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Spatial Dynamics of Mountain Pine Beetle
Epidemics with Optimal Forest Management
Charles Sims1, David Aadland2 & David Finnoff2
1Utah State University2University of Wyoming
Mountain Pine Beetle are endemic to the Rocky Mountains
Current epidemic covers more than 2 million acres in Colorado & Wyoming; another 40 million acres in British Columbia
Contributing factors are global warming, fire suppression and reduced timber harvesting◦Sims et al. 2010 JEDC◦Sims et al. 2011
Snowy Range Mountains, WY
Literature Sampling: Spatial Dynamics in Resource Economics
Sanchirico & Wilen (2005, JEEM)Costello & Polasky (2008, JEEM)Brock & Xepapadeas (2008, JEDC)Smith, Sanchirico & Wilen (2009, JEEM)Epanchin-Neil & Wilen (2011, RFF wp)
Space/Time tradeoff: human behavior; choice variables; number of species, patches, and time periods; migration; interdependencies
Model & Solution Procedure
Abstract model◦3 classifications for trees (X,Y,A)◦Mountain pine beetle stock (B)◦6 × 6 landscape grid◦Spatial heterogeneity◦Density-dependent MPB dispersal◦Forward-looking, rational forest managers
Solution procedure◦ Approx. linear harvest rule (Blanchard & Kahn 1980)◦ Linear harvest rule with nonlinear biological system◦ Linear quadratic approx. (Brock & Xepapadeas 2008)
Forest Dynamics on cell (i,j) Seed base:
Young trees:
Adult trees:
MPB risk:
),(),(),(),(1 1 ji
tAji
tYji
tXji
t AbYbXX
),(),(),(1 1 ji
tXji
tYji
t XYY
),(),(),(),(),(1
),(,
1 jit
A
jitY
jit
jit
jit hYAdA
jitH
22),(
2),(),(
aB
Bji
t
jitji
t
MPB Reproduction and migration
Reproduction:Migration
◦ Stage 1
◦ Stage 2
v
Nn
jipret
jijit
jipret
jijit
jit
nn BwBwB
),(,
),(),(),(,
),(),(),(
)exp(),(1
),(1
),(, t
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jit
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1),(0 )(),(),( 1
cjitAcjiji
t ew
Nn
jit
jitjiji
tjiji
t n
n
n
A
Aww ),(
),(),(),(),(),( 1
10 20 30 40 50 60 70 80 90 100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
300
310
320
330
340
0
0.2
0.4
0.6
0.8
1
Adult Stock (A)
Fra
cti
on o
f B
eetl
es R
em
ain
ing o
n O
wn C
ell
Local Optimal Management
1
),(,
),(1 ;,max1
),(t
NjitH
jit
t
h
AhUt
jit
),(2
),(2),(
1
),(22),(
1),(
1),(1
),(1),(
1),(1
1 jit
jitji
t
jitji
tji
tjit
jitji
ttjit
AA
AA
dEh
Benefit of harvesting
Opportunity cost of
harvesting
MPB migration
effect
MPB reproduction effect
First-order condition for ht(i,j)
where Ψi(i,j) is the marginal net benefit of an adult tree at time t on cell (i,j)
Reduced-form local harvest rulesFirst-order Taylor series approx.
Linearized system exhibits saddle path stability and implies linear harvest rules
Each rule is a function of 4X36=144 contemporaneous state variables
1
1~
~
~
~
t
tt
t
t
h
SEJ
h
S
tjit Shh ),(
Landscape Spatial MPB Externality
Local managers look to neighboring cells to estimate migration onto own forest
BUT local managers unconcerned about neighboring welfare
Spatial nature of externality may cause local managers to over or under harvest
Central Optimal Management
1
6
1
6
1
),(,
),(1
,...,
;,max1
)6,6()1,1(t
i j NjitH
jit
t
hh
AhUttt
First-order condition same as for local planner with addition of spatial MPB externality captured by appending two additional terms on the right side
Nn
jit
jitji
t
jitji
tji
tjit
jit
tnn
n
nn
n
AA
AA
E
),(2
),(2),(
1
),(22),(
1),(
1),(1
),(1
MPB migration externality
MPB reproduction externality
Steady State (Endemic Population) Analysis
Type 1: Centralized h 8% higher Externality impact = 2%
Type 2:Centralized h 5% higherExternality impact = 0.1%
Type 3:Centralized h 0.4% lowerExternality impact = -1%
Type 4:Centralized h 4% higherExternality impact = 0.1%
Type 5:Centralized h 5% higherExternality impact = 0.2%
Type 6:Centralized h 2% higherExternality impact = -0.3%
2 4 6 8 10 12 14
500010 00015 00020 00025 000
Cell 1,12 4 6 8 10 12 14
5000
10 000
15 000
Cell 1,22 4 6 8 10 12 14
2000400060008000
10 000Cell 1,3
2 4 6 8 10 12 14
2000
4000
6000
8000
