Starvation Dynamics of a Greedy Forager

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1 Marshall, J. (2017, July 12). Does greed help a forager survive? Retrieved from Starvation Dynamics of a Greedy Forager Bhat U, Redner S and Bénichou O 2017 J. Stat. Mech Zejun Liu, Chunyu Lu, Siddharth Mansingh, Robert McKay

2 Overview 1. Introduction to model 2. Formalisms of model 3. 1D results 4. 2D simulation 5. Analysis of previously existing work 6. Citation evaluation 7. Future outlook

3 Motivation Should one continue to exploit a rich localized cache in a desert? Or is it better to move to a region where resources are more plentiful overall? Learn about how organisms gain nourishment What does greed do to one s species and ecosystem?

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sha1_base64="m0/goxzjmfp2p74rz8hyvhoklkq=">aaab8nicbvdlssnafl2pr1pfvzdugkvwvrirdfl047jcx9cgmplo2qgtmtbzi5tqz3djqhg3fo07/8zjm4w2hhg4nhmvc+4je8enet63u9ry3nreke9w9vypdo+qxycdo1jnwzsqoxqvjiyjllkboqrwszqjcshyn5ze5373iwndlwzhlgfbtmasr5wstfj/ebocucky1nxyrxl1bwf3nfgfqugb5rd6nrgpmszmihxeml7vjrhkrcongs0rg9swhnapgbo+pzlezatzivlcvbdkyi2utk+iu1b/b2qknmywh3yyj2hwvvz8z+ungn0ggzdjikzs5udrklxubn6/o+kaurqzswjv3gz16yroqtg2vlel+ksnr5povd23/pg61rgr6ijdgzzdjfhwaw14gca0gykcz3ifnwedf+fd+violpxi5xt+wpn8ay0vkwo=</latexit> Investigating the Model Used For the Greedy Forager Model Components: Random Walk Modeling Parameterized by greed -1 G 1 metabolic capacity Measures of Model: N amount of food consumed average lifetime of forager T<latexit 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6 Formalism Used in this Work We assume a forager starts at 1 or L-1 thus it is on the boundary of desert( area has no food) and paradise (area filled with food). Food No Food Point 1 to Point L-1 fl(t) : Probability that an isotropic random walk first reaches either edge of the interval at time step t with this initial condition. F(t): Probability that the forager starts at either x = 1 or x = L 1, first reaches either x = 0 or x = L at step t

7 Formalism Used in this Work Point 1 to Point L-1 Food No Food Vk : probability that the forager has visited k distinct sites at the instant of starvation; F k (S) :probability that the forager eats before it starves in a desert of k sites; τ j : average time for a forager to escape a desert of length j by eating a unit of food at the desert edge before starvation

8 1D: semi-infinite desert geometry Initial Condition: X 0: Food X> 0: Desert

9 2. Generating function for isotropic random walk in SI geometry 1D: semi-infinite desert geometry Initial Condition: X 0: Food X> 0: Desert Asymptotic solution: 1. First passage formalism

10 1D: semi-infinite desert geometry Initial Condition: X 0: Food X> 0: Desert Asymptotic solution: 1. First passage formalism 2. Generating function for isotropic random walk in SI geometry For large S: Consumed food scale as S. Lifetime is proportional to S for strong greedy, S 2 for weak greedy.

11 Heuristic approach for 1D semi-infinite desert geometry Ballistic segment towards food Diffusive segment in the desert Continuum limit:

12 Heuristic Results. for 1D semi-infinite desert. geometry

13 Exact Discrete Solution of Survival Time Forager with capacity S Forager with capacity 2 Let ta be the average lifetime of the forager ta = p(1 + ta) + 2(1 p)

14 Exact Discrete Solution of Survival Time Forager with capacity S Forager with capacity S For General capacity S, split all trajectories into two categories: Set P, (all paths that return to food) Set Q,(all paths where the forager starves) How does one calculate P p?

15 1D Forager Simulation Position Time

16 2D Results Forager Desert Food If you are a forager with a higher capacity, absolute greed might not help you to live longer.

17 Comparisons with Previous Work Random walk (p=1/d) 1D finite desert: 2D simulation: Both average lifetime and consumed food scale as S 2.

18 Comparisons with Previous Work No consumption Aim: minimize the search time. No intelligence, random walk Tejedor V, Voituriez R, Bénichou O. Optimizing persistent random searches[j]. Physical review letters, 2012, 108(8):

19 Related work Introduce a model with both reproducing and nonreproducing consumers. Extension of starving random walk model of foraging (only consumption) Provides a framework to predict population abundances: - Damuth s law (larger animals have fewer individuals) - Cope s rule (larger body size tends to be more distinct)

20 Citation Evaluation and Progress Since Paper Since July 2017: Advancements since paper: Applied to chemotaxis in a porous medium (maze of resources without greed) 5 Diffusion of a population (many foragers without greed) 6 Extension to 2D models (with greed) 7 [5] Schilling, T., & Voigtmann, T. (2018, November). Clearing out a maze: A model of chemotactic motion in porous media. [6] Berbert, J., & Lewis, M. (2018, January). Superdiffusivity due to resource depletion in random searches. [7] Rager, C. L., Bhat, U., Bénichou, O., & Redner, S. (2018, July 16). The advantage of foraging myopically.

21 Critical Analysis Number of realizations to be averaged over was not mentioned. Too ideal a model. Improvements? Multiple Foragers Resource Replenishment Sparse Resources

22 Conclusion More greediness doesn t imply longer lifetime in 1D Lifespan tends to be longer for higher, positive greed values in 2D Many variations to this basic model yet to be explored