1. Are Forest Fires HOT? Jean Carlson, UCSB

2. Background Much attention has been given to “complex adaptive systems” in the last decade. Popularization of information, entropy, phase transitions, criticality, fractals, self-similarity, power laws, chaos, emergence, self- organization, etc. Physicists emphasize emergent complexity via self-organization of a homogeneous substrate near a critical or bifurcation point (SOC/EOC)

3. Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 18 Sep 1998 4 data sets

4. Criticality and power laws Tuning 1-2 parameters  critical point In certain model systems (percolation, Ising, …) power laws and universality iff at criticality. Physics: power laws are suggestive of criticality Engineers/mathematicians have opposite interpretation: –Power laws arise from tuning and optimization. –Criticality is a very rare and extreme special case. –What if many parameters are optimized? –Are evolution and engineering design different? How? Which perspective has greater explanatory power for power laws in natural and man-made systems?

5. Highly Optimized Tolerance (HOT) Complex systems in biology, ecology, technology, sociology, economics, … are driven by design or evolution to high- performance states which are also tolerant to uncertainty in the environment and components. This leads to specialized, modular, hierarchical structures, often with enormous “hidden” complexity, with new sensitivities to unknown or neglected perturbations and design flaws. “Robust, yet fragile!”

6. “Robust, yet fragile” Robust to uncertainties –that are common, –the system was designed for, or –has evolved to handle, …yet fragile otherwise This is the most important feature of complex systems (the essence of HOT).

7. Robustness of HOT systems Robust Fragile Robust (to known and designed-for uncertainties) Fragile (to unknown or rare perturbations) Uncertainties

8. Complexity Robustness Aim: simplest possible story Interconnection

9. Square site percolation or simplified “forest fire” model. The simplest possible spatial model of HOT. Carlson and Doyle, PRE, Aug. 1999

10. empty square latticeoccupied sites

11. connected not connected clusters

12. 20x20 lattice

13. A “spark” that hits an empty site does nothing. Assume one “spark” hits the lattice at a single site.

14. A “spark” that hits a cluster causes loss of that cluster.

15. Yield = the density after one spark yielddensityloss

16. density=.5 yield = Average over configurations.

17. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Y = (avg.) yield  = density “critical point” N=100 no sparks sparks

18. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 limit N   “critical point” Y = (avg.) yield  = density  c =.5927

19. Cold Fires don’t matter. Y 

20. Y Burned Everything burns. 

21. Critical point  Y

22. critical phase transition This picture is very generic and “universal.” Y 

23. Statistical physics: Phase transitions, criticality, and power laws

26. Power laws Criticality cluster size cumulative frequency

27. Average cumulative distributions clusters fires size

28. Power laws: only at the critical point low density high density cluster size cumulative frequency

29. Self-organized criticality (SOC) Create a dynamical system around the critical point yield density

30. Self-organized criticality (SOC) Iterate on: 1.Pick n sites at random, and grow new trees on any which are empty. 2.Spark 1 site at random. If occupied, burn connected cluster.

31. lattice fire distribution density yield fires

32. -.15

33. Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 18 Sep 1998 4 data sets

34. 10 -2 10 10 0 1 2 3 4 0 1 2 SOC FF Exponents are way off -1/2

35. Edge-of-chaos, criticality, self-organized criticality (EOC/SOC) yield density Essential claims: Nature is adequately described by generic configurations (with generic sensitivity). Interesting phenomena is at criticality (or near a bifurcation).

36. Qualitatively appealing. Power laws. Yield/density curve. “order for free” “self-organization” “emergence” Lack of alternatives? (Bak, Kauffman, SFI, …) But... This is a testable hypothesis (in biology and engineering). In fact, SOC/EOC is very rare.

37. Self-similarity?

38. What about high yield configurations? ? Forget random, generic configurations. Would you design a system this way?

39. Barriers What about high yield configurations?

40. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

41. Rare, nongeneric, measure zero. Structured, stylized configurations. Essentially ignored in stat. physics. Ubiquitous in engineering biology geophysical phenomena? What about high yield configurations?

42. critical Cold H ighly O ptimized T olerance (HOT) Burned

43. Why power laws? Almost any distribution of sparks Optimize Yield Power law distribution of events both analytic and numerical results.

