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Foraging on the move

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Abstract

Many animals (e.g. caribou, wildebeest) forage in groups while moving from one location to another. This means individuals have to simultaneously balance several demands: finding the best resources, maintaining the cohesion of the group, and migration in a certain direction. While there is a vast literature on both flocking and optimal foraging, there has been no work done to understand how animals should trade off the decision to flock or forage (since it is difficult to do both simultaneously) during migration. We develop an individual-based model to address this, and implement a genetic algorithm to find the best decision-rule for switching between foraging and flocking, under a variety of conditions.


Project Description

Background

GROUP LIVING: Animal groups are ubiquitous in nature and their formation comes with both costs and benefits. Groups provide individuals with a lower per-capita predation rate (dilution effect), shared information, and allow individuals to find both mates and resources more easily. However, groups tend to attract predators more easily than solitary individuals, can increase the transmission rate of diseases, and lead to competition for resources. In theory the size of an animal group should be determined by the payoff to an individual; at equilibrium group size, the benefit to an individual of joining a group should be equal to that of being a loner. An individual's specific position within a group can have significant fitness consequences (see Krause 1994); individuals in the center of the group are often the safest from predation (Hamilton 1971), but those on the edge are in the best position to discover and exploit new food resources [ref].

FLOCKING & FORAGING: There is a vast literature on flocking/swarming models (e.g. [refs]), and on optimal foraging behavior (e.g. Fretwell & Lucas 1969, Schoener 1971, Charnov 1976). Some theoretical work has been done on combining foraging and flocking, but this mostly relates to what has been termed "social foraging" (see Giraldeau & Caraco 2000) and examines how how individuals can gain information about foraging locations by flocking (e.g. Clark & Mangel 1984) or on how foraging strategy should vary with respect to location within the flock (Barta et al 1997). However there is no theoretical work on how foraging individuals should behave when they are in a group that is collectively moving, a behavior seen in wildebeest, caribou and many other species of migratory ungulates. Individuals in this situation face a trade-off between foraging and keeping up with the group. Individuals that fail to forage will starve, and those that fail to keep up with the group will likely be picked off by predators [ref]. This leads to a a mixed behavior at any moment, where some individuals are concentrating on foraging and others on flocking and moving towards their destination (e.g. Planet Earth video).

MAMMALIAN MIGRATION: A recent literature review found that the majority of papers that model migration focus on birds and fish, with almost no work done on modeling mammal migration (Bauer et al 2009). The only two papers from this review that modeled terrestrial mammal migration are:

  • Bergman et al (1999) found that correlated random walk model overpredicted long-range displacement of migrating caribou.
  • Boone et al (2006) found that migratory pathways of wildebeest derived using an evolutionary algorithm closely matched actual migration paths chosen, suggesting that rainfall and vegetation are key determinants in wildebeest migratory behavior.

WILDEBEEST: Wildebeest in Tanzania and Kenya move in a roughly circular clockwise direction, likely following the vegetation and rainfall across the seasons. During the dry season (May-Oct), they are found mostly in the north, and move southward as the rains start, and give birth synchronously in Feb/Mar. There is a rainfall gradient (and corresponding vegetation gradient) across the region with high rainfall in the northwest and low in the southeast. (Boone et al 2006)

CARIBOU: The search for food is one of the primary drivers of caribou migration. Caribou winter in areas with a large amount of vegatation biomass (mostly lichens). In the spring they move north towards areas with seasonal vegetation that is rich in nitrogen and minerals (necessarily for rapid growth and milk production), where they give birth. In Aug/Sept as the vegetation gradient becomes less pronounced, the herds disperse in small groups, coming back together only at the end of the growing season, to migrate back south, as lichens in the wintering area become the best source of energy again. (Fancy et al 1989)

Fancy and White (1987) found that caribou were the most efficient walkers (lowest net cost of locomotion) of all ungulates tested. Wildebeest were second most efficient, suggesting that migration is an important selective pressure on locomotion efficiency.

