supervised and helped by dr stephen hartley, dr marcus frean, marc hasenbank victoria university,...

Supervised and helped by Dr Stephen Hartley, Dr Marcus Frean, Marc Hasenbank

Victoria University, Wellington

Jim Barritt

© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington

Using a “Random Walk” to simulate the foraging behaviour of Pieris rapae

http://www.oulu.fi/

© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington2

Talk outline

• Background

• Theory

• Simulation

• Results

• Conclusion / Future work

http://www.oulu.fi/


Background

• Part of a project investigating insect foraging interactions (Pieris rapae)- Dr. Stephen Hartley, Marc Hasenbank

- Session 15 - “Egg laying on patchy resources and the importance of spatial scale”

• Simulation in conjunction with field studies


Theory - the Oviposition Question

Which cabbage ?

Pieris rapae


Theory - Resource concentration ?

• Is there a relationship between eggs per plant and plant density ?

- Concentration: Higher plant density should provide more information (e.g. olfactory cues) so animals are expected to locate dense patches easily and remain within them. This would lead to more eggs per plant on high density patches

- Root (1973)

- Dilution: Foraging animals may not remain in high density stands, instead moving around at a constant rate irrespective of plant density. This produces more eggs per plant on low density plants or egg “spreading” behaviour - Yamamura (1999)

- Ideal free distribution: Complete information / access leads to the eggs being distributed evenly

- Depends on patterns of movement

Resource concentration

Resource dilution

Ideal free distribution

Low Density High Density


Theory - Resource concentration ?



- Root (1973)

- Dilution: Foraging animals may not remain in high density stands, instead moving around at a constant rate irrespective of plant density. This produces more eggs per plant on low density plants or egg “spreading” behaviour - Yamamura (1999)




Resource dilution


Low Density High Density


Theory - Resource dilution ?



- Root (1973)

- Dilution: Foraging animals may not remain in high density stands, instead moving around at a constant rate irrespective of plant density. This produces more eggs per plant on low density plants or egg “spreading” behaviour

- Yamamura (1999)




Resource dilution


High DensityLow Density


Theory - Ideal free distribution ?



- Root (1973)

- Dilution: Foraging animals may not remain in high density stands, instead moving around at a constant rate irrespective of plant density. This produces more eggs per plant on low density plants or egg “spreading” behaviour

- Yamamura (1999)


• It depends on patterns of movement


Resource dilution



Simulation

• Two seasons of field observations (Marc Hasenbank)

• How do patterns of movement create the observed response ?- Published simulations: Jones (1977), Cain (1985), Byers

(2001)

- Random vs force of attraction (incorporate perceptual information?)

• How do we simulate ?- Quantify movement paths

- Create conceptual model - what is a random walk ?

- Run simulation experiment!


Quantifying movement paths

Start

Animal moves continuously in space



Start

Sample location in space over time1

2

3

4

5

6

7

8



Start

Join the dots to create “Steps” - an abstraction of the real path



Start

Measurements


Random walks - Step 1

• We can simulate the path:- Choose θ (angle of turn) at random (+/- 180°)

- Move 1 step length in that direction ...

+90°

0°

-90°

+/-180°



• We can simulate the path:- Choose θ (angle of turn) at random (+/- 180°)

- Move 1 step length in that direction

- Repeat

+90°

0°

-90°

+/-180°



• We can simulate the path:- Choose an heading at random (+/- 180°)

- Move 1 step length in that direction

- Repeat

+90°

0°-90°

+/-180°



• This is a “Random Walk” (or flight!)- It does not imply that the animal is behaving randomly but that

there is a random element involved in the interaction with the environment - Root & Kareiva (1984)

Start


Random walks - correlated steps

• Some animals may exhibit “pure” random walks - e.g. Whirligig Beetles (Gyrinus sp.)

• Butterflies have a “direction of travel” and so are more likely to turn with θ around 0°

• “Correlated”http://insects.tamu.edu

+90°

0°

-90°

+/-180°


Random walks - correlated steps

• We can simulate correlation of angle of turn by selecting θ from a probability distribution...