44. Special cases Singleton (a priori known spark) Uniform spark Optimize Yield Uniform grid Optimize Yield No fires

45. Special cases No fires Uniform grid In both cases, yields  1 as N .

46. Generally…. 1.Gaussian 2.Exponential 3.Power law 4.…. Optimize Yield Power law distribution of events

47. 51015202530 5 10 15 20 25 30 0.1902 2.9529e-016 2.8655e-0114.4486e-026 Probability distribution (tail of normal) High probability region

48. Grid design: optimize the position of “cuts.” cuts = empty sites in an otherwise fully occupied lattice. Compute the global optimum for this constraint.

49. Optimized grid density = 0.8496 yield = 0.7752 Small events likely large events are unlikely

50. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 random grid High yields. Optimized grid density = 0.8496 yield = 0.7752

51. “grow” one site at a time to maximize incremental (local) yield Local incremental algorithm

52. density= 0.8 yield = 0.8 “grow” one site at a time to maximize incremental (local) yield

53. density= 0.9 yield = 0.9 “grow” one site at a time to maximize incremental (local) yield

54. Optimal density= 0.97 yield = 0.96 “grow” one site at a time to maximize incremental (local) yield

55. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 random density Very sharp “phase transition.” optimized

56. 10 0 1 2 3 -4 10 -3 10 -2 10 10 0 grid “grown” “critical” size Cum. Prob. All produce Power laws

57. HOT SOC d=1 dd dd HOT  decreases with dimension. SOC  increases with dimension. SOC and HOT have very different power laws.

58. HOT yields compact events of nontrivial size. SOC has infinitesimal, fractal events. HOT SOC size infinitesimal large

59. A HOT forest fire abstraction… Burnt regions are 2-d Fire suppression mechanisms must stop a 1-d front. Optimal strategies must tradeoff resources with risk.

60. Generalized “coding” problems Fires Web Data compression Optimizing d-1 dimensional cuts in d dimensional spaces.

61. -6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Cumulative Frequency Decimated data Log (base 10) Forest fires 1000 km 2 (Malamud) WWW files Mbytes (Crovella) Data compression (Huffman) (codewords, files, fires) Los Alamos fire d=0d=1 d=2

62. -6-5-4-3-2012 0 1 2 3 4 5 6 Size of events Frequency Fires Web files Codewords Cumulative Log (base 10) -1/2

63. -6-5-4-3-2012 0 1 2 3 4 5 6 WWW DC Data + Model/Theory Forest fire SOC  =.15

64. -6-5-4-3-2012 0 1 2 3 4 5 6 FF WWW DC Data + PLR HOT Model

65. HOT SOC HOTData Max event sizeInfinitesimalLarge Large event shapeFractalCompact Slope  SmallLarge Dimension d  d-1  1/d SOC and HOT are extremely different.

66. SOC HOT & Data Max event sizeInfinitesimal Large Large event shapeFractal Compact Slope  Small Large Dimension d  d-1  1/d SOC and HOT are extremely different. HOT SOC

67. HOT: many mechanisms gridgrown or evolvedDDOF All produce: High densities Modular structures reflecting external disturbance patterns Efficient barriers, limiting losses in cascading failure Power laws

68. Robust, yet fragile?

69. Extreme robustness and extreme hypersensitivity. Small flaws

70. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

71. 00.20.40.60.81 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

72. If probability of sparks changes. disaster

73. Tradeoffs? Sensitivity to: sparks flaws assumed p(i,j)

74. Critical percolation and SOC forest fire models HOT forest fire models Optimized SOC & HOT have completely different characteristics. SOC vs HOT story is consistent across different models.

75. Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile Events/structureGeneric, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities

76. Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile. Events/structure Generic, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities Characteristics Toy models ? Power systems Computers Internet Software Ecosystems Extinction Turbulence Examples/ Applications

77. Power systems Computers Internet Software Ecosystems Extinction Turbulence But when we look in detail at any of these examples... …they have all the HOT features... Characteristic Critical HOT Densities Low High Yields LowHigh Robustness GenericRobust, yet fragile. Events/structure Generic, fractal Structured, stylized self-similarself-dissimilar External behavior Complex Nominally simple Internally Simple Complex Statistics Power laws Power laws only at criticalityat all densities

78. HOT features of ecosystems Organisms are constantly challenged by environmental uncertainties, And have evolved a diversity of mechanisms to minimize the consequences by exploiting the regularities in the uncertainty. The resulting specialization, modularity, structure, and redundancy leads to high densities and high throughputs, But increased sensitivity to novel perturbations not included in evolutionary history. Robust, yet fragile!