Cell 1,40 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 1,5
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 1,6
2 4 6 8 10 12 14
5000
10 000
15 000
Cell 2,12 4 6 8 10 12 14
5000
10 000
15 000Cell 2,2
0 2 4 6 8 10 12 14
2000400060008000
10 000Cell 2,3
2 4 6 8 10 12 14
2000
4000
6000
8000
Cell 2,40 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 2,5
0 2 4 6 8 10 12 140
2000
4000
6000
8000Cell 2,6
2 4 6 8 10 12 14
2000400060008000
10 000Cell 3,1
0 2 4 6 8 10 12 14
2000400060008000
10 000Cell 3,2
2 4 6 8 10 12 14
2000
4000
6000
8000
Cell 3,32 4 6 8 10 12 14
2000
4000
6000
8000Cell 3,4
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 3,5
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 3,6
2 4 6 8 10 12 14
2000
4000
6000
8000
Cell 4,12 4 6 8 10 12 14
2000
4000
6000
8000
Cell 4,22 4 6 8 10 12 14
2000
4000
6000
8000Cell 4,3
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 4,4
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 4,5
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 4,6
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,1
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,2
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,3
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,4
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,5
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 5,6
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 6,1
0 2 4 6 8 10 12 140
2000
4000
6000
8000Cell 6,2
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 6,3
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 6,4
0 2 4 6 8 10 12 14
2000
4000
6000
8000Cell 6,5
0 2 4 6 8 10 12 140
2000
4000
6000
8000Cell 6,6
0 2 4 6 8 10 12 142.02.53.03.54.04.55.05.5
Cell 1,12 4 6 8 10 12 14
2.2
2.4
2.6
2.8
3.0Cell 1,2
2 4 6 8 10 12 14
2.72.82.93.03.13.2
Cell 1,32 4 6 8 10 12 14
2.752.802.852.902.953.00
Cell 1,42 4 6 8 10 12 14
2.45
2.50
2.55
2.60
Cell 1,50 2 4 6 8 10 12 14
3.20
3.25
3.30
3.35
3.40Cell 1,6
2 4 6 8 10 12 14
2.2
2.4
2.6
2.8
3.0Cell 2,1
2 4 6 8 10 12 14
1.81.92.02.12.22.3
Cell 2,22 4 6 8 10 12 14
2.2
2.3
2.4
2.5
Cell 2,32 4 6 8 10 12 14
2.25
2.30
2.35
2.40
Cell 2,42 4 6 8 10 12 14
1.982.002.022.042.062.082.10
Cell 2,52 4 6 8 10 12 14
2.45
2.50
2.55
2.60
Cell 2,6
2 4 6 8 10 12 14
2.72.82.93.03.13.2
Cell 3,12 4 6 8 10 12 14
2.2
2.3
2.4
2.5
Cell 3,22 4 6 8 10 12 14
2.52.62.72.82.9
Cell 3,30 2 4 6 8 10 12 14
2.5
2.6
2.7
2.8
2.9Cell 3,4
2 4 6 8 10 12 14
2.222.242.262.282.30
Cell 3,50 2 4 6 8 10 12 14
2.80
2.85
2.90
2.95Cell 3,6
2 4 6 8 10 12 14
2.752.802.852.902.953.00
Cell 4,12 4 6 8 10 12 14
2.25
2.30
2.35
2.40
Cell 4,20 2 4 6 8 10 12 14
2.5
2.6
2.7
2.8
2.9Cell 4,3
2 4 6 8 10 12 14
2.5
2.6
2.7
2.8
Cell 4,42 4 6 8 10 12 14
2.22
2.24
2.26
2.28
Cell 4,52 4 6 8 10 12 14
2.80
2.85
2.90
Cell 4,6
2 4 6 8 10 12 14
2.45
2.50
2.55
2.60
Cell 5,12 4 6 8 10 12 14
1.982.002.022.042.062.082.10
Cell 5,22 4 6 8 10 12 14
2.222.242.262.282.30
Cell 5,32 4 6 8 10 12 14
2.22
2.24
2.26
2.28
Cell 5,42 4 6 8 10 12 14
1.982.002.022.042.062.08
Cell 5,52 4 6 8 10 12 14
2.45
2.50
2.55
2.60Cell 5,6
0 2 4 6 8 10 12 143.20
3.25
3.30
3.35
3.40Cell 6,1
2 4 6 8 10 12 14
2.45
2.50
2.55
2.60
Cell 6,20 2 4 6 8 10 12 14
2.80
2.85
2.90
2.95Cell 6,3
2 4 6 8 10 12 14
2.80
2.85
2.90
Cell 6,42 4 6 8 10 12 14
2.45
2.50
2.55
2.60Cell 6,5
2 4 6 8 10 12 14
3.223.243.263.283.303.323.34
Cell 6,6
Preliminary Findings
Local managers anticipate MPB spread; anticipatory harvesting
Harvesting can accelerate spread & dampen MPB outbreaks/cycles
Spatial externality comprised of two parts that work in opposite directions
Central managers may harvest more or less than local managers
Internalizing spatial externality requires combination of taxes and subsidies
Future work
Is the location of the outbreak important?How do results change with different
spatial size?Consider landscape with different
management objectivesEventually overlay on actual forested
areas
The End.
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