402
Annual Grass Invasion and Tree Invasion Disturbances
The temporal multipliers for invasive annual grass expansion and tree expansion
were calculated from two heuristic Gompertz equations (not including the CO 2 fer-
tilization). The Gompertz equation is highly flexible for curve fitting and a special
case of it is the negative exponential:
Yearlyannualgrassexpansionvariabilityfactor=
́e- ́^20 exp(-..^75 ́+()PDSI^1 ) ́TMCO (^2)
(13.4)
Yearlytreeexpansionvariabilityfactor
PD
- ́e- ́^20 exp(- ́.(^75 SSI+^1 )) ́()TMCO 205.
(13.5)
where TMCO 2 is the temporal multiplier for CO 2 levels, which is <2 for any yearly
value with climate change and equal to one without climate change. In accordance
with our hypothesized relationship between species expansion and soil moisture
and CO 2 levels, the effect of CO 2 levels as expressed by its temporal multiplier
(between 0 and 1) on variability is proportional, whereas the effect of PDSI is expo-
nential (i.e., greater). We arbitrarily dampened the effect of CO 2 fertilization on
trees by taking the square root of the CO 2 temporal multiplier. The Gompertz equa-
tions allow for some expansion during even dry years (PDSI < 0), average expansion
(temporal multiplier close to 1) during average moisture years, and a rapid rise of
expansion (multiplier increasing to 4.5 and 2.5), respectively, for invasive annual
grass expansion and tree expansion during very wet years. The parameters 4.5 and
2.5 were chosen to match values from the initial Park’s study by Provencher et al.
( 2013 ). Equations 13.4 and 13.5 are not temporal multipliers, however, because they
are not divided by their averages. In the absence of climate change effects, yearly
values of Eqs. (13.4) and (13.5), respectively, were each divided by their temporal
average over 75 years, whereas each yearly value of Eqs. (13.4) and (13.5) with
climate change, respectively, was divided by the no-climate change average to
reflect the hypothesis of altered levels.
Fire
The shrubland–woodland fire temporal multipliers considered the roles of 3 years of
PDSI, more specifically that fine fuels will more likely burn in the current dry year
immediately following two previous and consecutive wetter-than-average years
where fine fuels accumulated. The equation to calculate the temporal multipliers from
shrubland fire contained two Gompertz functions to account for 3 years of PDSI:
Yearlyshrublandwoodlandareaburnedvariabilityfactor
MaxFir- = ee ́ ́ee- ́^30 exp(- ́.(^71 PDSItt-+^10 - ́.)^72 PDSI--)e( 1 -^32 ́-xp( ́PDSIIt))
(13.6)L. Provencher et al.