79. Tong Zhou HOT and evolution : mutation and natural selection in a community Barriers to cascading failure: an abstraction of biological mechanisms for robustness Begin with 1000 random lattices, equally divided between tortoise and hare families Each parent gives rise to two offspring Small probability of mutation per site Sparks are drawn from P(i,j) Fitness= Yield (1 spark for hares, full P(i,j) for tortoises) Death if Fitness<0.4 Natural selection acts on remaining lattices Competition for space in a community of bounded size

80. Genotype (heritable traits): lattice layout Phenotype (characteristics which can be observed in the environment): cell sizes and probabilities Fitness (based on performance in the organisms lifetime): Yield Fast mutators (hares) Slow mutators (tortoises) Hares: -noisy patterns -lack protection for rare events

81. hares tortoises (Primitive) Punctuated Equilibrium: Hares win in the short run. But face episodic extinction due to rare events (niche protects 50). Tortoises take over, and diversity increases. Until hares win again.

82. Tortoise population exhibits power laws Hares have excess large events

83. Convergent Evolution: Species which evolve in spatially separate, but otherwise similar habitats develop similar phenotypic traits. They are not genetically close, but have developed similar adaptations to their environmental niches. Our analogy: different runs with the same P(i,j) evolve towards phenotypically similar, genotypically dissimilar lattice populations

84. The five great extinctions are associated with a rate of disappearance of species well in excess of the background, as deduced from the fossil record. Paleontologists attribute these to rare disturbances, such as meteor impacts. Robust, yet Fragile!

85. Punctuated Equilibrium (left) vs. Gradualism (right): PE: rapid, bursts of change (horizontal lines), followed by extended periods of relative stability (vertical lines), followed by extinction. Our analogy: after a transient period of rapid evolution lattices have barrier patterns, which are relatively stable until extinction

86. Large extinction events are typically followed by increased diversity. The recovery period is the time lapse between the peak extinction rate, and the maximum rate of origination of new species. Our analogy: extinction of the hares is are followed by diversification of both families

87. The current mass extinction is frequently attributed to overpopulation and causes which can be attributed to humans, such as deforestation Our analogy: large events can be due to rare disturbances, especially if they are not not part of the evolutionary history of the (vulnerable) species. Robust, yet Fragile!

88. Uniform Sparks Skewed Sparks Evolution by natural selection in coupled communities with different environments: Fitness based on a single spark. Eliminate protective niches. Fixed maximum capacity for each habitat. Fast and slow mutation rates (rate subject to mutation).

89. Coupled Habitats: Fast and slow mutators compete with each other in each habitat, with a small chance of migration from one habitat to the other. Efficient barrier patterns develop in the uniform habitat. After an extinction in the skewed habitat, uniform lattices invade, and subsequently lose their lower right barriers: a successful strategy in the short term, but leads to vulnerability on longer time scales

90. Patterns of extinction, invasion, evolution Over an extended time window, spanning the two previous extinctions, we see the long term fitness initially increases as the invading lattices adapt to their new environment. This is followed by a sudden decline when the lattices lose a barrier. This adaptation is beneficial for common events, but fatal for rare events.

91. HOT Disturbance Evolution and extinction Specialization density fitness

92. In a model which retains abstract notions of genotype, phenotype, and fitness, highly evolved lattices develop efficient barriers to cascading failure, similar to those obtained by deliberate design. HOT and Evolution Robustness in an uncertain environment provides a mechanism which leads to a variety of phenomena consistent with observations in the fossil record (large extinctions associated with rare disturbance, punctuated equilibrium, genotypic divergence, phenotypic convergence). Robustness barriers are central in natural and man made complex systems. They may be physical (skin) or in the state space (immune system) of a complex, interconnected system.

