Mechanistic-statistical modeling of pest invasions
Lionel Roques
(http://lionel.biosp.org)
with Jérôme Rousselet & Samuel Soubeyrand (INRA)
INRA Biostatistics and Spatial Processes (BioSP) Lab – Avignon – France
Biostatistique
& Processus Spatiaux
SEMINAR ON PEST RISK ASSESSMENT - October 3rd 2012
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Part I
Differential equations as mechanistic
models of population dynamics
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Ordinary differential equations
Non-spatial models. Dynamics of the population size N(t):
0
N (t) = f (N), t > 0,
N(0) = N0 .
Derivation:
- Step 1:
N(t + τ ) − N(t)
= (births-deaths) during τ,
= N(t) × (birth rate-death rate) × τ.
- Step 2:
τ → 0 ⇒ N 0 (t) = N(t) × (birth rate-death rate).
Define r (N) = (birth rate-death rate) = per capita growth rate =
f (N)
.
N
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Usual growth terms f (N)
• Constant growth rate r : Malthusian growth
f (N) = r N, and N(t) = N0 e r t .
• Density dependence r (N) = r0 (1 − N/K ) : logistic growth
r0N
f (N) = r0 N(1 −
N
)
K
K
• Allee effect r (N) = (1 − N/K )(N − ρ), with Allee threshold ρ
f (N) = N (1 −
N
K )(N
− ρ)
0
ρ
K
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Diffusion equation (=heat equation)
Spatio-temporal model without reproduction. Population density u(t, x ):
(
∂u
(t, x ) = D ∆u, t > 0, x ∈ Rn ,
∂t
u(0, x ) = u0 (x ).
∆u : n-dimensional Laplace dispersal operator. ∆u =
n
X
∂2 u
k=1
∂ x2
.
Random walk derivation (1D case), cf. Turchin (1998). Assumptions:
• individuals are independent;
• each time step τ, move left or right at a distance λ.
1/2
x0-λ
1/2
x0
x0+λ
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Derivation of the diffusion equation
Assume that there is 1 individual.
p(t, x ): probability of finding the individual at (t, x ).
p(t + τ, x ) =
1
1
p(t, x − λ) + p(t, x + λ).
2
2
Series expansions:
2
p(t + τ, x ) = p(t, x ) + τ ∂p
∂t + O(τ 2 ),
∂p
λ2 ∂ p
p(t, x + λ) = p(t, x ) + λ ∂x + 2 ∂x 2 + O(λ3 ),
2
λ2 ∂ p
3
p(t, x − λ) = p(t, x ) − λ ∂p
∂x + 2 ∂x 2 + O(λ ),
Finally,
∂p
∂2p
λ2
= D 2 , with D = lim
.
τ →0,λ→0 2τ
∂t
∂x
Law of large numbers →
∂u
∂2u
= D 2 for pop. density u(t, x ).
∂t
∂x
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Reaction-diffusion equations in heterogeneous environments
Spatio-temporal models with reproduction. Population density u(t, x ):
∂u
(t, x ) = D(t, x , u) + f (t, x , u), t > 0, x ∈ Rn .
| {z }
| {z }
∂t
dispersal
growth
D(t, x , u) : local dispersal operator; ex.: ∆u, ∆(D(t, x )u), . . . ...
f (t, x , u) : growth function; ex.: logistic growth, space-time dependent
intrinsic growth rate:
f (t, x , u) = r (t, x ) u − γ u 2 ,
r (t, x ) > 0 : favorable regions; r (t, x ) < 0 : unfavorable regions.
Reaction-diffusion models for species invasions and spread: Shigesada
and Kawasaki (1997), Turchin (1998), Murray (2002), Okubo and Levine
(2002)...
Ordinary differential equations
Diffusion equation
Reaction-diffusion equations
Integro-differential equations
Integro-differential equations
Spatio-temporal models with reproduction. Population density u(t, x ):
∂u
(t, x ) = D(t, x , u) + f (t, x , u), t > 0, x ∈ Rn .
| {z }
| {z }
∂t
dispersal
growth
D(t, x , u) : nonlocal dispersal operator; ex.:
Z
D(t, x , u) = D(u)(t, x ) =
J(|x − y |) (u(t, y ) − u(t, x )) dy .
