In every sawtooth pattern that I've seen before, the population is usually large, in the sense that for any positive number P, the density of the set of times when the population is less than P is zero. (P.S., May 11, 2016: No, that's not true. For the quadratic sawtooth the population is small for long periods of time. The set of such times does not have an asymptotic density.)
For the sawtooth below, the opposite is true: The density of the set of times when the population exceeds 1581 is zero. Most of the time, the stationary part of the pattern is just a p240 oscillator, whose population fluctuates between 1351 and 1581. But at times around 3840*2^n - 1760, it receives a glider from a caber tosser, which triggers one cycle of an e.f. 7/6 sawtooth. During that time, the population grows and then shrinks; after that the stationary part becomes an oscillator again.
If I've computed correctly, at the times T when the population is maximal, it's about C*T^a, where a = log(7/6)/log(2) ~ 0.22239. (I haven't computed C yet.)
Code: Select all
#C Usually small sawtooth
#C Dean Hickerson, 5/2/2016
x = 339, y = 387, rule = B3/S23
305bo$297bo7b2o$296bobo2b3obobo$297bo3b2o3b3o$298bobob2obobo$302b2ob2o
$297bob5obo$285b2o11b4o$285b2o7$277b2o$277b2o23bo$290bobo6bo$288bobobo
11bo$286b2o10bo2bo3bo$288bo10bo2b3o$299bo3bo$300b2o$301bo10b2o$304bo4b
o2bo$303bobo2bob3o2bo$304b4o4bo2bo$312bo2bo$305bo6bo$310bob2o$305bo29b
o$306b2o19bo7b2o$326bobo2b3obobo$327bo3b2o3b3o$328bobob2obobo$332b2ob
2o$327bob5obo$328b4o12$335b2o$335b2o7$327b2o$327b2o71$105bo$103b3o$
102bo$102b2o3$98b2o$97bobo$99bo6b2o5b2o$106b2o5b2o2$89bo19b2o$88bobo
18b2o$90bo$34b2o$34b2o$55bo31bobo$8b2o43b3o30bo3bo$8b2o42bo33bo4bo$52b
2o28b2o8bo$82b2o3b2o2bo$11b2o59bo16b2o$11b2o57b3o$69bo$8b2o47bo11b2o
11bo$8b2o23bo22b2o24b3o$2o29bo3bo19bo2bo26bo22bo$bo28bo4bo20b2o26b2o
20b3o$bobo31bo63b2o4bo$2b2o20b2o7b2o64bo5b2o$19bo4b2o2bo67b2obo$17b2ob
2o37b2o34bo2bo3bo$17b2ob2o37b3o24b2o8b2o4bo$16b3obo7bob2o25b2o2bo24bob
2o$19b3o10bo12b2o10bo2b2o26b2o$29bo3bo11bobo9bo2bo25bob2o10b2o3b2o$29b
o2bo14bo10b3o25b2o13b5o$29bo2bo2b2o10b2o53b3o$29bo2bo2bobo30b2o3bo29bo
$29bo2b2o2b2o30bo3bobo$28bo3bo2bo33bo3bobo$19b2o10bo2b2o20b2o12bo3bobo
b2o$19b2o7b2o4bo22bo10bob4o2bob2o$54b3o10bobo3bobo$16b2o36bo12bobo2bo
2b2ob2o$16b2o50bo3b2o2bobo25bo$76bobo10b2o14b2o$77bo11b2o13b2o$19b2o
17bo$19b2o15bobo$37b2o85b3o$126bo$41bo56b2o3b2o20bo$42bo57b3o8bobo31bo
$40b3o56bo3bo8b2o31b3o$100bobo9bo35bo$101bo45b2o2$94bo8b2o$94b3o6bo$
97bo6b3o35bo$96b2o8bo29b2o3bobob2o$136b2o2bob3ob2o$141b6o$98b3o44bo$
