Animal Interactions and the Emergence of Territorial Patterns
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Transcript of Animal Interactions and the Emergence of Territorial Patterns
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Animal Interactions and the Emergence of Territorial Patterns
Jonathan R. PottsBristol Centre for Complexity Sciences & School of
Biological Sciences29 April 2010
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Outline
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Outline• Introduce the main problem: how territorial
and home-range patterns emerge from animal movements and interactions
![Page 4: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/4.jpg)
Outline• Introduce the main problem: how territorial and
home-range patterns emerge from animal movements and interactions
• Describe a model we’ve built to tackle this problem
![Page 5: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/5.jpg)
Outline• Introduce the main problem: how
territorial and home-range patterns emerge from animal movements and interactions
• Describe a model we’ve built to tackle this problem
• Results and analysis of the model
![Page 6: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/6.jpg)
Outline• Introduce the main problem: how
territorial and home-range patterns emerge from animal movements and interactions
• Describe a model we’ve built to tackle this problem
• Results and analysis of the model• Application to data set on the red fox
(Vulpes vulpes)
![Page 7: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/7.jpg)
Outline• Introduce the main problem: how
territorial and home-range patterns emerge from animal movements and interactions
• Describe a model we’ve built to tackle this problem
• Results and analysis of the model• Application to data set on the red fox
(Vulpes vulpes)• Questions
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How do home range and territory patterns emerge?
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How do home range and territory patterns emerge?
• Definitions: – An animal’s home range (HR) is the area in which it spends it’s time during
“everyday” activities.– An animal’s territory is a defended area from which conspecifics are excluded.
![Page 10: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/10.jpg)
How do home range and territory patterns emerge?
• Definitions: – An animal’s home range (HR) is the area in which it spends it’s time during “everyday”
activities.– An animal’s territory is a defended area from which conspecifics are excluded.
• Idea: They must both emerge somehow from the movements and interactions of the animals.
![Page 11: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/11.jpg)
How do home range and territory patterns emerge?
• Definitions: – An animal’s home range (HR) is the area in which it spends it’s time during “everyday”
activities.– An animal’s territory is a defended area from which conspecifics are excluded.
• Idea: They must both emerge somehow from the movements and interactions of the animals.
• Question: How does this happen?
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
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7.76x 10
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How do home range and territory patterns emerge?
• Definitions: – An animal’s home range (HR) is the area in which it spends it’s time during “everyday”
activities.– An animal’s territory is a defended area from which conspecifics are excluded.
• Idea: They must both emerge somehow from the movements and interactions of the animals.
• Question: How does this happen?
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
7.58
7.6
7.62
7.64
7.66
7.68
7.7
7.72
7.74
7.76x 10
4
Approach:• Build a model using features of the animals’ movements and interactions.• See which features are important by analysing the model’s output against HR patterns from the data.
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The red fox: a model example• Our model is based on
the behaviour of the urban red fox (Vulpes vulpes).
• Over 30 years of movement data in Bristol (collected by Steve Harris and co-workers).
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The red fox: a model example• Our model is based on
the behaviour of the urban red fox (Vulpes vulpes).
• Over 30 years of movement data in Bristol (collected by Steve Harris and co-workers).
Key features used in model:• Hinterland marker. Scents homogeneously as it moves.• Conspecific avoidance. On encountering the scent of a neighbour, the animal does not advance into the neighbouring territory.
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The red fox: a model example• Our model is based on
the behaviour of the urban red fox (Vulpes vulpes).
• Over 30 years of movement data in Bristol (collected by Steve Harris and co-workers).
Key features used in model:• Hinterland marker. Scents homogeneously as it moves.• Conspecific avoidance. On encountering the scent of a neighbour, the animal does not advance into the neighbouring territory.
The model can be applied to any animal with these two behavioural features.
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The model
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The model• Individuals exist in a lattice with periodic boundary conditions.
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Periodic boundary conditions???
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The model• Individuals exist in a lattice with periodic boundary conditions.
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The model• Individuals exist in a lattice with periodic boundary conditions.• They deposit scent at every site they visit.
