Predator-Prey Oscillations in Space (again)

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Predator-Prey Oscillations in Space
(again) Sandi Merchant D-dudes
meeting November 21, 2005
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Review of the Model ODE I

• Starts with a standard ODE predator-prey model
• Logistic prey growth
• Constant predator death rate B
• Type II predator functional response
• Behaviour of this model is fairly well-understood

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Review of the Model ODE II

• Equilibria and Stability

Extinction of both species Always unstable
(saddle)
Extinction of predator Prey at carrying
capacity Stable node or unstable saddle
Coexistence of predator and prey Can be any
type of steady state
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Review of the Model ODE III

• Division of Parameter Space -- C0.2
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
• Sample Parameter Path fix A0.5, vary B
  (predator death rate)

Red prey only eq. stable (coexistence
unstable) Blue stable coexistence (no limit
cycle) White stable oscillations
(coexistence unstable)
B
A
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Review of the Model ODE IV

• B 0.6 --- predator goes extinct (prey to
  carrying capacity)

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Review of the Model ODE IV

• B 0.4 --- convergence to coexistence equilibrium

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Review of the Model ODE IV

• B 0.345 --- damped oscillations to coexistence
  equilibrium

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Review of the Model ODE IV

• B 0.33 --- small amplitude high frequency
  oscillations

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Review of the Model ODE IV

• B 0.1 --- large amplitude oscillations

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Review of the Model ODE IV

• B 0.025 --- large amplitude long period
  oscillations

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Review of the Model PDE I

• What happens if there is a biological invasion in
  this type of system?
• Invasion speed
• Spatial and temporal pattern after invasion
  
• Sherratt et al. (1997) studied this.
• Added diffusion to make PDE model
• Numerically simulated invasion of predator into
  prey at carrying capacity

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PDE Model Behaviour

• Sherratt et al. found surprising spatiotemporal
  patterns developed when simulating invasion in
  this way.
• Travelling wavetrains/plane waves
• Spatiotemporal Chaos
  
• Implications of applying such a model to real
  systems
• Predators can cause prey populations to oscillate
  or even behave chaotically after invasion!
• ODE and PDE model predictions do not agree.
  
• Sherratt et al. simply showed that these
  behaviours existed... no (compelling) explanation
  of why or how these patterns emerge.
  

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PDE Model Behaviour

• I set out to understand this model better
  
• Simulated the same system as Sherratt et al., for
  a variety of parameter values.
  
  
• Example Same parameter path as ODE graphs
  
  
  
  
  
• Initial Condition for all simulations
  
• Prey density 1 everywhere
• Predator density 0 everywhere, except
• Predator density 0.5 for x in the interval
  0,5

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PDE Model Behaviour

• B0.35
  
• Ordinary Travelling Wavefront
  
  
  
  
  
  
  
  
  
• Wavefront Speed 0.494

1.5 1
Prey Density
0 x
1000
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PDE Model Behaviour

• B0.32
  
• Small Amplitude Wavetrains behind front
  
  
  
  
  
  
  
  
  
• Wavefront Speed 0.605

1.5 1
Prey Density
0 x
1000
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PDE Model Behaviour

• B0.25
• Large Amplitude Wavetrain behind damped
  oscillatory front
  
  
  
  
  
  
  
  
  
• Wavefront Speed 0.791

1.5 1
Prey Density
0 x
1000
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PDE Model Behaviour

• B0.1
• Behaviour getting a little less predictable
  
  
  
  
  
  
  
  
  
  
  
• Wavefront speed 1.088

1.5 1
Prey Density
0 x
1000
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PDE Model Behaviour

• Can get quite chaotic-looking
  
  
  
  
  
  
  
  
  
  
  
  
  

1.5 1
0 x
1000
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Patterns Observed

• Invasion Speed (speed of front) increases as B
  decreases
  

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Patterns Observed

• More complex spatiotemporal pattern as B
  decreases
• Nothing ---gt wavefront ---gt wavetrain ---gt chaos
  
• Counterintuitive?
• Invasion speed increases with increasing
  complexity
• Wavetrain-type solutions sub-patterns
• Fixed damped oscillation wavefront moving at
  constant speed forwards
• Wavetrains moving at different speed, usually in
  reverse direction.
• Amplitude and frequency of wavetrains increases
  as B is decreased
  

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Patterns Observed

• Relationship between wavetrain and wavefront
• There appears to be some interaction between the
  damping behind the wavefront and the wavetrains
• Wavetrains originate at the tail of the
  wavefront
• More oscillatory wavefronts seem to produce
  larger amplitude wavetrains why?
  
• Is there a relationship between the speed of the
  wavetrain and the speed of the wavefront?
  
• Have not measured wavetrain speed

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Two Big Questions

• Are these patterns real?
• Always possible that simply result of numerical
  scheme/method of simulation
• Might be only transitional behaviour
• Could be result of boundary conditions (no-flux)
  
• Where do these patterns come from?
• Can we show that certain solutions bifurcate from
  other solutions?
• Is the behaviour of the model predictable? (ie.
  relatively insensitive to parameters)
• Often in chaotic systems this is not the case
• Is there a predictable set of transitions before
  chaos?
• Otherwise, maybe useless for applications
  
  

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Plan to Answer (1)

• Are these patterns real?
• Analytically finding these solutions is likely
  impossible
• Getting the same solutions using alternative
  numerical schemes would help verify their
  existence
• I plan to show that the wavetrains alone are
  solutions (in a non-invasion scenario)
• Make a new simulation with periodic boundary
  conditions and a domain length of one spatial
  period see if same solutions arise.
  
  
  
• Use two different packages for the numerical
  simulation
• Matlab as for invasion simulations
• AUTO a numerical continuation package

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Plan to Answer (2)

• Where do these patterns come from?
• My Hypothesis as B is decreased
• Prey only solution (no invasion) loses stability
  to coexistence eq. solution (ordinary wavefront)
• Coexistence solution loses stability to wavetrain
  solution (Hopf bifurcation)
• Wavetrain solution loses stability eventually
  resulting in spatiotemporal chaos
• How to test hypothesis
• If the solutions are real (question (1)), then
  examine their stability
• Compute the spectrum of the various steady
  states
• If the spectrum of a particular solution crosses
  the imaginary axis (real part zero), then it
  loses stability
• If this happens precisely where the new solution
  types appear, then it is evidence
• Pattern of the crossing may also lend support
  
  

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How to Compute the Spectrum?

• Unlike ODE models, spectrum can be a continuous
  curve of values
• Substantially more difficult to compute than
  eigenvalues for ODEs
• Most common methods involve discretizing with
  finite differences and computing the eigenvalues
  of a HUGE matrix
• I will use a new method developed by Rademacher
  et al (2005)
• Convert eigenvalue problem to a BVP
• Use AUTO to path-follow solution to the BVP
• Very efficient )
• Involves programming in FORTRAN (
  

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Spectrum of Wavetrain Solution

• Wavetrain solution found using XPP/AUT
  
• Unstable because crosses imaginary axis
  
• I have yet to find a stable wavetrain solution
  
  

Imag. part
Real part
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Challenges

• All the outlined methods for computing stability
  only determines the linear stability
• It is possible that true (nonlinear) stability
  does not necessarily follow
• Need to learn and apply PDE dynamical systems
  theory
  
• What if my hypotheses are wrong?
• All of the above work may not end up telling us
  anything about how the spatial patterns emerge
• The application of the new method of computing
  essential spectra is still a novel and useful
  exercise

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Challenges

• Need to be able to compute the speed and period
  of wavetrains accurately in order for AUTO work
  to be possible