Parameter Settings from Literature

MIGRATION RATES/DISTANCES:

  • caribou migrate up to 40km/day (Fancy et al 1989)
  • caribou have longest terrestrial animal migration: 1200km (Akesson 2007)
  • mean migratory movement for caribou 14-26 km/day (Murray 1991)
  • wildebeest migration is ~200km (Akesson 2007)
  • The speed at which animal migrations can travel is dependent on three factors: speed of locomotion, rate of energy acquisition, rate of energy consumption (Akesson 2007)
  • migration range shows diminishing return function with more fuel runners pay cost of extra weight (Akesson 2007)

ENERGY EXPENDITURE:

  • energy cost for uphill locomotion in caribou (note: doesnt include cost of walking in snow): 22.6 (1.3 SE) kJ/(kg*m) (Fancy & White 1987)
  • energy cost for horizontal caribou locomotion in snow: 1.696 KJ/(Kg*km) (Johnson et al. 2002)
  • energy cost for wildebest: 4.23 kg/(individual *day) (Sinclair 1975)
  • caribou are ~272kg (http://www.amnh.org/nationalcenter/Endangered/caribou/caribou.html)
  • caribou are ~2m long (Kuzyk et al 1999)
  • CALCULATION: (1.696kJ/kg/km)*(1km/1000m)*(272kg/ind)*(2m/body)= 0.922kJ/ind/body

ENERGY INTAKE:

  • maximum consumption rate for wildebeest: 4 kg/day (Note: simulation parameter not measured) (Fryxell et al. 1988)
  • energy from foraging: 169.1 KJ/(kg*day) (Murray 1991)
  • 0.3 g per bite for wildebeest (Murray 1991; Figure 3/4)
  • ~7.96 MJ/kg dried matter for plants (Murray 1991; Table 1)
  • CALCULATION: (0.3g/bite)*(1kg/1000g)*(7.96MJ/kg)*(1000kJ/MJ) = 2.388 kJ/bite

Herd Size and Density

  • Elk winter herd size (max,min): (14,253) (Creel and Winnie 2005)
  • Elk density: 13.7 individuals/ km^2 (Creel and Winnie 2005)
  • Elk distance apart with in herd: <5 body lengths (Creel and Winnie 2005)
  • Wildebeest main herd size in Serengeti: 467 individuals (Fryxell et al. 1988)
  • Herd size in 1960 of Caribou in Newfoundland: 450 individuals (Mahoney and Schafer 2002)
  • Herd size in 2000 of Caribou in Newfoundland: 6,102 individuals (Mahoney and Schafer 2002)

Model Details

INPUTS/ASSUMPTIONS:

  • simple NetLogo flocking model (flocking.nlogo) where individuals move based on two rules: 1) "separate" (if other agents are too close, move away from them), and 2) "align" and "cohere" (otherwise move in the same direction as nearby agents, and move towards them)
  • add in foraging element: agents have energy reserves that deplete slowly as they move, and increase if they stop to forage
  • agents have behavioral rules to decide when to forage and when to flock (only one can be done at at time)
  • agents die if they fail to forage (starve) or fail to flock (are subject to predation)
  • agents are constantly moving, i.e. not everyone can just stop and forage (to match concept that migrating organisms are moving under time constraints)


OUTPUTS:

  • group metrics -- how does adding foraging behavior compare to just flocking?
  • how are stopping time / group size / migration rate interrelated?
  • adaptive dynamics -- what is the ideal balance between foraging and flocking activities?
  • do we get different movement patterns under different parameter settings -- e.g. stringy 'wildebeest' movement vs 'flowing' caribou movement?


EXTENSIONS:

  • evolve flocking vs foraging decision rule
  • build in interaction with the environment (local depletion of resources)
  • look at how habitat structure affects group movement

Results/Observations/Predictions

OBSERVATIONS:

  • FlockingForaging_evolve_v2: continuous replacement means that population that isn't stable can be maintained by new individuals coming in
  • FlockingForaging_evolve_v3: agents stay mostly sedentary
    • this is fixed by changing the starting density of the simulation
  • FlockingForaging_evolve_v4.nlogo: agents evolve bimodal distribution in flocking parameter
    • this seems to go away when flocking takes priority over foraging (instead of the reverse)
  • FlockingForaging_evolve_v4.nlogo: with mutation agents are evolving threshold for flocking larger than flock size (!) -- not sure why these are staying around since they should be unfit...


