Animal Interactions and the Emergence of Territorial Patterns

Jonathan R. PottsBristol Centre for Complexity Sciences & School of

Biological Sciences29 April 2010

Outline

Outline• Introduce the main problem: how territorial

and home-range patterns emerge from animal movements and interactions

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

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

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)

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

How do home range and territory patterns emerge?

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.

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.

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?

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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

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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.

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).

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 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.

The model

The model• Individuals exist in a lattice with periodic boundary conditions.

Periodic boundary conditions???

The model• Individuals exist in a lattice with periodic boundary conditions.

The model• Individuals exist in a lattice with periodic boundary conditions.• They deposit scent at every site they visit.

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.

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.

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).

Model Demo Movie

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.

Quantifying the relationship between territories and home ranges

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).

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.

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.

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.

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

Mean square displacement???

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

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

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

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

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

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

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

Quantifying the boundary movement

• The value of k decreases exponentially with the product of TAS and ρ2.

2/12)( kttb

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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!!!

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

To summarise (avec sans maths)…

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

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

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

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

But does the theory fit with the data?

Verifying the model with fox data

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

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

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

Conclusions

Conclusions• Our model explains the mechanisms that cause

“macroscopic” home range and territorial patterns to emerge from “microscopic” animal movements and interactions.

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).

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.

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)!

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