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The model• Individuals exist in a lattice with periodic boundary conditions.• They deposit scent at every site they visit.• Scent remains for a fixed number of timesteps: the Active Scent
Time, TAS.
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The model• Individuals exist in a lattice with periodic boundary conditions.• They deposit scent at every site they visit.• Scent remains for a fixed number of timesteps: the Active Scent
Time, TAS.• If an individual is at a lattice site that does not contain foreign scent then it moves to a neighbouring lattice site at random.
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The model• Individuals exist in a lattice with periodic boundary conditions.• They deposit scent at every site they visit.• Scent remains for a fixed number of timesteps: the Active Scent
Time, TAS.• If an individual is at a lattice site that does not contain foreign scent then it moves to a neighbouring lattice site at random.• If an individual is at a lattice site that does contain foreign scent then it moves to a neighbouring lattice site that does not contain foreign scent (chosen at random).
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Model Demo Movie
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Model output – position density plots
• Left plot: the position densities of 8 animals after running the 2D simulation.• Additional feature: Boundary-dependent correlation (BDC). The random walk changes to a correlated RW after reaching the territory boundary. This correlation decays as the walker moves away.• Top right: 1D walkers with no BDC.• Below: 1D walkers with BDC.
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Quantifying the relationship between territories and home ranges
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Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
![Page 28: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/28.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.
![Page 29: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/29.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.
![Page 30: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/30.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.
• Random walker, constrained by nearby random walkers.
![Page 31: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/31.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
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Mean square displacement???
![Page 33: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/33.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
![Page 34: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/34.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
displacement
![Page 35: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/35.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
displacementsquare
![Page 36: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/36.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
displacementsquaremean
![Page 37: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/37.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
2/12)( kttb
displacementsquaremean
time
![Page 38: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/38.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
– The probability distribution is Gaussian (a.k.a. Normal), which means that the X% Minimum Convex Polygon (MCP) can be derived from the MSD of the distribution (e.g. if X=90, the width of 90% MCP of the boundary is 1.645*2*MSD1/2).
2/12)( kttb
![Page 39: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/39.jpg)
Quantifying the relationship between territories and home ranges
• To quantify this relation, we look at the movement of the territory boundaries (for which we use the 1D model).
• The boundaries obey a single file diffusion process.• Well understood in physics literature.• Key features:
– The mean square displacement (MSD) scales asymptotically as t1/2 (MSD = variance of the probability distribution) so that, at long times
where b(t) is the position of the boundary, k is a type of “(sub)-diffusion constant”, dependent on TAS and the population density, ρ.
– The probability distribution is Gaussian (a.k.a. Normal), which means that the X% Minimum Convex Polygon (MCP) can be derived from the MSD of the distribution (e.g. if X=90, the width of 90% MCP of the boundary is 1.645*2*MSD1/2).
2/12)( kttb Key quantity to understand
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Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2.
2/12)( kttb
![Page 41: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/41.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2.• TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps
2/12)( kttb
![Page 42: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/42.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2.• TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps • Why the product of ρ2 and TAS?
2/12)( kttb
![Page 43: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/43.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2.• TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps • Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
2/12)( kttb
![Page 44: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/44.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
![Page 45: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/45.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 46: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/46.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 47: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/47.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 48: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/48.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 49: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/49.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 50: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/50.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 51: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/51.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
First passage time = TFP
![Page 52: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/52.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 53: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/53.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 54: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/54.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 55: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/55.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
still active?
![Page 56: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/56.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 57: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/57.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 58: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/58.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 59: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/59.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 60: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/60.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
scent
![Page 61: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/61.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps• Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).
Territory
No active scent!!!
![Page 62: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/62.jpg)
Quantifying the boundary movement
• The value of k decreases exponentially with the product of TAS and ρ2. • TAS↔k ↔ distribution of b(t) ↔ boundary MCP ↔ HR size and overlaps • Why the product of ρ2 and TAS?• For large TAS, it turns out that ρ2 is approximately 1/TFP, where TFP is the time it takes, on average, for the individual to go from one boundary to the other (the first-passage time).• If we increase TAS/TFP, and hence TASρ2, we would expect to wait longer between successive boundary movements, so k decreases.