93. Forest Fires: a case where a common disturbance type (fires) Acts over a broad range of scales (terrestrial ecosystems) Power law statistics describing the distribution of fire sizes. Exponents are consistent with the simplest HOT model involving optimal allocation of resources (suppress fires). Evolutionary dynamics are much more complex.

94. Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 18 Sep 1998 4 data sets

95. 10 -4 10 -3 10 -2 10 10 0 1 0 1 2 3 4 All four data sets are fit with the PLR model with α=1/2. Size (1000 km 2 ) Rank order

96. Forest fires dynamics Intensity Frequency Extent Weather Spark sources Flora and fauna Topography Soil type Climate/season

97. Los Padres National Forest Max Moritz

98. Yellow: lightning (at high altitudes in ponderosa pines) Red: human ignitions (near roads) Ignition and vegetation patterns in Los Padres National Forest Brown: chaperal Pink: Pinon Juniper

99. Santa Monica Mountains Max Moritz and Marco Morais

100. SAMO Fire History

101. Fires 1991-1995 Fires 1930-1990 Fires are compact regions of nontrivial area.

102. 10 -4 10 -3 10 -2 10 10 0 1 0 1 2 3 4 4 Science data sets +LPNF + HFIREs (SA=2) PLR SM Rescaling data for frequency and large size cutoff gives excellent agreement, except for the SM data set Cumulative P(size) size

103. We are developing a realistic fire spread model HFIREs: GIS data for Landscape images

104. Models for Fuel Succession Regrowth modeled by vegetation succession models

105. 1996 Calabasas Fire Historical fire spread Simulated fire spread Suppression?

106. HFIREs Simulation Environment Topography and vegetation initialized with recent observations (100 m GIS resolution) for Santa Monica Mountains Weather based on historical data (SA rate treated as a separate parameter) Fire spread modeled using Rothermel equations Fuel regrowth based on succession models 8 ignitions per year Weather sampled stochastically from distribution (4 day SA at prescribed rate) Fire terminates in a cell when rate of spread (RoS) falls below a specified value Generate 600 year catalogs, omit data for first 100 years in our statistics

107. Preliminary results from the HFIRES simulations (no extreme weather conditions included) (we have generated many 600 year catalogs varying both extreme weather and suppression)

108. Data: typical five year periodHFIREs simulations: typical five year period Fire scar shapes are compact

109. 10 -4 10 -3 10 -2 10 10 0 1 0 1 2 3 4 LPNF PLR HFIRE SA=2, RoS=.033 m/s, FC=46 yr Excellent agreement between data, HFIREs and the PLR HOT model

110. 10 -4 10 -3 10 -2 10 10 0 1 0 1 2 3 4 4 Science +LPNF + Hfire (SA=2) PLR SM small: incomplete large: short catalog, or aggressive human intervention (inhomogeneous) SM discrepancy?

111. 10 0 1 2 3 4 5 0 1 2 3 Deviations from typical regional values for suppression (RoS) and the number of extreme weather events (SA), lead to deviations from the α=1/2 fit, and unrealistic values of the fire cycle (FC) SA=0, vary stopping rate SA= 1, 2, 4, 6

112. SA=2,  =.5 SA=4,  =.5 SA=6,  =.3 SA=0,  =.65 10 2 3 4 5 0 1 2 3 Increased rate of SA leads to flatter curves, shorter fire cycles Type conversion!

113. What is the optimization problem? Fire is a dominant disturbance which shapes terrestrial ecosystems Vegetation adapts to the local fire regime Natural and human suppression plays an important role Over time, ecosystems evolve resilience to common variations But may be vulnerable to rare events Regardless of whether the initial trigger for the event is large or small (we have not answered this question for fires today) We assume forests have evolved this resiliency (GIS topography and fuel models) For the disturbance patterns in California (ignitions, weather models) And study the more recent effect of human suppression Find consistency with HOT theory But it remains to be seen whether a model which is optimized or evolves on geological times scales will produce similar results Plausibility Argument: HFIREs Simulations:

114. The shape of trees by Karl Niklas L: Light from the sun (no overlapping branches) R: Reproductive success (tall to spread seeds) M: Mechanical stability (few horizontal branches) L,R,M: All three look like real trees Simulations of selective pressure shaping early plants Our hypothesis is that robustness in an uncertain environment is the dominant force shaping complexity in most biological, ecological, and technological systems