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December  2014, 19(10): 3319-3340. doi: 10.3934/dcdsb.2014.19.3319

Evolutionarily stable diffusive dispersal

Alex Potapov 1, , Ulrike E. Schlägel 1, and  Mark A. Lewis 2,

1. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada, Canada
2. 

Department of Mathematical and Statistical Sciences, Department of Biological Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada

Received  July 2013 Revised  December 2013 Published  October 2014

We use an evolutionary approach to find ``most appropriate'' dispersal models for ecological applications. From a random walk with locally or nonlocally defined transition probabilities we derive a family of diffusion equations. We assume a monotonic dependence of its diffusion coefficient on the local population fitness and search for a model within this class that can invade populations with other dispersal type from the same class but is not invadable itself. We propose an optimization technique using numerically obtained principal eigenvalue of the invasion problem and obtain two candidates for evolutionary stable dispersal strategy: Fokker-Planck equation with diffusion coefficient decreasing with fitness and Attractive Diffusion equation (Okubo and Levin, 2001) with diffusion coefficient increasing with fitness. For FP case the transition probabilities are defined by the departure point and for AD case by the destination point. We show that for the case of small spatial variability of the population growth rate both models are close to the model for ideal free distribution by Cantrell et al. (2008).
Keywords: evolutionarily stable strategy, Diffusion, ideal-free distribution., random walk, fitness.
Mathematics Subject Classification: Primary: 92D40; Secondary: 92D5.
Citation: Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319
References:
[1]

P. A. Abrams and L. Ruokolainen, How does adaptive consumer movement affect population dynamics in consumer-resource metacommunities with homogeneous patches?, J. Theor. Biol., 277 (2011), 99-110. doi: 10.1016/j.jtbi.2011.02.019.

[2]

D. G. Aronson, The role of diffusion in mathematical biology: Skellam revisited, in Mathematics in Biology and Medicine (eds. V. Capasso, E. Grosso, S.L. Paaveri-Fontana) Springer, Berlin, (1985), 2-6. doi: 10.1007/978-3-642-93287-8_1.

[3]

J. E. Brittain and T. J. Eikeland, Invertebrate drift - a review, Hydrobiologia, 166 (1988), 77-93. doi: 10.1007/BF00017485.

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, The Atrium, Southern Gate, 2003. doi: 10.1002/0470871296.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024.

[6]

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. doi: 10.1007/s002850050120.

[7]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds I. Theoretical development, Acta Biotheoretica, 19 (1969), 16-36. doi: 10.1007/BF01601953.

[8]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[9]

V. Křivan, R. Cressman and C. Schneider, The ideal free distribution: A review and synthesis of the game-theoretic perspective, Theor. Population Biol., 73 (2008), 403-425.

[10]

Y. Lou, Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, in Tutorials in Mathematical Biosciences IV Lecture Notes in Mathematics Vol. 1922, Springer, Berlin Heidelberg (2008), 171-205. doi: 10.1007/978-3-540-74331-6_5.

[11]

D. W. Morris, Adaptation and habitat selection in the eco-evolutionary process, Proc. Roy. Soc. B, 278 (2011), 2401-2411. doi: 10.1098/rspb.2011.0604.

[12]

D. W. Morris and P. Lundberg, Pillars of Evolution, Oxford Univ. Press, Oxford, 2011. doi: 10.1093/acprof:oso/9780198568797.001.0001.

[13]

L. Ni, A Perron type theorem on the principal eigenvalue of nonsymmetric elliptic operators,, to appear in American Mathematical Monthly. Avalable online at URL: , (): 1210. 

[14]

A. Okubo and S. Levin, Diffusion and Ecological Problems, Springer, NY, 2001. doi: 10.1007/978-1-4757-4978-6.

[15]

O. Ovaskainen and S. J. Cornell, Biased Movement at a Boundary and Conditional Occupancy Times for Diffusion Processes, J. Appl. Prob., 40 (2003), 557-580. doi: 10.1239/jap/1059060888.