+90°

0°

-90°

+/-180°

Cain (1985) - demonstrated similar distribution for observed Pieris angles of turn


Random walks - Correlated steps

• After 10 steps ...

Start


Correlated random walk

• More “directional” than “pure” random walk

Start

Start


From theory to simulation

• Take the standard deviation (s.d.) of the probability distribution and the step length ...

0°

-90°

+/-180°

+90°


From theory to simulation

• Provides two simulation parameters, L and A

+90°

0°

-90°

+/-180°

A

L


Simulation

• Visual demonstration with simple cabbage layout

• Experiment Parameters

• Results


Visual demonstration - Step 0

L=10

A=20

Cabbage

Butterfly

Release boundary



L=10

A=20

When move outside “world”, removed



L=10

A=20



L=10

A=20

When intersect a cabbage, lay egg and “die”


Visual demonstration - Step 12 (End)

L=10

A=20


Experiment Parameters

• L = step length (0.5m to 2m)

• A = s.d angle of turn (20° to 100°)

• Cabbage radius = 20cm, spacing = 25cm

• Large L / Small A = more directional

• Small L / Large A = more “wiggle”

• 12, 000 butterflies

• 10 replicates

Directional “Wiggle”


Experimental Cabbage layout


Results Simulation


Results Simulation vs Field



Resource dilution


Results Log Linear Regression

✔

H0 - β=0Field p-value = 0.037Simulation p-value = 0.0425

r2 (Field+Simulation) = 0.9

Simulation

Field




Resource dilution

Results Log Linear Regression

✔

H0 - β=0Field p-value = 0.037Simulation p-value = 0.0425

r2 (Field+Simulation) = 0.9

Simulation

Field


Consistent with literature (Yamamura, 1999)


Results - varying parameters

L=50 L=100 L=150 L=200

Plant Density

Eggs

Per

Pla

nt

(mean +

/std

err

)

A=20

A=60

A=100


Results - varying parameters

L=50 L=100 L=150 L=200

Plant Density

Eggs

Per

Pla

nt

(mean +

/std

err

)

A=20

A=60

A=100

L=200A=100


Conclusions

• With a simple correlated random walk we can predict egg distributions for Pieris - No attractive force

• With no attractive force we observe resource dilution

• Attractive force could potentially produce resource concentration ...

✔

?

✔


Conclusions




✔

?

✔


Future Work

• Deterministic attraction- Force of attraction (similar to gravity)

- Perceptual ranges

- Information gradients / matrix

• Random walk influenced by Environment- Move length and Angle of turn as functions of information

• Lifecycle: multiple eggs, migration vs birth

• Multi-species- Co-existance by having different movement patterns?

• Fractal (Levy) Walks and landscape


Acknowledgements

• Thanks to- Dr Stephen Hartley- Dr Marcus Frean- Marc Hasenbank- Victoria University Bug Group

- Special thanks to John Clark and the staff of Woodhaven Farm (Levin)

- Funded by a Royal Society Marsden grant

http://www.oulu.fi/


Questions ?

• With a simple correlated random walk we can predict egg distributions for Pieris



✔

?

✔


ReferencesAldrich, J. (1997). R.A. Fisher and the making of maximum likelihood 1912-1922. Statistical Science 12, pp.162-176.

Bukovinszky, T., R. P. J. Potting, Y. Clough, J. C. van Lenteren, and L. E. M. Vet. (2005). The role of pre- and post-alighting detection mechanisms in the responses to patch size by specialist herbivores. Oikos 109, pp. 435-446.

Byers, J. A. (2001). Correlated random walk equations of animal dispersal resolved by simulation. Ecology 82, pp.1680-1690.

Cain, M. L. (1985). Random Search by Herbivorous Insects: A Simulation Model. Ecology 66, pp. 876-888.

Finch, S., and R. H. Collier. (2000). Host-plant selection by insects - a theory based on 'appropriate/inappropriate landings' by pest insects of cruciferous plants. Entomologia Experimentalis Et Applicata 96, pp. 91-102.

