dcdsb.2018208

[2]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

[3]

Robert Stephen Cantrell, Chris Cosner, Yuan Lou. Evolution of dispersal and the ideal free distribution. Mathematical Biosciences & Engineering, 2010, 7 (1) : 17-36. doi: 10.3934/mbe.2010.7.17

[4]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989

[5]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841

[6]

Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733

[7]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[8]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[9]

Linfeng Mei, Xiaoyan Zhang. On a nonlocal reaction-diffusion-advection system modeling phytoplankton growth with light and nutrients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 221-243. doi: 10.3934/dcdsb.2012.17.221

[10]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176

[11]

Zhiguo Wang, Hua Nie, Yihong Du. Asymptotic spreading speed for the weak competition system with a free boundary. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5223-5262. doi: 10.3934/dcds.2019213

[12]

Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060

[13]

Benlong Xu, Hongyan Jiang. Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4255-4266. doi: 10.3934/dcdsb.2018136

[14]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 859-875. doi: 10.3934/dcdsb.2019193

[15]

Nancy Azer, P. van den Driessche. Competition and Dispersal Delays in Patchy Environments. Mathematical Biosciences & Engineering, 2006, 3 (2) : 283-296. doi: 10.3934/mbe.2006.3.283

[16]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

[17]

Chang-Hong Wu. Spreading speed and traveling waves for a two-species weak competition system with free boundary. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2441-2455. doi: 10.3934/dcdsb.2013.18.2441

[18]

Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139

[19]

Shi-Liang Wu, Wan-Tong Li, San-Yang Liu. Exponential stability of traveling fronts in monostable reaction-advection-diffusion equations with non-local delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 347-366. doi: 10.3934/dcdsb.2012.17.347

[20]

Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]