Joint simulation of stand dynamics and landscape evolution using a tree-level model for mixed uneven-aged forests

Abstract

• Context

A century and more after a major reforestation program, large areas in the French Southern Alps have moved to a landscape mosaic of old pine plantations and new, heterogeneous and uneven-aged, mixed stands. These conditions are challenging foresters in silvicultural practices and management choices.

• Aims

The aims of this study are to understand, analyze, model, and simulate the ongoing phenomena, and to propose a decision-making tool.

• Methods

An individual-based forest dynamics model considering recruitment, growth, and mortality (as related to the spatial arrangement of stands and species, to site conditions and competition) and a simulation system including spatial sampling are designed and calibrated to allow simulation of both silviculture treatments at the stand level and management strategies at the forest or landscape level.

• Results

By keeping track of the trees while simulating at the forest level, they offer an alternative to upscaling strategies and a suitable tool for prediction of stand and forest characteristics in situations influenced by strong driving forces such as colonization and forest maturation.

• Conclusion

This approach is a straightforward means for adjusting forest management to trends such as expansion of shade-tolerant species; as spatial and temporal variation in site conditions are accounted for, it is also a promising way towards predicting their warming-induced upward shift.

Appendix: Parameter estimates and regression statistics for each submodel and species

Hereafter, “R ²” denotes the coefficient of determination, “mean” is the mean of dependent variable, “CVR” is the residual coefficient of variation (standard deviation of the residuals divided by the mean of the dependent variable), in percent, and “N” is the sample size.

Recruitment submodel (see also Fig. 1)

$$ Log(smallsdgs + 1.1) = r1 + r2 \cdot {\left( {neardist + 15} \right)^{{ - r3}}} $$

(1a)

Abies alba

(R ² = 0.67, mean = 0.55/year, CVR = 105%, N = 59)

Parameter	Value	Standard error
r1	−0.51	0.56
r2	11.65	3.69
r3	0.40	0.15

Pinus nigra

(R ² = 0.13, mean = 0.39/year, CVR = 155%, N = 179)

Parameter	Value	Standard error
r1	−0.046	0.129
r2	5	fixed
r3	0.75	0.09

Pinus sylvestris

(R ² = 0.06, mean = 0.17/year, CVR = 137%, N = 214)

Parameter	Value	Standard error
r1	0.10	0.06
r2	1.04	2.13
r3	0.70	0.84

Pinus uncinata

(R ² = 0.05, mean = 0.21/year, CVR = 162%, N = 188)

Parameter	Value	Standard error
r1	0.10	fixed
r2	1.02	0.88
r3	0.56	0.28

$$ Log(smallsdgs + 1.1) = r1 \cdot {e^{{ - r2 \cdot {{\left( {neardist + 15} \right)}^{{r3}}}}}} $$

(1b)

Fagus sylvatica

(R ² = 0.27, mean = 1.02/year, CVR = 90%, N = 214)

Parameter	Value	Standard error
r1	4.17	4.28
r2	0.30	0.63
r3	0.34	0.29

Diameter Growth submodel for Trees

Pinus nigra, P. sylvestris, P. uncinata

$$ \begin{array}{*{20}{c}} {DINC5 = 0.1 \cdot a1 + \frac{{a2}}{{1 - a3 \cdot Log\left( {HdomINC5} \right)}}} \\ { \cdot \left( {1 - a4 \cdot {e^{{ - a5 \cdot NB{A^{{ - a6}}}}}}} \right)} \\ { \cdot \left( {1 + a7 \cdot ICS} \right)} \\ { \cdot min\left[ {1.025,\left( {1 + a8 \cdot \left( {Hdom50 - 15} \right)} \right)} \right]} \\ \end{array} $$

(2a)

(R ² = 0.88, mean = 0.36 cm/year, CVR = 22%, N = 654)

Parameter	Value	Standard error
a1	−45.98	11.68
a2	64.10	12.46
a3	0.0212	0.0093
a4	0.401	0.060
a5	19.72	6.42
a6	1.23	0.13
a7	0.232	0.054
a8	0.015	0.0033

Equation 2a was fitted only for P. nigra. As P. sylvestris and P. uncinata have many traits in common with P. nigra, we used a correspondence (set up using D2; not presented) between diameter increment of each three species, as depending on HdomINC5, NBA, and ICS.

Abies alba

$$ \begin{array}{*{20}{c}} {DINC5 = 100 \cdot b1 \cdot 1 /\left( {1 - b2 \cdot Log\left( {H/Age} \right)} \right) \cdot Ag{e^{{ - b3}}}} \\ { \cdot 1/\left( {1 - b4 \cdot {e^{{ - b5 \cdot NBA}}}} \right)} \\ { \cdot \left( {1 - {e^{{ - b6 \cdot ICS}}}} \right)} \\ \end{array} $$

(2b)

N.B. : In this case, ICS is computed using dbh under bark, and DINC5 is the increment under bark. The switch between under bark and over bark dbh values is achieved using bark thickness relationships (established using D2, not presented). NBA is over bark.

