SLIDE 1
Outline
- Insect outbreaks
- Front propagation
Khain, Lin, and Sander, EPL (2011)
Role of fluctuations in front propagation: the insect outbreak model - - PowerPoint PPT Presentation
Role of fluctuations in front propagation: the insect outbreak model - - PowerPoint PPT Presentation
Role of fluctuations in front propagation: the insect outbreak model Evgeniy Khain Department of Physics Oakland University Michigan, USA Collaborators Len Sander, Nat Lin Physics Department, University of Michigan Outline Insect
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Spruce Budworm outbreaks
Western Spruce Budworm is the most destructive defoliator of coniferous forests (spruces, hemlocks, pines and firs) in Western North America. The eastern version of the Spruce Budworm is Choristoneura fumiferana, which is one of the most destructive native insects in the northern spruce and fir forests of the Eastern United States and Canada. The budworm outbreaks occur every 30–60 years, whereby populations jump suddenly from endemic to epidemic levels. Western spruce budworm branch damage
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The system of equations describes both spruce budworm dynamics and tree dynamics (total surface area of the branches and health of the trees). (89 citations) (297 citations)
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Red dashed line: leaf area Blue solid line: insect population Now I will consider a much shorter time scale and focus only on spruce budworm dynamics An example based on the model presented by Robert May (Nature, 1977)
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The simplest insect outbreak model
J.D. Murray, Mathematical Biology,
- I. An Introduction, page 7.
2 2 2
1 N A BN K N N r dt dN
B B
logistic growth predation by birds (“type III response”) birth rate carrying capacity
predation term
N A B
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The simplest outbreak model (cont.)
2 2
1 1 u u q u ru dt du
In dimensionless form: There is a parameter region in (r,q) plane, where there are two stable states: u1 – normal state (small population size) u3 – outbreak state
u3 u1
u ξ
2 2
x u D
f(u) u
u1 u3 f(u)
) ( vt x u u
SLIDE 8
Front velocity
u3 u1
u ξ
f(u) u
u1 u3
) ( vt x u u
2 2
) ( x u D u f t u
u D u f u v ) (
d u du u f v
u u 2
) ( ) (
3 1
front velocity
d u equation ) (
front velocity can be found by using a standard shooting procedure
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Discrete stochastic lattice model
every site can be occupied by any number of insects A randomly picked particle can jump to a neighboring site (to the right or to the left), proliferate or die with probabilities related to the diffusion, birth and death rates on the site
) 2 /( D r p
b death
) 2 /( D r D p p
b left right
2 2 i i b i b
N A BN K N r
the death rate
2 2 2 2 2
1 x N D N A BN K N N r t N
B B
i
) 2 /( D r r p
b b birth
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Computing front velocity in simulations of discrete stochastic model
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Front velocity: continuum and discrete
The continuum stall point r* does not coincide with the discrete stall point rd* ! For r* < r < rd* : Continuum case: v > 0, the outbreak (N3) state wins Discrete case: v < 0, the normal (N1) state wins
N3 N1
N x
v < 0 v > 0
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The reason for stochastic correction: spontaneous jumps between the two states
discrete N3 discrete N1 Single site spontaneous jumps N3 N1 Single site spontaneous jumps N1 N3
- E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)
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Verifying the effect of spontaneous transitions
A characteristic time for spontaneous transition N1 N3
3 1
t
1 3
t
A characteristic time for spontaneous transition N3 N1
The times can be computed using the “rare events” theory,
Doering, Sargsyan, and Sander (2005); Doering, Sargsyan, Sander, and Vanden-Eijnden (2007)
- E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)
1
1 3 3 1
t t k
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discrete N3 discrete N1
The role of population size The role of diffusion is less clear
The transition time scales exponentially with A – a scaling factor for number of particles in a single site
1 1 3 3
u N u N A
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Summary
- I showed an example of a bistable system, where the direction
- f front propagation can be reversed by fluctuations.
- This effect results from spontaneous transitions between the
two (meta)stable states, which are not taken into account in the continuum approach.
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Role of fluctuations in front propagation: the insect outbreak model Abstract: Propagating fronts arising from bistable reaction diffusion equations are a purely deterministic effect. Stochastic reaction diffusion processes also show front propagation which coincides with the deterministic effect in the limit of small fluctuations (usually, large populations). However, for larger fluctuations propagation can be affected. We give an example, based on the classic spruce-budworm model, where the direction of wave propagation, i.e., the relative stability of two phases, can be reversed by fluctuations.
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) ( ) ( exp
3 2 1 3
u u A t t
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Relaxation oscillations
SLIDE 20
The simplest outbreak model (cont.)
2 2
1 1 u u q u ru dt du
In dimensionless form: There is a parameter region in (r,q) plane, where there are two stable states: u1 – normal state (small population size) u3 – outbreak state
u3 u1
u ξ
2 2
x u D
f(u) u
u1 u3 f(u)
) ( vt x u u
d u du u f v
u u 2
) ( ) (
3 1
front velocity
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Discrete stochastic lattice model
every site can be occupied by any number of insects A randomly picked particle can jump to a neighboring site (to the right or to the left), proliferate or die with probabilities related to the diffusion, birth and death rates on the site
) 2 /( D r p
b death
) 2 /( D r r p
b b birth
) 2 /( D r D p p
b left right
2 2 i i b i b
N A BN K N r
the death rate
2 2 2 2 2
1 x N D N A BN K N N r t N
B B
i
SLIDE 22
Computing front velocity in discrete simulations
SLIDE 23
Front velocity: continuum and discrete
The continuum stall point r* does not coincide with the discrete stall point rd* ! For r* < r < rd* : Continuum case: v > 0, the outbreak (N3) state wins Discrete case: v < 0, the normal (N1) state wins
SLIDE 24
The reason for stochastic correction: spontaneous jumps between the two states
discrete N3 discrete N1 Single site spontaneous jumps N3 N1 Single site spontaneous jumps N1 N3
- E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)
SLIDE 25
Verifying the effect of spontaneous transitions
A characteristic time for spontaneous transition N1 N3
3 1
t
1 3
t
A characteristic time for spontaneous transition N3 N1
The times can be computed using the “rare events” theory,
Doering, Sargsyan, and Sander (2005); Doering, Sargsyan, Sander, and Vanden-Eijnden (2007)
- E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)
1
1 3 3 1
t t k
SLIDE 26
discrete N3 discrete N1
The role of population size The role of diffusion is less clear
The transition time scales exponentially with A – a scaling factor for number of particles in a single site
1 1 3 3
u N u N A
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Summary III
- I showed an example of a bistable system, where the direction
- f front propagation can be reversed by fluctuations.
- This effect results from spontaneous transitions between the