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

jit

jit

jipret AB

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.

Comments?