Física Estatística e o Andar dos Animais

Transcription

1 Outline Física Estatística e o Andar dos Animais Gandhimohan M. Viswanathan Universidade Federal de Alagoas 15 Encontro Sergipano de Física Julho de 2010

2 Outline Lévy Walks in a biological context Refs: M. F. Shlesinger and J. Klafter, in On Growth and Form, edited by H. E. Stanley and N. Ostrowsky (Nijhoff, Dordrecht, 1986), p M. Levandowsky, J. Klafter and B. S. White, Bull. Marine Sci. 43, 758 (1988). B. J. Cole, Anim. Behav. 50, 1317 (1995). F. L. Schuster and M. Levandowsky, J. Euk. Microbio. 43, 150 (1996). M. Levandowsky, B. S. White and F. L. Schuster, Acta Protozoologica 36, 237 (1997).

3 1. G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, H. E. Stanley, of wandering albatrosses, Nature 381, (1996). 2. G. M. Viswanathan, S. V. Buldyrev, S. Havlin, M. G. da Luz, E. Raposo, and H. E. Stanley, Optimizing the Success of Random Searches, Nature 401, (1999). 3. F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra, G. M. Viswanathan, Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies, Physical Review Letters 88, (2002); 89, (2002). 4. F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra, G. M. Viswanathan, Erratum: Optimizing the Encounter Rate in Biological Interactions: Lévy versus Brownian Strategies [Phys. Rev. Lett. 88, (2002)], Physical Review Letters 5. E. P. Raposo, S. V. Buldyrev, M. G. E. da Luz, M. C. Santos, H. E. Stanley and G. M. Viswanathan, Dynamical Robustness of Lévy Search Strategies, Physical Review Letters 91, (2003). 6. M. C. Santos, E. P. Raposo, G. M. Viswanathan and M. G. E. da Luz, Optimal random searches of revisitable targets: Crossover from superdiffusive to ballistic random walks, Europhys. Lett., 67 (5), (2004). 7. Frederic Bartumeus, M.G.E. da Luz, G.M. Viswanathan and Jordi Catalan, Animal search strategies: a quantitative random walk analysis, Ecology 86 (11), (2005). 8. G. M. Viswanathan, E. P. Raposo, F. Bartumeus, J. Catalan, M. G. E. da Luz, Necessary criterion for distinguishing true superdiffusion from correlated random walk processes, Phys. Rev. E 72, (2005). 9. M. C. Santos, G. M. Viswanathan, E. P. Raposo, and M. G. E. da Luz, Optimization of random searches on regular lattices, Phys. Rev. E 72, (2005). 10. J. C. Cressoni, M. A. A. da Silva, and G.M. Viswanathan, Amnestically Induced Persistence in Random Walks, Phys. Rev. Lett. 98, (2007). 12. C. L. Faustino, L. R. da Silva, M. G. E. da Luz, E. P. Raposo and G. M. Viswanathan, Search dynamics at the edge of extinction: Anomalous diffusion as a critical survival state Europhys. Lett (2007). 13. M. C. Santos, D. Boyer, O. Miramontes, G. M. Viswanathan, E. P. Raposo, J. L. Mateos and M. G. E. da Luz, Origin of power-law distributions in deterministic walks: The influence of landscape geometry, Phys. Rev. E 75, (2007). 14. A. M. Edwards, R. A. Phillips, N. W. Watkins, M. P. Freeman, E J. Murphy, V. Afanasyev, S. V. Buldyrev, M. G. E. da Luz, E. P. Raposo, H. E. Stanley and G. M. Viswanathan, Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and deer, Nature 449, (2007). 15. G.M. Viswanathan, E.P. Raposo and M.G.E. da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches [review article], Physics of Life Reviews 5, (2008). 16. F. Bartumeus, J. Catalan, G. M. Viswanathan, E. P. Raposo and M.G.E. da Luz The influence of turning angles on the success of non-oriented animal searches, Journal of Theoretical Biology 252, (2008). 17. E. P. Raposo, S. V. Buldyrev, M. G. E. da Luz, G. M. Viswanathan and H. E. Stanley, Lévy flights and random searches, J. Phys A 42, (2009). 18. M. C. Santos, E. P. Raposo, G. M. Viswanathan and M. G. E. da Luz, Can collective searches profit from Lévy walk strategies? J. Phys. A 42, (2009). 19. Marcos G. E. da Luz, Alexander Grosberg, Ernesto P. Raposo and G. M. Viswanathan (Guest Editors), The random search problem: trends and perspectives, J. Phys. A 42, (2009). 20. G. M. Viswanathan, Fish in Lévy flight foraging, Nature (2010).