Rn
Dispersal kernel J(λ) : probability distribution of moving at a distance λ.
• Thin-tailed kernels ' local dispersal operators;
• fat-tailed kernels:
- take long-distance dispersal events into account;
- fast spreading with acceleration of the population range (Kot and
Lewis 1996; Garnier 2011).
Fitting models based on differential equations
Part II
Mechanistic-statistical modeling
Fitting models based on differential equations
Motivation and difficulties
Mechanistic model uθ = M(θ) parametrized by θ (coefficients, initial
condition, ...)
Aim: infer the parameter θ, based on observations O.
Usual method: compare the observations O with uθ̃ .
Difficulties: related with the data O:
• noisy data, binarized signal;
• strong spatial and temporal censorship;
• indirect observation of uθ (attack rates, ...).
→ How to compare O and uθ̃ ?
Fitting models based on differential equations
Main idea of the mechanistic-statistical approach
O: observation of a process J(uθ ) (attack rates, ...) with uθ = M(θ).
• Build a statistical model of the observation process: O ∼ S(J(uθ )).
• Compute a likelihood L(θ̃) = P(O|uθ̃ ) = P(O|θ̃).
• Various estimation methods can be used: maximum likelihood,
Bayesian method, ...
Remark: mechanistic-statistical models ⊂ state-space models.
Examples of use:
- Environmental sciences: Berliner (2003), Wikle (2003a) and
Campbell (2004).
- Ecology: Wikle (2003b), Buckland et al. (2004), Rivot et al.
(2004), Soubeyrand et al. (2009).
Mechanistic model
Statistical model
Part III
A toy example: a reaction-diffusion
model for the spread of an invasive
species
Validation
Mechanistic model
Statistical model
Mechanistic model
Assumption: an invasive species arrives into a heterogeneous landscape.
Chosen approach: reaction-diffusion, spatially heterogeneous coefficients:

∂u


= D ∆ u + r (x ) u − γu 2 , t > 0, x ∈ Ω = (0, 1)2 ,
∂t
(Mθ )
reflecting boundary conditions,


−kx −x0 k
initial condition: u(0, x ) = e
, x ∈ Ω.
Unknown coefficients: θ = (D, γ, r (x )) :
• D, γ are positive constants.
• r (x ) is piecewise constant : r = (r1 , r2 , r3 , r4 ).
Validation
Mechanistic model
Statistical model
Mechanistic model
“Patchy environment": mosaic of habitats
Figure: 4 types of habitat where r (x ) is constant r1 , r2 , r3 , r4 .
Validation
Mechanistic model
Statistical model
Mechanistic model
“Patchy environment": mosaic of habitats
Figure: Introduction point.
Validation
Mechanistic model
Statistical model
Mechanistic model
“Patchy environment": mosaic of habitats
Figure: Obervation sites ωi .
Validation
Mechanistic model
Statistical model
Validation
Statistical model for the observations
Typical example of indirect observations:
Impact of the pest = number of attacked hosts
Spatial and temporal censorship:
- observations in small regions ωi ;
- at some fixed times τi,j .
Model assumptions:
• 1. impact in ωi at time τi,j proportional to the number of
“pest-months":
Z
τi,j
Z
J(uθ )(ωi , τi,j ) = α
u(t, x ) dx dt;
0
ωi
• 2. observations in ωi at time τi,j follow indep. Poisson distributions
(mean value ρ J(uθ )(ωi , τi,j )):
Oi,j ∼indep Poisson{ρ J(uθ )(ωi , τi,j )}.
Mechanistic model
Statistical model
Validation
Statistical model for the observations
Typical example of indirect observations:
Impact of the pest = number of attacked hosts
Spatial and temporal censorship:
- observations in small regions ωi ;
- at some fixed times τi,j .