98b3o8b3o4bo$97bo3bo9bo4b3o$110bo8bo$96b2o3b2o15b2o75b2o$163b2o30b2o$
107b3o53b2o$107bo$102b2o4bo51b2o5b2o23b2o$101bobo56b2o5b2o23b2o$103bo
14b2o3b2o27bobo$118bobobobo28b2o40b2o$119b5o29bo41b2o$103bo11b2o3b3o
46b2o8b2o22b2o$68bo46bobo3bo47b2o8bo23bo$66bobo46bo61bobo21bobo$67b2o
108b2o22b2o2$71bo28b3o66b2o14bo$72bo26bo3bo64bo2bo12b3o$70b3o25bo5bo
63bo2bo12bo2bo$98bo5bo62b2ob2o14b2o$101bo16b3o47bo2bo14bo$99bo3bo13b2o
b2o35b2o10b2o12b2o$100b3o14b2ob2o36bo24b4o$101bo15b5o36bobo23bo2bo$
116b2o3b2o36b2o24bo2$101b2o81bobo$101b2o81b2o2$187b2o$187b2o$116b2o$
117bo$114b3o67b2o$114bo69b2o$120b3o$120b3o$109bo10b3o$109b3o5b3o$112bo
4b3o$111b2o4b3o2$98bo15bo45b2o$96bobo14bobo44b2o2b2o$97b2o10b2o2b2o49b
obo$109b2o54bo$101bo$102bo23b2o34b2o4b3o$95b2o3b3o23bo36bo4b3o$95b2o
13b2o12bobo12bo20b3o5b3o$111b2o11b2o13bobo18bo10b3o$110bo28b2o4bo25b3o
$145b3o23b3o$88bo59bo$87bob2o4bo51b2o$87bo2b2o2b2o$90bo2b2o30b2o23bo$
89b2ob2o31b2o$90bo$91b2o43b2o5b2o$136b2o5b2o$217b3o$85bo54b2o78bo$83b
2o55b2o74bo3bo$84b2o129bobo2bo$213bo2bobo$137b2o74bo3bo$137bo75bo$78b
2o30b2o23bobo18b2o4b3o49b3o$77bobo29bobo23b2o18bo2bo3bob2obo43bo$77bo
33bo42b2o2bo9bo41bobo$76b2o77b2o2bo4bo3bo41b2o$84b2o70b4o4b4o46b2o$84b
2o128bobo$209b2o5bo$87b2o120b2o5b2o$87b2o78b2o$111b2o5b2o47b2o$111b2o
5b2o$84b2o$71b2o11b2o28b2o60bo$71b2o41b2o59bobo$176bo2$74bobo$78b2o11b
2o$74bo5bo10b2o11b4o62b3o$71b2o7bo27bo61bo$71b2o2bob2ob2o5b2o5b2o8bo4b
o62bo4b2o$63b2o10bo2bob2o5b2o5b2o9b2obo59bob2o5bobo$64bo11b4o27bo60bob
o8bo$64bobo9b3o85b2o13b2o$65b2o98bo$81bo80b3o6b2o$82b2o14b2o62bo8b2o$
81b2o16bo$96b3o$96bo8bo$105b2o$104bobo5$82b2o$82b2o$105b2o$79b2o24bo$
79b2o22bobo$103b2o2$82b2o4b2o$82b2o4b2o44bobo$134b2o$135bo$105b2o20bo$
98b2o5b2o19bobo$98b2o14b2o10b2obo29b2o$114b2o10b2ob2o28b2o$93bo32b2obo
45b2o$92bobo31bobo46b2o$93bo6b2o25bo$100bo$101b3o$103bo$129b3o$99b2o
27b2ob2o$99b2o27b2ob2o$128b5o15b3o$127b2o3b2o16bo$102b2o45bo$102b2o2$
99b2o$99b2o30b2o$91b2o$92bo$92bobo$93b2o34b2o$129b2o3$99b2o67bo$99bobo
66b3o$99bo71bo$170b2o6bo$177bobo$106b2o70bo$105bobo$105bob2o63b2o$105b
o2bo50b3o3b2o5b2o$102b2ob2o2bobo46bo2bo3b2o$102b3o5bo22b2o26bo$133bo
27bo$131bobo24bobo21b2o4b2o$126bo4b2o40b2o7b2o4b2o$124bobo46bobo$110b
2o4b2o7b2o40b2o4bo$110b2o4b2o48bobo$166bo$165b2o22bo5b3o$133b2o53bobo
2b2ob2o$126b2o5b2o56bo2bo$126b2o63b2obo$192bobo$121bo70b2o$120bobo$
121bo6b2o$128bo71bo$129b3o66bobo$131bo67b2o2$146bo$146b2o$137b2o2b2o4b
2o56b2o$137b2o2b2o4b3o55bobo$141b2o4b2o58bo$146b2o10bo48b2o$146bo10b3o
39b2o$157b3o39b2o2$155b2o3b2o34b2o$155b2o3b2o34b2o3$158bo40b2o$157bobo
39b2o$156b2o$156b2o$156b3o$157bobo$158b2o!