(R ² = 0.61, mean = 0.38 cm/year, CVR = 32%, N = 644)

Parameter	Value	Standard error
b1	0.122	0.061
b2	4.09	2.56
b3	0.151	0.053
b4	0.786	0.052
b5	0.014	0.007
b6	3.55	0.55

Fagus sylvatica

$$ \begin{array}{*{20}{c}} {DINC5 = 100 \cdot c1 \cdot {{\left( {1 - {e^{{ - c2 \cdot H/Age}}}} \right)}^{{c3}}} \cdot Ag{e^{{c4}}}} \\ { \cdot 1/\left( {1 + {e^{{c5 \cdot NBA}}}} \right)} \\ { \cdot \left( {1 - {e^{{ - c6 \cdot ICS}}}} \right)} \\ \end{array} $$

(2c)

N.B.: underbark and overbark values: see Eq. 2b

(R ² = 0.69, mean = 0.25 cm/year, CVR = 29%, N = 686)

Parameter	Value	Standard error
c1	1.28	0.18
c2	0.1	fixed
c3	0.86	0.05
c4	0.056	0.053
c5	0.020	0.002
c6	0.84	0.07

Diameter Growth submodel Seedlings and saplings

$$ \begin{array}{*{20}{c}} {DINC5 = 5 \cdot 0.1 \cdot d1 \cdot {e^{{ - {e^{{d2 - d3 \cdot Dlag}}}}}}} \\ { \cdot {e^{{ - \left( {d4 - {{H} \left/ {{H\max }} \right.}} \right)}}}^2} \\ { \cdot 1/[1 + d5 \cdot ((SDLp + 1)/100)]} \\ { \cdot 1/[1 + d6 \cdot {{((SDLfa + 1)/100)}^{{d7}}}]} \\ { \cdot 1/[1 + d8 \cdot {{((NBA + 0.01)/60)}^{{d9}}}]} \\ \end{array} $$

(3)

When parameter d4 is non-significant (“NS”), the modifier ${e^{{ - \left( {d4 - {{H} \left/ {{H\max }} \right.}} \right)}}}^2$is not included (i.e. it is replaced by 1).

Abies alba

(R ² = 0.83, mean = 0.27 cm/year, CVR = 40%, N = 864)

Parameter	Value	Standard error
d1	12.99	0.80
d2	0.1	fixed
d3	1.24	0.19
d4	-	NS
d5	−0.071	0.024
d6	0.72	0.08
d7	2.08	0.15
d8	1.38	0.16
d9	2	fixed

N.B.: d5 < 0 : i.e. SDLp is favorable to diameter growth of Abies alba seedlings/saplings.

Fagus sylvatica

(R ² = 0.51, mean = 0.25 cm/year, CVR = 47%, N = 1,065)

Parameter	Value	Standard error
d1	6.34	0.58
d2	0.25	0.07
d3	5.01	1.10
d4	-	NS
d5	0.048	0.029
d6	0.51	0.067
d7	1.59	0.20
d8	0.46	0.12
d9	2	fixed

Pinus nigra

(R ² = 0.45, mean = 0.46 cm/year, CVR = 35%, N = 448)

Parameter	Value	Standard error
d1	8.57	0.62
d2	−0.056	0.333
d3	14.28	4.97
d4	0.19	0.05
d5	0.40	0.07
d6	1.47	0.53
d7	1.65	0.44
d8	13.59	6.65
d9	8	fixed

D3 contains too few growth samples or site/stand conditions for Pinus sylvestris or P. uncinata; for simulation, we used the relationship fitted for Pinus nigra.

Height–Diameter submodel for Trees

$$ H = Hdom \cdot \frac{{(1 - {e^{{ - s \cdot DBH}}})}}{{(1 - {e^{{ - s \cdot Ddom}}})}} $$

(4)

With $s = s0 + s1 \cdot {\rm Re} lSpcgMod$ (see § 2.4.4)

Abies alba

(R ² = 0.97, mean = 17.34 m, CVR = 12%, N = 5,689)

Parameter	Value	Standard error
s1	0.000257	0.000024
s0	0.0130	0.00214

Fagus sylvatica

(R ² = 0.92, mean = 12.55 m, CVR = 14%, N = 16,081)

s1	0.000535	0.000016
s0	0.0300	0.00161

Pinus nigra

(R ² = 0.97, mean = 12.54 m, CVR = 11%, N = 9,846)

s1	0.000131	0.000015
s0	0.0659	0.00175

Pinus sylvestris

(R ² = 0.94, mean = 10.28 m, CVR = 14%, N = 24,397)

s1	0.000342	9.175E-6
s0	0.0276	0.00129

Pinus uncinata

(R ² = 0.93, mean = 9.76 m, CVR = 15%, N = 3,696)

s1	0.000089	0.000021
s0	0.0505	0.00304

Height Growth submodel for Seedlings and saplings (see also Fig. 2)

$$ \begin{array}{*{20}{c}} {HINC5/{\text{HdomINC5}} = } \hfill \\ {{1/[1} + {\text{t1}} \cdot {\text{Hla}}{{\text{g}}^{\text{t2}}} \cdot {\text{((1 - H / Hmax}}{{)}^{\text{t3}}} + {0}{.01)]}} \hfill \\ { \cdot {1/[1} + {\text{t4}} \cdot {{{\text{( (SDLfa}} + {1) /100)}}^{\text{t5}}}{]}} \hfill \\ { \cdot {1/[1} + {\text{t6}} \cdot {\text{(NBA}} + {0}{.1) /50}{{)}^{\text{t7}}}{]}} \hfill \\ \end{array} $$