Model assumptions:
• 1. impact in ωi at time τi,j proportional to the number of
“pest-months":
Z
τi,j
Z
J(uθ )(ωi , τi,j ) = α
u(t, x ) dx dt;
0
ωi
• 2. observations in ωi at time τi,j follow indep. Poisson distributions
(mean value ρ J(uθ )(ωi , τi,j )):
Oi,j ∼indep Poisson{ρ J(uθ )(ωi , τi,j )}.
Mechanistic model
Statistical model
Validation
Simulated data set
θ = (D, γ, r) with:
D = 5 · 10−2 , γ = 1, r = (r1 , r2 , r3 , r4 ) = (4, 2, 0, −4), α = 104 , ρ = 0.8.
Solution uθ of the mechanistic model Mθ :
1.7
0
(a) t = 0
Mechanistic model
Statistical model
Validation
Simulated data set
θ = (D, γ, r) with:
D = 5 · 10−2 , γ = 1, r = (r1 , r2 , r3 , r4 ) = (4, 2, 0, −4), α = 104 , ρ = 0.8.
Solution uθ of the mechanistic model Mθ :
1.7
0
(b) t = 0.2
Mechanistic model
Statistical model
Validation
Simulated data set
θ = (D, γ, r) with:
D = 5 · 10−2 , γ = 1, r = (r1 , r2 , r3 , r4 ) = (4, 2, 0, −4), α = 104 , ρ = 0.8.
Solution uθ of the mechanistic model Mθ :
1.7
0
(c) t = 0.5
Mechanistic model
Statistical model
Validation
Simulated data set
θ = (D, γ, r) with:
D = 5 · 10−2 , γ = 1, r = (r1 , r2 , r3 , r4 ) = (4, 2, 0, −4), α = 104 , ρ = 0.8.
Solution uθ of the mechanistic model Mθ :
1.7
0
(d) t = 4
Mechanistic model
Statistical model
Validation
Simulated data set
θ = (D, γ, r) with:
D = 5 · 10−2 , γ = 1, r = (r1 , r2 , r3 , r4 ) = (4, 2, 0, −4), α = 104 , ρ = 0.8.
Observations are carried out with the statistical model:
400
350
300
250
Oi,j 200
150
100
50
0
0
100
200
300
400
500
J(uθ)(ωi,τi,j)
Figure: Real impact vs observation: J(uθ )(ωi , τi,j ) vs Oi,j .
Mechanistic model
Statistical model
Validation
Computation of the likelihood
Likelihood L(θ̃) of a parameter θ̃: P(O|J(uθ̃ )) :
L(θ̃) =
Y
ωi ,τi,j
e
−ρ J(uθ̃ )(ωi ,τi,j )
ρ J(uθ̃ )(ωi , τi,j )
Oi,j !
Oi,j
,
where:
• Oi,j : number of observed attacked hosts in ωi at time τi,j ;
• uθ̃ solution of the mechanistic model with parameter θ̃.
Mechanistic model
Statistical model
Bayesian estimation of the parameters
Aim: to estimate θ using the observations Oi,j .
Bayesian method
- Uniform prior distributions:
π(θ̃) =
1(10−2 ≤ D ≤ 1, 0.1 < γ ≤ 10, −5 ≤ r1 , . . . , r4 ≤ 5)
.
0.99 × 9.9 × 104
- Posterior distribution:
P(θ̃ | O) = R
L(θ̃)π(θ̃)
.
L(ρ) π(ρ)dρ
R∗ ×R∗ ×R4
+
+
Estimation of the posterior distribution by MCMC algorithm.