4 Cambridge Univ. Press (2010)

5 Outline Outline 1 Problem and Movitation 2 3 4

6 Problem and Movitation Why study this problem Why study this problem? Problem: How to describe quantitatively random search processes (e.g., biological foraging)? Motivation: Eating and mating: search processes ubiquitous What is the optimal search strategy? Technological applications Abundance of experimental data Complexity: free will vs. physical/biological constraints Neither 100% deterministic nor 100% stochastic

7 Problem and Movitation Diomedea exulans : Wingspan of 3.3 m Weigh 6 12 kg Flies 4000 km in 8 days (500 km /day) Eat squid, small fish Grounded when eat too much! Easily circumambulates the earth at low latitudes

8 Problem and Movitation Finding: non-gaussian behavior 3.0 Data: Time intervals spent outside H20 (Flying) by albatrosses. Finding: Histogram N(t) of flight times follows an inverse square power law. Hypothesis: Lévy flights. Test: look at other animals log 10 N(t) µ=2 (b) log 10 t Nature 381, 413 (1996).

9 Problem and Movitation Examples of LF patterns µ=2.5 µ=2.0 µ=1.5 P(l) l µ, l > l 0

10 Problem and Movitation Why bother with Lévy flights? Brownian walk Lévy flight Brownian walker returns many times to the same place Find virgin territory. LF reduce oversampling.

11 Problem and Movitation Lévy flyers fly very far (a) Short range dispersal (b) Long range dispersal How far is far?

12 Problem and Movitation Lévy flights scale-free w/ characteristic scale scale free f(x) f(x) = exp ( -a x ) g(x) g(x) = x -b g(λx) = g(x) λ -b (a) (b) x x log f log g log f log g (c) (d) x log x

13 Problem and Movitation Lévy flight exponents µ=1 µ 1 Ballistic limit log P(l) µ>3 Brownian regime µ=3 1>µ>3 Levy flights µ=3 Brownian+log correction log l LF superdiffusive

14 Problem and Movitation Lévy flight have Hurst exponents H > 1/2 Slope defines the Hurst exponent H log <x 2 > Slope = 2H log t Superdiffusion: superlinear growth of MSD.

15 Problem and Movitation Lévy flight have Hurst exponents H > 1/2 log <x 2 > H>1 Super-ballistic regime H=1 Ballistic limit H>1/2 Superdiffusion H=1/2 Normal diffusion H<1/2 Subdiffusion 0 H=0 Confinement and localization log t Superdiffusion: superlinear growth of MSD.

16 Problem and Movitation Question: Why would animals follow LW? Answer: study a model of animal locomotion (a) (b) (A) if target within radius of vision, go straight to target (B) otherwise pick random orientation and distance from a power law distribution and move incrementally 2 r v looking for targets. If no target is found, pick new orientation and distance, else go to (A). l j

17 Analytical solution Problem and Movitation P(l) l µ l = η = λ r v dx x 1 µ + λ λ x µ dx r v x µ dx ( ) ( ) µ 1 λ 2 µ rv 2 µ 2 µ rv 1 µ + λ2 µ rv 1 µ 1 N l N d (λ/r v ) µ 1 (5) N n (λ/r v ) (µ 1)/2 (6) µ opt = 2 δ (1) (2) (3) (4) (7)

18 Simulation results Problem and Movitation Analytical (a) λ=10 λ=10 2 λ=10 3 λ=10 4 Simulations Nature 401, 911 (1999). λη (b) 1 D µ Conclusion: LW optimize random nondestructive searches

19 Problem and Movitation The ultra-long flights A closer, more careful, re-examination wet records per 10 min intervals Wandering albatross (chick rearing) Wandering albatross (brood guard) time [days] Notice the ultra-long initial flights! (Sergey Buldyrev) New Test: Filter out the initial and final flights.

20 Problem and Movitation Re-analysis of albatross data New PTT Data: confirms spurious nature of ultra-long flights Finding: Filtering eliminates the long tails Question: What about other studies? N(t) complete data filtered data complete data log binned filtered data log binned µ= t [1]. Nature 449, 1044 (2007). doi: /nature06199 [2]. This figure unpublished.