CHECK FOR (PREDICTIONS):

  • Past theoretical work suggests that fragmentation of a herd results more easily for a individuals are heterogeneous in their walking speeds (Gueron et al 1993)
  • Individual position within the group should change over time, as individuals change their preference for different group locations as their conditions (e.g. energy reserves) change (Parrish 1999)
  • Gueron et al (1996) found that the geometry of the group was dependent on the average speed of individuals, showing bands at slow speeds and columns at fast speeds
  • if cost of moving is too high, migration should 'fail'

References

  1. Åkesson, S. & Hedenström, A. 2007. How migrants get there: migratory performance and orientation. BioScience, 57, 123-133.
  2. Barta, Z. 1997. Geometry for a selfish foraging group: a genetic algorithm approach. Proceedings of the Royal Society B: Biological Sciences, 264, 1233-1238.
  3. Bauer, S., Barta, Z., Ens, B., Hays, G., Mcnamara, J. & Klaassen, M. 2009. Animal migration: linking models and data beyond taxonomic limits. Biology Letters, 1-4.
  4. Bergman, C., Schaefer, J. & Luttich, S. 2000. Caribou movement as a correlated random walk. Oecologia, 123, 364-374.
  5. Boone, R., Thirgood, S. & Hopcraft, J. 2006. Serengeti wildebeest migratory patterns modeled from rainfall and new vegetation growth. Ecology, 87, 1987-1994.
  6. Charnov, E. 1976. Optimal foraging, the marginal value theorem. Theoretical Population Biology, 9, 129-136.
  7. Clark, C. & Mangel, M. 1984. Foraging and flocking strategies: information in an uncertain environment. American Naturalist, 626-641.
  8. Creel, S. & Winnie, J.A. Responses of elk herd size to fine-scale spatial and temporal variation in the risk of predation by wolves. Animal Behaviour 69, 1181-1189(2005).
  9. de Knegt, H.J. et al. Patch density determines movement patterns and foraging efficiency of large herbivores. Behavioral Ecology (2007).
  10. Fancy, S. & White, R. 1987. Energy expenditures for locomotion by barren-ground caribou. Revue canadienne de zoologie, 65, 122-128.
  11. Fancy, S., Pank, L., Whitten, K. & Regelin, W. 1989. Seasonal movements of caribou in arctic Alaska as determined by satellite. Canadian Journal of Zoology, 67, 644-650.
  12. Focardi, S. & Pecchioli, E. Social cohesion and foraging decrease with group size in fallow deer (Dama dama). Behavioral Ecology and Sociobiology 59, 84-91(2005).
  13. Focardi, S., Marcellini, P. & Montanaro, P. Do ungulates exhibit a food density threshold? A field study of optimal foraging and movement patterns. Journal of Animal Ecology 606-620(1996).
  14. Frair, J.L. et al. Scales of movement by elk (Cervus elaphus) in response to heterogeneity in forage resources and predation risk. Landscape Ecology 20, 273-287(2005).
  15. Fretwell, S. & Lucas, H. 1969. On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheor, 19, 16-36.
  16. Fryxell, J.M. et al. Landscape scale, heterogeneity, and the viability of Serengeti grazers. Ecology Letters 8, 328-335(2005).
  17. Fryxell, J.M. et al. Why are migratory ungulates so abundant?. American Naturalist 131, 781-798(1988).
  18. Giraldeau, Luc-Alain and Thomas Caraco. Social foraging theory. Princeton: Princeton University Press, 2000.
  19. Gueron, S., Levin, S. A. & Rubenstein, D. I. 1996. The Dynamics of Herds: From Individuals to Aggregations. Journal of Theoretical Biology, 182, 85-98.
  20. Hamilton, W. 1971. Geometry for the selfish herd. Journal of Theoretical Biology, 31, 295-311.
  21. Johnson, C.J. et al. Movement parameters of ungulates and scale-specific responses to the environment. Journal of Animal Ecology 225-235(2002).
  22. Krause, J. 1994. Differential Fitness Returns in Relation to Spatial Position on Groups. Biological Reviews, 69, 187-206.
  23. Parrish, J. 1999. Complexity, Pattern, and Evolutionary Trade-Offs in Animal Aggregation. Science, 284, 99-101.
  24. Mueller, T. et al. In search of forage: predicting dynamic habitats of Mongolian gazelles using satellite-based estimates of vegetation productivity. Journal of Applied Ecology 45, 649-658(2008).
  25. Murray, M.G. Maximizing Energy Retention in Grazing Ruminants. Journal of Animal Ecology 60, 1029-1045(1991).
  26. Schoener, T. 1971. Theory of feeding strategies. Annual Review Of Ecology And Systematics, 2, 369-404.
  27. Sinclair, A.R.E. The Resource Limitation of Trophic Levels in Tropical Grassland Ecosystems. Journal of Animal Ecology 44, 497-520(1975).