Validation
Mechanistic model
Statistical model
Validation
Results: marginal distributions of the parameters D, γ and r
70
60
50
40
30
20
10
0
0
0.05
0.1
D
0.15
0.2
Mechanistic model
Statistical model
Validation
Results: marginal distributions of the parameters D, γ and r
3
2.5
2
1.5
1
0.5
0
0
1
2
γ
3
4
Mechanistic model
Statistical model
Validation
Results: marginal distributions of the parameters D, γ and r
4
3
2
1
0
−5 −4 −3 −2 −1
0
r
1
2
3
4
5
Mechanistic model
Statistical model
Validation
Conclusions
• Identifiability and good estimation of the diffusion, competition and
growth parameters;
• works with strongly censored data: observation set < 10% of Ω;
no measurement in the r4 region;
• fast numerical computation of the solution of Mθ ,
→ allows estimation by MCMC algorithm;
• other parameters could have been estimated (e.g. introduction point
x0 ).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Part IV
Application: pine processionary moth
Statistical inference
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
The pine processionary moth (PPM) Thaumetopoea pityocampa
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Range expansion towards northern France
(a) 1980
Mainly caused by the recent increase in winter temperatures (URTICLIM
project).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Range expansion towards northern France
(b) 2005
Mainly caused by the recent increase in winter temperatures (URTICLIM
project).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Range expansion towards northern France
(c) 2011
Mainly caused by the recent increase in winter temperatures (URTICLIM
project).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Can attack several types of pines
(d) Host trees
(e) A nest
Statistical inference
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Also harmful to human beings, pets and cattle
Statistical inference
Introduction: the PPM
Data
Difficulties
Statistical model
Highly allergenic bristles
Mechanistic model
Statistical inference
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Simplified one-year life cycle
Adults:
Emergence
Dispersal
Laying
Caterpillars
build nest
Caterpillars:
Procession
Adults:
Emergence
Dispersal
Laying
OVO-LARVAL STAGE
J
A S
O N D J
F
M A M J
DIAPAUSE
- The life cycle depends on the location.
- Possible existence of prolonged diapause.
J
A
Statistical inference
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Main objectives
• Estimation of the population density in the Paris Basin;
• estimation of the diffusivity (mobility) of the adults;
• estimation of the “local fitness".
Local fitness F (x ) : number of adults who might emerge during year
n + 1 – in the absence of demographic constraint – for one (unit of adult
density×unit of time) at the position x during year n (“PPM-months").
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Data
Binary (and incomplete) observations: presence/absence of PPM nests:
+ Paris
+ Chartres
+ Fontainebleau
20 km
(f) 2006-2007
Blue regions: PPM nests have not been detected.
Red regions: PPM nests have been detected.
Other regions: not observed.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Data
Binary (and incomplete) observations: presence/absence of PPM nests:
+ Paris
+ Chartres
+ Fontainebleau
20 km
(g) 2007-2008
Blue regions: PPM nests have not been detected.
Red regions: PPM nests have been detected.
Other regions: not observed.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Data
Binary (and incomplete) observations: presence/absence of PPM nests:
+ Paris
+ Chartres
+ Fontainebleau
20 km
(h) 2008-2009
Blue regions: PPM nests have not been detected.
Red regions: PPM nests have been detected.
Other regions: not observed.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Main difficulties
Linked with the data:
• binary (and incomplete) data indicating the presence or not of PPM
nests;
• classical models of population dynamics deal with continuous data;
→ can be overcome by using a statistical model for the observation.
Linked with the life cycle and expansion process:
• PPM nest range evolves through a discrete time process;
• consequence of dispersal and laying of adult PPMs: a continuous
time process;
→ mechanistic model has to link nest density, adult density, and
parameters.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Statistical model for the observation process
Based on the observation protocol.
Study site Ω: divided into I spatial units ωi of area % = 4 km2 .
Discrete time indexed by n = 2007, 2008, 2009. One unit of time = one
year = one life cycle of PPM.
Observation variable Oi,n (site i, year n)
- Oi,n = 1 ↔ detection of PPM nests;
- Oi,n = 0 ↔ no PPM nest detected.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Statistical model for the observation process
Assumption:
Observation variable Oi,n depends on the average PPM nest density in
site i during year n;
→ unobserved variable Ui,n .
Observation model:
Oi,n |Ui,n ∼ Bernoulli{d(Ui,n )}.
Assuming that, in each unit area, each nest is independently detected
with probability p yields:
d(s) = 1 − (1 − p)% s .
(Measurement of p with another data set: p = 0.1)
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Mechanistic model: adult dispersal and death
Diffusion equation for the (female) adult pop. density v (t, x ):
v
∂v
= D∆v − .