21 Other organisms Problem and Movitation Reindeer & Jackals Dinoflagellates & Spider monkeys Sharks & Bees Very recent results Reindeer: Semidomesticated female reindeer (Rangifer tarandus tarandus L.), found discrepancies between the actual reindeer foraging paths and a (non-lévy) CRW model. A. Mårell, J. P. Ball and A. Hofgaard, Canadian J. of Zoology 80, 854 (2002). Jackals: radio-tracking of trajectories of a species of African side-striped jackal. Evidence for superdiffusion, complexity of actual foraging trajectories, which include curvature. R. P. D. Atkinson, C. J. Rhodes, D. W. Macdonald, and R. M. Anderson, Oikos 98, 134 (2002).

22 Other organisms Problem and Movitation Reindeer & Jackals Dinoflagellates & Spider monkeys Sharks & Bees Very recent results Dinoflagellates: Oxyrrhis marina. Distribution of flight times switched from an exponential to an inverse square power-law distribution (µ = 2) when the prey (Rhodomonas sp.) decreased in abundance. F. Bartumeus, F. Peters, S. Pueyo, C. Marrase, and J. Catalan, Proc. Natl. Acad. Sci. (USA) 100, (2003). Spider monkeys: Ateles geoffroyi, in the forests of the Yucatan Peninsula. Power law tailed distribution of steps, consistent with LW. G. Ramos-Fernandez, J. L. Mateos, O. Miramontes, G. Cocho, H. Larralde, and B. Ayala-Orozco, Behav. Ecol. Sociobiol. 55, 223 (2004). D. Boyer, O. Miramontes, G. Ramos-Fernandez, J. L. Mateos, and G. Cocho, Physica A 342, 329 (2004).

23 Other organisms Problem and Movitation Reindeer & Jackals Dinoflagellates & Spider monkeys Sharks & Bees Very recent results Sharks Basking sharks may use search tactics structured across multiple scales. D. W. Sims, M. J. Witt, A. J. Richardson, E. J. Southall, and J. D. Metcalfe, Proc. Royal Soc. B273, 1195 (2006). Fruit flies: Trajectories in air µ 2 for PDF of distances. Also: evidence of intermittent searches. Perhaps 1st study of intermittent AND scale-free search? A. M. Reynolds and M. A. Frye, PLoS One 4 e354 (2007). Honey bees: Harmonic radar to record the flight paths of honey bees searching for their hives. Finding: scale invariant walks with a µ = 2. A. M. Reynolds, A. D. Smith, R. Menzel, U. Greggers, D. R. Reynolds and J. R. Riley, Ecology 88, 1955 (2007). A. M. Reynolds, A. D. Smith, D. R. Reynolds, N. L. Carreck and J. L. Osborne, J. Experimental Biology 210, 3763 (2007).

24 Very recent results Problem and Movitation Reindeer & Jackals Dinoflagellates & Spider monkeys Sharks & Bees Very recent results Sharks, bony fish, sea turtles and penguins: Analysis of data sets comprising more than 10 7 data points. Rank-frequency plots Comparison of different models Answers criticisms of double log plots Impressive amount of data Convincingly shows Lévy behavior D.W. Sims et al. Nature 451, 1098 (2008). N. E. Humphries et al. Nature 465, 1066 (2010). G. M. Viswanathan, Nature (2010). Strong support for Lévy Flight hypothesis!

25 Discussion Problem and Movitation Discussion Conclusions Small range of scaling ( 3 decades) Log-log plots over 1 2 decades Superdiffusion LW (e.g., fbm) Locally SD, globally diffusive (e.g., Markovian CRW)

26 Discussion Real trajectories curved Problem and Movitation Discussion Conclusions (a) rw model 1 rw model 2 (b) rw model 1 rw model 2 Both models have same distribution of turning angles, but one model is Markovian, other non-markovian, generated from LW skeleton. Phys. Rev. E 72, (2005).

27 Conclusions Problem and Movitation Discussion Conclusions Strong support for Lévy Flight hypothesis. Even Stronger evidence of superdiffusion (H > 1/2). LW optimize random nondestructive searches. LF foraging conceptual advance. Experimental data: curved trajectories, higher complexity. LW applied to biological foraging remains interesting! Emergent behavior vs. evolutionary adaptation.!! Obrigado!!