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 outbreaks • Front propagation Khain, Lin, and Sander, EPL (2011) Methods: discrete approach and continuum description

Spruce Budworm outbreaks Western Spruce Budworm is the most destructive defoliator of coniferous forests (spruces, hemlocks, pines and firs) in Western North America. Western spruce budworm branch damage 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.

(297 citations) (89 citations) The system of equations describes both spruce budworm dynamics and tree dynamics (total surface area of the branches and health of the trees).

An example based on the model presented by Robert May (Nature, 1977) 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

The simplest insect outbreak model J.D. Murray, Mathematical Biology, I. An Introduction, page 7.    2 dN N BN     r N 1    B 2 2   dt K A N B predation logistic growth birth rate by birds (“type III response”) B carrying capacity predation term N A

The simplest outbreak model (cont.)     2 2 du u u u      D  In dimensionless form: ru 1    2 2   dt q 1 u x f(u) There is a parameter region in (r,q) plane, where there are two stable states: f(u) u 1 – normal state (small population size) u 3 – outbreak state u u u 3 u 3 u 1     u u x vt ( ) ξ u 1

Front velocity u u 3 f(u) u     u u x vt ( ) ξ u 3 u 1 u 1    2 u u       f ( u ) D ( equation ) u d   2 t x   u 3        f u du v u f ( u ) D u ( )  u v 1 front velocity front velocity can be found    by using a standard shooting procedure  2 ( u ) d  

Discrete stochastic lattice model every site can be occupied by any number of insects i 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     the death rate p r /( r 2 D ) birth b b      r N BN p /( r 2 D )    b i i death b  2 2 K A N      p p D /( r 2 D ) b i right left b      2 2 N N BN N      r N 1 D      B 2 2 2   t K A N x B

Computing front velocity in simulations of discrete stochastic model

Front velocity: continuum and discrete N N 3 v > 0 v < 0 N 1 x The continuum stall point r* does not coincide with the discrete stall point r d * ! Continuum case: v > 0, the outbreak (N 3 ) state wins For r* < r < r d * : Discrete case: v < 0, the normal (N 1 ) state wins

The reason for stochastic correction: spontaneous jumps between the two states discrete N 3 Single site Single site spontaneous jumps spontaneous jumps N 3 N 1 N 1 N 3 discrete N 1 E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)

Verifying the effect of spontaneous transitions t A characteristic time for spontaneous transition N 1 N 3 1  3 t 3  A characteristic time for spontaneous transition N 3 N 1 1 The times can be computed using the “rare events” theory, Doering, Sargsyan, and Sander (2005); Doering, Sargsyan, Sander, and Vanden-Eijnden (2007) t    1 3 k 1 t  3 1 E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)

The role of population size discrete N 3 discrete N 1 The transition time scales exponentially with A – N N   3 1 A a scaling factor for number of particles in a single site u u 3 1 The role of diffusion is less clear

Summary • I showed an example of a bistable system, where the direction of 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.

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.

        t t exp A ( u ) ( u )  3 1 0 2 3

Relaxation oscillations

The simplest outbreak model (cont.)     2 2 du u u u      D  In dimensionless form: ru 1    2 2   dt q 1 u x f(u) There is a parameter region in (r,q) plane, f(u) where there are two stable states: u 1 – normal state (small population size) u 3 – outbreak state u u u 3 u 3 u 1 u 3      f ( u ) du u u x vt ( )  u v 1 front velocity  ξ   u 1  2 u d ( )  

Discrete stochastic lattice model every site can be occupied by any number of insects i 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     p r /( r 2 D ) the death rate birth b b      r N BN p /( r 2 D )    b i i death b  2 2 K A N      p p D /( r 2 D ) b i right left b      2 2 N N BN N      r N 1 D      B 2 2 2   t K A N x B

Computing front velocity in discrete simulations

Front velocity: continuum and discrete The continuum stall point r* does not coincide with the discrete stall point r d * ! Continuum case: v > 0, the outbreak (N 3 ) state wins For r* < r < r d * : Discrete case: v < 0, the normal (N 1 ) state wins

The reason for stochastic correction: spontaneous jumps between the two states discrete N 3 Single site Single site spontaneous jumps spontaneous jumps N 3 N 1 N 1 N 3 discrete N 1 E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)

Verifying the effect of spontaneous transitions t A characteristic time for spontaneous transition N 1 N 3 1  3 t 3  A characteristic time for spontaneous transition N 3 N 1 1 The times can be computed using the “rare events” theory, Doering, Sargsyan, and Sander (2005); Doering, Sargsyan, Sander, and Vanden-Eijnden (2007) t    1 3 k 1 t  3 1 E. Khain, Y. T. Lin, and L.M. Sander, EPL (2011)

The role of population size discrete N 3 discrete N 1 The transition time scales exponentially with A – N N   3 1 A a scaling factor for number of particles in a single site u u 3 1 The role of diffusion is less clear

Summary III • I showed an example of a bistable system, where the direction of 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.