3-radius flocking model where individual movement is a weighted combination of the direction they inherently want to go and the influence of individuals around them
wildebeest migration clearly driven by rainfall gradient
model suggests that wildebeest are maximizing green grass intake (rate of intake of high-quality food)
original Boids model
video of caribou migration

Participation

Meetings

  • 14 July 2009 7pm (GMT-4)

Tasks (Current)

BIG PICTURE:

  1. setup a base ABM of flocking and foraging (done, yay!)
  2. see which foraging and flocking thresholds evolve
  3. make sure these thresholds are 'stable' under realistic conditions for caribou/wildebeest
  4. see how these thresholds vary with key model parameters (population size, energy from forage, predation)
  5. incorporate landscape
  • TASK A: Since #1 is basically done above, we should write up a clear description of the model, explaining all the parameters and logic behind them, before we forget it all!!
  • TASK B: try varying ticks per generation and number of generations per run (ideally on a computer cluster to avoid killing someones laptop), to see if we get tighter convergence if we run long enough
  • TASK C: If B doesn't produce good results, then come up with a better metric AND figure out how to export this from NetLogo (It may just make sense to export the threshold parameter values for all individuals and not just the average/stdev)
  • TASK D: Related to this, it would be good to come up with an efficient way of analyzing the NetLogo data in another program. I'm partial to MATLAB but would be absolutely ok using another program if we think that would be easier. Kate and I figured out that if you delete the header lines and replace all the quotes with commas, then the csv files are easily imported into MATLAB. However some of the data imports in as text instead of numbers. The other annoying thing is that all the stats are calculated at every tick instead of every generation. It would be great if there were someway to get just stats at every generation...
  • TASK E: We should play around with comparing the continuous vs discrete methods. This would probably just involve running a few simulations with each to make sure we're getting similar results. It may be that things evolve faster in continuous time, so that may be the way to go for our mass simulations. There's an output graph that shows the age distribution of turtles in the population, so that you can check to make sure individuals aren't dying each tick.
  • TASK F: For #3 I think we basically need to know 1) average herd size, 2) average distance/speed and 3) average mortality for caribou/wildebeest migrations. Then we can setup a population of this size and our evolved parameters and see if it can make it that distance with that mortality.
  • TASK G: Related to #3 -- we should think about the form of our predation mortality term and determine the best value for sigma in the function (or perhaps use a different function). It seems that there were lots of individuals dying from flocking in the simulations.
  • TASK H: Once we do at least B/C/D above, we should run simulations again varying they key model parameters that we're interested in (for #4). In the meantime though, someone could run simulations with different values to get a feel for how these change the model -- perhaps they don't change the mean population value of the thresholds but do affect the distribution, or maybe they just affect migration speed, etc.
  • TASK I: Figure out exactly what we want to know by incorporating landscape/resources into the model. Do we want to add resource factors as more key parameters in the model to vary -- then see how thresholds vary under different resource distributions? If so, what would key resource parameters be? Or do we just want to see how a group would move differently across different landscape types? This would be simpler but gets away from our central question.

Tasks (Old)

LITERATURE

  • migration models lit review (Allison)
  • basic description of wildebeest & caribou migration cycles (Allison)
  • appropriate foraging parameters / energy function for ungulates: "energy-forage" and "energy-move" in NetLogo model, also probability of forage as a function of energy (Liliana)
  • appropriate flocking parameters / flocking lit review: "minimum-separation", "max-align-turn", "max-cohere-turn", "max-separate-turn", and "vision" in NetLogo model (Kate)
  • collective behavior lit review (Allison)
  • foraging lit review (Liliana)
  • look for flocking metrics -- e.g. group 'coherence' or group structure/dynamics

MODEL TWEAKING (SIMPLE)