∂t
ν
v
- Mortality term: − , life expectancy ν: 1 day;
ν
- boundary conditions: reflecting (west, east, south) and absorbing
(north);
- initial condition: depends on the nest density of the previous year
(next slide).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Mechanistic model: nest creation (=laying)
Assumption 1: nest creation in ωi
- is proportional to the number
of Z“pest-months" in ωi
Z ∞
= cumulative density wi =
v (s, x )dx ds;
0
ωi
- is proportional to a local favorability parameter fi in ωi ;
- is limited by environmental carrying capacity = number of hosts χi
in ωi .
→ average nest density
Ui,n = min{wi fi , χi }.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Mechanistic model: reproduction
Assumption 2 on initial adult density v (0, x ):
- it depends on nest density at n − 1: v (0, x ) = r (Ui,n−1 )Ui,n−1 for
x ∈ ωi ;
U
(Allee effect), R: max number of (female) adult
- r (U) = R 1+U
individuals per nest volume unit.
→ nest density Ui,n can be computed recursively:
Ui,n = min{(wi |Un−1 )Fi , χi },
Un−1 = (U1,n−1 , . . . , UI,n−1 ): nest density over the whole study site
during year n − 1.
Fi = R fi : local fitness.
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Statistical inference: computation of a likelihood
Unknown parameters:
- diffusion coefficient of the moths (D);
- local fitness Fi . F = (F1 , . . . , FI ).
Assuming that the observations (conditionally on nest densities) are
independent:
Y
L(D, F) =
P(Oi,n |Ui,n ),
years n, obs. sites ωi
i.e.,
Y
L(D, F) =
years
n,
obs. sites
Oi,n d(Ui,n ) + (1 − Oi,n ) (1 − d(Ui,n )).
ωi
Estimation carried out with MCMC algorithm (uniform prior distributions
in [0, 150] for the Fi and 30 km2 /day for D).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Posterior distribution of the diffusion D
0.25
Probability density
0.2
0.15
0.1
0.05
0
0
2
4
6
8
10
12
D
- Clearly different from the prior distribution;
- posterior median 9.3 km2 /day: not too far from previously
computed value (5.1 km2 /day, Robinet, 2006);
- higher than usually observed for Lepidoptera (Kareiva, 1983);
- may indicate that the dispersal is not purely diffusive (otherwise, the
individuals would cross around 164 km).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Posterior distribution of the local fitness F
150
x2
0
x1
3 rd q
uart
ile
x2
0
x1
x2
0
x1
med
ia
1 st q
n
uart
ile
0
- strongly depends on the location and non-uniform;
- spatially structured (close regions tend to resemble each other).
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Computation of the nest density
Solution of the mechanistic model, with modal values of the parameters
D, F :
1
0
(i) 2007
Introduction: the PPM
Data
Difficulties
Statistical model
Mechanistic model
Statistical inference
Computation of the nest density
Solution of the mechanistic model, with modal values of the parameters
D, F :
1
0
(j) 2008
Conclusions
Part V
Conclusions
Conclusions
Conclusions
• Reaction-diffusion equations as mechanistic models:
- can easily incorporate spatial/temporal heterogeneities;
- good identifiability of the parameters;
- fast computation of the solution → allow parameter estimation by
MCMC;
- local diffusion can be a problem → use integro-differential eqs (lower
computational speed).
• Mechanistic-statistical models:
- bridge the gap between continuous models/discrete data sets;
- can incorporate various type of information;
- allow the estimation of unobserved processes;
- lengthy estimation processes → require easy to solve mechanistic
models.
Conclusions
Perspectives
• Use integro-differential mechanistic models → estimation of the
dispersal kernel;
• use genetic data (available for the PPM).
(k) cytochrome c oxydase distribution (from Rousselet et al., 2010)
Conclusions
Kiitos! / Thank you!
References:
PPM expansion:
- Roques, Soubeyrand, Rousselet (2011) J Theor Biol 274, 43-51.
Mechanistic-statistical models in ecology:
- Soubeyrand, Laine, Hanski, Penttinen (2009) Am Nat 174, 308-320;
- Soubeyrand, Neuvonen, Penttinen (2009) Bull Math Biol 71, 318-338.