Figure 1.
Individual fire simulations in the absence of suppression for (A) mild, and (B) extreme conditions.
The total number of parcels is divided between unburned (green), burning (red), and burned (black) according to the stochastic dynamics of the birth-death process. The fire terminates when the number of burning firelets j first reaches zero. In (A) the final fire size F is roughly
, while in (B)
.
Figure 2.
Fire size distributions calculated for varying growth and suppression rates.
Results are based on simulation runs for a fire in a region containing burnable substrate of
parcels. In (A), growth rate
varies in the absence of suppression. Dangerous conditions are described by a pure power law with exponent
. Extreme conditions
lead to excess weight in the tail, which can be interpreted as fires that have surpassed a threshold value and can no longer be controlled. Finally, mild conditions
result in a sharp (exponential) cutoff
, which decreases as
decreases. In (B), the suppression rate
varies in dangerous conditions
. Pure power laws are obtained for a range of
, with an increasing exponent (steepening slopes) as suppression is increased.
Figure 3.
Burn probabilities as a function of initial size s and suppression rate
.
Conditions are mild in (A), dangerous in (B), and extreme in (C). As the conditions become increasingly severe, the transition from low to high shifts to smaller initial sizes. Results shown represent averages for 100 fires.
Figure 4.
Risk curves as a function of initial size s and suppression rate.
Conditions are mild in (A), dangerous in (B), and extreme in (C). In each case the line style indicates suppression rate: (solid), 2 (dashed), and 3 (dotted). Burn probability
corresponds to
, with
. In mild conditions risk generally remains low, unless the initial size is large. In dangerous conditions suppression plays a significant role in reducing risk across the full range of initial sizes. In extreme conditions suppression is most effective when the initial size is small. Beyond a certain size, the fire is increasingly likely to grow out of control.
Figure 5.
Tradeoffs between time delay and suppression rate
represented by contours of constant burn probability
.
At small time delays, must exceed some minimum value to keep
low. As
increases, initially suppression rate must increase roughly exponentially with delay to maintain a constant
. At a larger value of
, there is a sharp transition where the contours become essentially vertical, corresponding to a delay, determined by the underlying rates and system size, beyond which the fire is likely to have either burned out or grow out of control. Results shown represent dangerous conditions
and averages over 1000 simulations and all fires are initialized with size
Figure 6.
Resource optimization for two simultaneous fires, Fire A and Fire B, subject to the constraint .
Fire A has dangerous conditions and initial size . In panel (A) the optimization procedure is shown explicitly for
(extreme case), and varying values of
. Optimal solutions correspond to the minimum mean total size
for each curve, and begin with a small allocation to Fire B when
is small, rapidly shifting resources to Fire B as
increases, and then shifting resources back to Fire A when
becomes sufficiently large. Results represent averages over 1000 realizations. In panel (B) these results are extended to values of
extending from mild to extreme conditions. The trends in panel (A) persist for all cases where Fire B is extreme. When Fire B has dangerous or mild conditions, resources are not shifted back to Fire A when
becomes large. Panel (C) illustrates the value of
associated with the optimal solution in panel (B). Here
serves as a proxy for total damage. Figure (D) shows the size difference between solutions corresponding to the maximum (worst case) and minimum (optimal) value of
. Regions where the difference is elevated correspond to regimes where optimal decision making has the most significant impact.