  • how to initially distribute agents?
currently are all started in roughly same area and orientation
alternative would be to give them all the same preferred direction (a la Couzin et al 2005)
  • how to step model forward (appropriate time step)? ("stepsize" and "steprepeats" in NetLogo model)
    • Have just stepsize = 1 and steprepeats (which was only smoothing) now removed SteveLade 04:59, 25 June 2009 (UTC)
  • problem: why do agents die from bad-foraging but do not die from bad-flocking? (or have we just not found those parameter settings?)
    • note: starvation rate depends on relative values of "energy-forage" and "energy-move"
maybe have agents die from bad-flocking when too few individuals in their sight radius (instead of none)
Your sigmoidal forage-probability curve has threshold at 0.5 energy units. If energy-move is large compared to this value (and you currently have it at 0.3 by default), it will take only a couple of steps to go from 'stomach full' to 'dead'. Why there are no flocking deaths I don't know. SteveLade 04:17, 24 June 2009 (UTC)
  • how to prevent flock from wrapping around across the boundary?
    • Currently not a problem -- flock not long enough
  • should we stick with 2-zone model or change to 3-zone one (a la Couzin et al 2005)?
  • NOTE: energy levels become synchronized over time and move like a wave through the population
    • only happens for high "vision" values
    • depends on "energy-move" values
    • Doesn't seem to happen any more SteveLade 04:59, 25 June 2009 (UTC)
    • yeah, this was a due to a bug in how I sent initial conditions -Allison

MODEL DEVELOPMENT (MORE INVOLVED)

  • develop/implement metrics to describe group
    • flock density
    • average flock speed (Done SteveLade 04:59, 25 June 2009 (UTC))
    • average fraction of time individual spends foraging (This is just from the ratio of energies received/spent from flocking/foraging SteveLade 04:59, 25 June 2009 (UTC))
    • group size stability threshold (below which group fails to flock/survive)
    • group 'coherence' (check literature)
    • group structure/dynamics (check lit)
  • adaptive dynamics framework to evolve parameters (Andrew & Steve)
  • design behavior rules to determine when to forage and when to flock (Steve & Andrew)
should decision to forage be independent at each time step or are foraging individuals more likely to keep foraging than to start flocking?
Have behaviour rules but currently no hysteresis for foraging SteveLade 04:59, 25 June 2009 (UTC)
  • couple foraging to changes in resource distribution -- e.g. a patch is depleted by a foraging agent and must grow back after some time
  • design different landscape resource distributions to have agents moving across

Members

Original Discussion

Snapshot of caribou migration.

Allison Shaw: Many animals forage in groups while moving from one location to another. This means individuals have to simultaneously balance several demands: finding the best resources, maintaining the cohesion of the group, and in some cases moving in a certain direction. Can we develop an agent-based model with a simple set of individual movement rules that would allow for all these demands to be met?

This was inspired by a piece of Planet Earth footage on caribou: go to http://dsc.discovery.com/convergence/planet-earth/video-player/video-player.html, scroll down in the video clips to "Planet Earth: Plains: Following the Caribou" and watch the dynamics at about 1:30-2:00. (If anyone has a hard copy of this segment or knows how to get one, please let me know!). In this case each individual caribou pauses to eat along the way but the group never fragments and in fact it seems to almost 'flow' through an area. My guess is that one of the physicists could provide some interesting insight on how to model this.


Daniel Wuellner: Cool idea. Most importantly: I actually brought the Planet Earth DVDs with me which I'll happily lend; maybe we can organize a viewing w/ a projector somewhere.

  • Roozbeh Daneshvar: Daniel, I am absolutely in for such a preview. Can you upload it in After Hours so that we all watch it together?

I think there's some swarm literature out there for ideas on rules you could extend to incorporate foraging (or any other caribouish behavior). The one I know is Flocks, herds and schools: A distributed behavioral model (this actually might be the 'original' swarm paper).

Kate Behrman: I also interested in this. One possible extension could be to consider how the structure of the landscape between the two locations affects the movement of the group.

Murad Mithani: It sounds similar to what happens when the cognitive processes are focused on a particular problem to come up with ideas. The initiation of problem solving is a conscious mechanism that flourishes when that initial push is taken away. If you guys are planning to model this in some way, count me in.

Steven Lade: I like the sound of this too. Dare I suggest a meeting, perhaps one lunchtime, to flesh out plans a little more? Allison, since it was your idea, would you like to call it?

Allison Shaw: Sure, that would be great! How about lunch tomorrow (Tuesday the 16th)? Let's try to synchronize sitting together. I talked to JP about doing a Planet Earth showing and he said we could use the projector for the lectures, but we'd need to get a decent set of speakers (as far as I know there isn't a working TV/DVD combination in any of the lounges and we'd have to pay to use the more advanced media system in the lecture room).

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