(5)

When parameter t3 is non-significant (“NS”), the term ${\text{ ((1 - H / Hmax)}}{{ }^{\text{t3}}}{ } + { 0}{.01)}$ is not included (i.e. it is replaced by 1).

Abies alba

(R ² = 0.67, mean = 0.38 m/m, CVR = 65%, N = 1,173)

Parameter	Value	Standard error
t1	34.1	3.5
t2	8	fixed
t3	-	NS
t4	0.29	0.06
t5	2.18	0.30
t6	0.12	0.05
t7	0.50	0.39

Fagus sylvatica

(R ² = 0.30, mean = 0.75 m/m, CVR = 56%, N = 1,161)

Parameter	Value	Standard error
t1	6.98	1.39
t2	2	fixed
t3	5.32	1.23
t4	0.15	0.05
t5	1.67	0.35
t6	0.009	0.053
t7	2	fixed

Pinus nigra, Pinus sylvestris and P. uncinata

(R ² = 0.65, mean = 0.62 m/m, CVR = 33%, N = 760)

Parameter	Value	Standard error
t1	6.69	1.11
t2	2.43	0.30
t3	14.81	1.57
t4	1.39	0.46
t5	2.39	0.53
t6	5.06	2.56
t7	10.48	2.80

For Pinus sylvestris or P. uncinata, D3 contains too few individuals with height growth measurements and the corresponding range of site/stand conditions is quite narrow; for simulation, we used the relationship fitted for P. nigra.

Mortality submodels

Trees

$$ {\text{probMort5}} = {\text{(m1}} + {\text{m2}} \cdot N{\text{BA)}} \cdot {{\text{e}}^{{{\text{(m3}} + {\text{m4}} \cdot N{\text{BA)}} \cdot {\text{DBH/Ddom}}}}} $$

(6)

DBH and Ddom are under bark values. N is the number of combinations of classes of NBA (8 classes ranging between 10 and 60 m²/ha) by classes of DBH/Ddom (7 ranging between 0.3 and 1.1). The mean is the mortality rate for 5 years.

Abies alba

(R ² = 0.32, mean = 0.0186, CVR = 93%, N = 48)

Parameter	Value	Standard error
m1	−0.0088	0.035
m2	0.0022	0.0012
m3	−2.09	1.25
m4	0.0021	0.0178

Fagus sylvatica

(R ² = 0.40, mean = 0.0037, CVR = 72%, N = 56)

m1	−0.0014	0.0016
m2	0.00026	0.00008
m3	−0.33	0.70
m4	−0.011	0.015

Pinus nigra

(R ² = 0.55, mean = 0.0060, CVR = 65%, N = 49)

m1	0.0028	0.0089
m2	0.00086	0.00033
m3	−2.48	1.08
m4	−0.0012	0.0257

Pinus sylvestris

(R ² = 0.76, mean = 0.0186, CVR = 22%, N = 42)

m1	0.0066	0.0064
m2	0.0013	0.0003
m3	−0.48	0.32
m4	−0.028	0.009

Pinus uncinata

(R ² = 0.14, mean = 0.0195, CVR = 54%, N = 53)

m1	0.00051	0.00523
m2	0.00082	0.00026
m3	0.92	0.53
m4	−0.037	0.014

Seedlings/Saplings (see also Fig. 3)

$$ probMort5 = 5 \cdot \left[ {1 - {e^{{ - m0 \cdot NB{A^2} \cdot \left[ {1/min(l,H/H\max ) - 1} \right]}}}} \right] $$

(7)

N is the number of combinations of classes of NBA (from 10 to 60 with classes of width 10 m²/ha) by classes of H/Hmax (from 0.1 to 1.0 with classes of width 0.1). The mean is the mortality rate for 5 years.

Abies alba

(R ² = 0.85, mean = 0. 0175, CVR = 115%, N = 27)

Parameter	Value	Standard error
m0	2.803E-7	8.569E-8

Fagus sylvatica

(R ² = 0.30, mean = 0.0095, CVR = 210%, N = 29)

m0

4.607E-7

1.147E-7

Pinus nigra

(R ² = 0.56, mean = 0. 0395, CVR = 122%, N = 25)

m0

1.408E-6

3.186E-7

Pinus sylvestris

(R ² = 0.94, mean = 0. 0295, CVR = 51%, N = 11)

m0

1.007E-6

3.027E-7

Dataset D3 contains too few samples for Pinus uncinata; for simulation, we used the relationship fitted for Pinus nigra.