[16]

A. Potapov, Stochastic model of lake system invasion and its optimal control: neurodynamic programming as a solution method, Nat. Res. Mod., 22 (2009), 257-288. doi: 10.1111/j.1939-7445.2008.00036.x.

[17]

R. Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2007. ISBN 3-900051-07-0, http://www.R-project.org.

[18]

H. E. Romeijn and R. L. Smith, Simulated annealing for constrained global optimization, J. Global Optimization, 5 (1994), 101-126. doi: 10.1007/BF01100688.

[19]

P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, MA., 1998.

show all references

References:
[1]

P. A. Abrams and L. Ruokolainen, How does adaptive consumer movement affect population dynamics in consumer-resource metacommunities with homogeneous patches?, J. Theor. Biol., 277 (2011), 99-110. doi: 10.1016/j.jtbi.2011.02.019.

[2]

D. G. Aronson, The role of diffusion in mathematical biology: Skellam revisited, in Mathematics in Biology and Medicine (eds. V. Capasso, E. Grosso, S.L. Paaveri-Fontana) Springer, Berlin, (1985), 2-6. doi: 10.1007/978-3-642-93287-8_1.

[3]

J. E. Brittain and T. J. Eikeland, Invertebrate drift - a review, Hydrobiologia, 166 (1988), 77-93. doi: 10.1007/BF00017485.

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley, The Atrium, Southern Gate, 2003. doi: 10.1002/0470871296.

[5]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024.

[6]

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. doi: 10.1007/s002850050120.

[7]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds I. Theoretical development, Acta Biotheoretica, 19 (1969), 16-36. doi: 10.1007/BF01601953.

[8]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge Univ. Press, Cambridge, 1998. doi: 10.1017/CBO9781139173179.

[9]

V. Křivan, R. Cressman and C. Schneider, The ideal free distribution: A review and synthesis of the game-theoretic perspective, Theor. Population Biol., 73 (2008), 403-425.

[10]

Y. Lou, Some Challenging Mathematical Problems in Evolution of Dispersal and Population Dynamics, in Tutorials in Mathematical Biosciences IV Lecture Notes in Mathematics Vol. 1922, Springer, Berlin Heidelberg (2008), 171-205. doi: 10.1007/978-3-540-74331-6_5.

[11]

D. W. Morris, Adaptation and habitat selection in the eco-evolutionary process, Proc. Roy. Soc. B, 278 (2011), 2401-2411. doi: 10.1098/rspb.2011.0604.

[12]

D. W. Morris and P. Lundberg, Pillars of Evolution, Oxford Univ. Press, Oxford, 2011. doi: 10.1093/acprof:oso/9780198568797.001.0001.

[13]

L. Ni, A Perron type theorem on the principal eigenvalue of nonsymmetric elliptic operators,, to appear in American Mathematical Monthly. Avalable online at URL: , (): 1210. 

[14]

A. Okubo and S. Levin, Diffusion and Ecological Problems, Springer, NY, 2001. doi: 10.1007/978-1-4757-4978-6.

[15]

O. Ovaskainen and S. J. Cornell, Biased Movement at a Boundary and Conditional Occupancy Times for Diffusion Processes, J. Appl. Prob., 40 (2003), 557-580. doi: 10.1239/jap/1059060888.

[16]

A. Potapov, Stochastic model of lake system invasion and its optimal control: neurodynamic programming as a solution method, Nat. Res. Mod., 22 (2009), 257-288. doi: 10.1111/j.1939-7445.2008.00036.x.

[17]

R. Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2007. ISBN 3-900051-07-0, http://www.R-project.org.

[18]

H. E. Romeijn and R. L. Smith, Simulated annealing for constrained global optimization, J. Global Optimization, 5 (1994), 101-126. doi: 10.1007/BF01100688.

[19]

P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, MA., 1998.

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