2/12)( kttb
![Page 63: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/63.jpg)
To summarise (avec sans maths)…
![Page 64: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/64.jpg)
To summarise….
• We have a quantitative predictive theory that relates the active scent time TAS to the home range patterns.
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
7.58
7.6
7.62
7.64
7.66
7.68
7.7
7.72
7.74
7.76x 10
4
![Page 65: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/65.jpg)
To summarise….
• We have a quantitative predictive theory that relates the active scent time TAS to the home range patterns.
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
7.58
7.6
7.62
7.64
7.66
7.68
7.7
7.72
7.74
7.76x 10
4maths
![Page 66: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/66.jpg)
To summarise….
• We have a quantitative predictive theory that relates the active scent time TAS to the home range patterns.
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
7.58
7.6
7.62
7.64
7.66
7.68
7.7
7.72
7.74
7.76x 10
4
physiological
maths
![Page 67: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/67.jpg)
To summarise….
• We have a quantitative predictive theory that relates the active scent time TAS to the home range patterns.
5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74
x 104
7.58
7.6
7.62
7.64
7.66
7.68
7.7
7.72
7.74
7.76x 10
4
physiological ecological
maths
![Page 68: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/68.jpg)
But does the theory fit with the data?
![Page 69: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/69.jpg)
Verifying the model with fox data
![Page 70: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/70.jpg)
Verifying the model with fox data
• Applying our theory to fox data on home range patterns derived from position fixes, we find that
daysTAS0.35.17.2
![Page 71: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/71.jpg)
Verifying the model with fox data
• Applying our theory to fox data on home range patterns derived from position fixes, we find that
• Is this realistic?
daysTAS0.35.17.2
![Page 72: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/72.jpg)
Verifying the model with fox data
• Applying our theory to fox data on home range patterns derived from position fixes, we find that
• Is this realistic?• During the mange epizootic in Bristol 1994-5, there
was a time-lag of about 3-4 days between territories being vacated and then being taken over by other foxes, suggesting that our prediction is roughly correct.
daysTAS0.35.17.2
![Page 73: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/73.jpg)
Conclusions
![Page 74: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/74.jpg)
Conclusions• Our model explains the mechanisms that cause
“macroscopic” home range and territorial patterns to emerge from “microscopic” animal movements and interactions.
![Page 75: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/75.jpg)
Conclusions• Our model explains the mechanisms that cause “macroscopic”
home range and territorial patterns to emerge from “microscopic” animal movements and interactions.
• We have a quantitative predictive theory that relates a physiological property (TAS) of an animal to a macroscopic ecological property of the animal (the home range patterns).
![Page 76: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/76.jpg)
Conclusions• Our model explains the mechanisms that cause
“macroscopic” home range and territorial patterns to emerge from “microscopic” animal movements and interactions.
• We have a quantitative predictive theory that relates a physiological property (TAS) of an animal to a macroscopic ecological property of the animal (the home range patterns).
• Analysis of red fox data suggests that our predictive theory is realistic.
![Page 77: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/77.jpg)
Conclusions• Our model explains the mechanisms that cause
“macroscopic” home range and territorial patterns to emerge from “microscopic” animal movements and interactions.
• We have a quantitative predictive theory that relates a physiological property (TAS) of an animal to a macroscopic ecological property of the animal (the home range patterns).
• Analysis of red fox data suggests that our predictive theory is realistic.
• Since our model makes few assumptions, it can readily be extended– as a basis for analysing territorial defence strategies (e.g.
hinterland vs. borderland)– to factor in underlying geography/resource distribution– to try to explain core-area emergence – and probably more (insert your idea here)!
![Page 78: Animal Interactions and the Emergence of Territorial Patterns](https://reader035.fdocuments.net/reader035/viewer/2022062323/56816050550346895dcf7aa2/html5/thumbnails/78.jpg)
Acknowledgements, Questions
• Thanks to – my supervisors, Luca Giuggioli and Stephen Harris– the Mammal Group at Bristol– BCCS – EPSRC
• Thank you for listening. Any questions?
These slides are on the web: http://www.bio.bris.ac.uk/research/mammal/spaceuse.html