Fretwell, S. D., and H. L. Lucas. (1970). On territorial behaviour and other factors influencing habitat distribution in birds. Acta Biotheoretica 19, pp. 16-36.

Grez, A. A., and R. H. Gonzalez. (1995). Resource Concentration Hypothesis - Effect of Host-Plant Patch Size on Density of Herbivorous Insects. Oecologia 103, pp. 471-474.

Holmgren, N. M. A., and W. M. WGetz. (2000). Evolution of host plant selection in insect under perceptual constraints: A simulation study. Evolutionary Ecology Research 2, pp. 81-106.

Jones, R. E. (1977). Movement Patterns and Egg Distribution in Cabbage Butterflies. The Journal of Animal Ecology 46, pp. 195-212.

Olden, J. D., R. L. Schooley, J. B. Monroe, and N. L. Poff. ( 2004). Context-dependent perceptual ranges and their relevance to animal movements in landscapes. Journal of Animal Ecology 73, pp. 1190-1194.

Otway, S. J., A. Hector, and J. H. Lawton. (2005). Resource dilution effects on specialist insect herbivores in a grassland biodiversity experiment. Journal of Animal Ecology 74, pp. 234-240.

Root, R. B. (1973). Organization of a Plant-Arthropod Association in Simple and Diverse Habitats: The Fauna of Collards (Brassica Oleracea). Ecological Monographs 43, pp. 95-124.

Root, R. B., and P. M. Kareiva. (1984). The search for resources by cabbage butterflies (Pieris rapae): ecological consequences and adaptive significance of markovian movements in a patchy environment. Ecology 65:147-165.

Tilman, D., and P. M. Kareiva. (1997). Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions. Monographs In Population Biology 30

Yamamura, K. 1999. Relation between plant density and arthropod density in cabbage. Researches on Population Ecology 41:177-182.


Mean Squared Displacement


Why simulate ?

• Wide range of existing research modelling behaviour of Pieris rapae- Jones (1970), Cain (1985), Byers(2001)- Are these a good fit to our field observations?- Validation of current theory

• Provide a conceptual model to aid interpretation of field data- Use simple model and compare to field data- Reveal intrinsic patterns

• Assess potential behaviour mechanisms affecting egg distribution- How do the butterflies move ?


Logr regression details


Published Parameters

•Published data:-Byers(2001) - derived from Root & Kareiva (1984)-A ≈ 50 degrees-L ≈ 2.5 m


Statistical tests

• H0 Field egg distribution = Simulation- observed → field, expected → simulation

• H0 Field regression slope (β) = Simulation


Statistical tests

• H0 Field egg distribution = Simulation- observed → field, expected → simulation

- X2 H0 - observed = expected : p<0.001 (3e-11) ∴ significant difference

• H0 Field regression slope (β) = Simulation- t-test H0 - β simulation = β field : p=0.839 ∴ no significant difference

• All results show resource dilution - Negative β- p<0.05 that β = 0 (Ideal free distribution)

✔

✘


Conclusions

• Simulation reproduces effects observed in the field- Resource dilution In both simulation and field results

- Suggests random walk is good basis for representing Pieris movement

- Consistent with literature

• But...

• Does not yet represent field results accurately- Saw change in effect for lower step length

- Need to explore more parameters

- Change behaviour algorithm e.g. more than 1 egg

- Future work ...

✔

✘


Field results

• Which do we observe in our field experiments ?


Resource dilution



Field results


Resource dilution




Resource dilution


Field results - log transformation

✔

✘

✘

H0 - β=0p-value = 0.037

r2 = 0.9

Field



Resource dilution

Field results - log transformation

✔

H0 - β=0p-value = 0.037

r2 = 0.9

Field


Consistent with literature (Yamamura, 1999)


Theory

• Is there a relationship between plant density and eggs per plant ?

1 plant

20 plants, density = 0.2

High Density


Low Density


Theory - Response to plant density

• Three possible responses ➔

1 plant


High Density


Low Density


Yamamura 1999 Results

supervised and helped by dr stephen hartley, dr marcus frean, marc hasenbank victoria university,...

Documents

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