Post on 26-Mar-2015
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/
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington3
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington4
Theory - the Oviposition Question
Which cabbage ?
Pieris rapae
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington5
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington6
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington7
Theory - Resource dilution ?
• 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
High DensityLow Density
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington8
Theory - Ideal free distribution ?
• 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
• It depends on patterns of movement
Resource concentration
Resource dilution
Ideal free distribution
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington9
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!
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington10
Quantifying movement paths
Start
Animal moves continuously in space
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington11
Quantifying movement paths
Start
Sample location in space over time1
2
3
4
5
6
7
8
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington12
Quantifying movement paths
Start
Join the dots to create “Steps” - an abstraction of the real path
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington13
Quantifying movement paths
Start
Measurements
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington14
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°
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington15
Random walks - Step 2
• 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°
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington16
Random walks - Step 3
• We can simulate the path:- Choose an heading at random (+/- 180°)
- Move 1 step length in that direction
- Repeat
+90°
0°-90°
+/-180°
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington17
Random walks - Step 10
• 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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington18
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°
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington19
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington20
Random walks - Correlated steps
• After 10 steps ...
Start
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington21
Correlated random walk
• More “directional” than “pure” random walk
Start
Start
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington22
From theory to simulation
• Take the standard deviation (s.d.) of the probability distribution and the step length ...
0°
-90°
+/-180°
+90°
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington23
From theory to simulation
• Provides two simulation parameters, L and A
+90°
0°
-90°
+/-180°
A
L
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington24
Simulation
• Visual demonstration with simple cabbage layout
• Experiment Parameters
• Results
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington25
Visual demonstration - Step 0
L=10
A=20
Cabbage
Butterfly
Release boundary
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington26
Visual demonstration - Step 1
L=10
A=20
When move outside “world”, removed
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington27
Visual demonstration - Step 2
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington28
Visual demonstration - Step 3
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington29
Visual demonstration - Step 4
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington30
Visual demonstration - Step 6
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington31
Visual demonstration - Step 8
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington32
Visual demonstration - Step 10
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington33
Visual demonstration - Step 11
L=10
A=20
When intersect a cabbage, lay egg and “die”
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington34
Visual demonstration - Step 12 (End)
L=10
A=20
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington35
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”
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington36
Experimental Cabbage layout
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington37
Results Simulation
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington38
Results Simulation vs Field
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington39
Results Simulation vs Field
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington40
Resource concentration
Resource dilution
Ideal free distribution
Results Log Linear Regression
✔
H0 - β=0Field p-value = 0.037Simulation p-value = 0.0425
r2 (Field+Simulation) = 0.9
Simulation
Field
Ideal free distribution
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington41
Resource concentration
Resource dilution
Results Log Linear Regression
✔
H0 - β=0Field p-value = 0.037Simulation p-value = 0.0425
r2 (Field+Simulation) = 0.9
Simulation
Field
Ideal free distribution
Consistent with literature (Yamamura, 1999)
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington42
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington43
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington44
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 ...
✔
?
✔
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington45
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 ...
✔
?
✔
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington46
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 ...
✔
?
✔
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington47
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
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington48
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/
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington49
Questions ?
• With a simple correlated random walk we can predict egg distributions for Pieris
• With no attractive force we observe resource dilution
• Attractive force could potentially produce resource concentration ...
✔
?
✔
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington50
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.
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington51
Mean Squared Displacement
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington52
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 ?
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington53
Logr regression details
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington54
Published Parameters
•Published data:-Byers(2001) - derived from Root & Kareiva (1984)-A ≈ 50 degrees-L ≈ 2.5 m
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington55
Statistical tests
• H0 Field egg distribution = Simulation- observed → field, expected → simulation
• H0 Field regression slope (β) = Simulation
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington56
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)
✔
✘
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington57
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 ...
✔
✘
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington58
Field results
• Which do we observe in our field experiments ?
Resource concentration
Resource dilution
Ideal free distribution
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington59
Field results
Resource concentration
Resource dilution
Ideal free distribution
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington60
Resource concentration
Resource dilution
Ideal free distribution
Field results - log transformation
✔
✘
✘
H0 - β=0p-value = 0.037
r2 = 0.9
Field
Ideal free distribution
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington61
Resource dilution
Field results - log transformation
✔
H0 - β=0p-value = 0.037
r2 = 0.9
Field
Ideal free distribution
Consistent with literature (Yamamura, 1999)
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington62
Theory
• Is there a relationship between plant density and eggs per plant ?
1 plant
20 plants, density = 0.2
High Density
5 plants, density = 0.05
Low Density
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington63
Theory - Response to plant density
• Three possible responses ➔
1 plant
20 plants, density = 0.2
High Density
5 plants, density = 0.05
Low Density
© Jim Barritt 2006School of Biological Sciences, Victoria University, Wellington64
Yamamura 1999 Results