Mostly Musing – Rebuilding the Logistic map


I love building Excel files. In the grand scheme of things I’m not great at it, but it is something I very much enjoy and I get to use it on my job from time to time. Because of my hike It’s been over a year since I have done any real analysis work, and I can feel how much my blades have dulled. It’s true – If you don’t use it you lose it. Recently, a friend showed me a this YouTube video which talks about a basic population model. Before reading this post, go watch the video; it’s only about ten minutes, and is very well done! It seemed simple enough to recreate, so I figured it was the perfect opportunity to resharpen my blades and have a little fun while doing it. 

The ultimate goal of the exercise was to produce a plot like [Figure 1]. This plot describes how the population will settle (or not) given the Growth rate. There is nothing particularly interesting between the growth rates of 0 and 3 — The population stabilizes at a single value over time. But Growth rates 3-4 are where the real action is. As the growth rate ascends from three, the population progresses to a bistable state, that is the population flips back and forth between two values each iteration. Then – at about a growth rate of 3.45 the stabilization splits again into a quad-stable cycle, and then once again into an 8-stable cycle. After that are a few more splits and then – complete chaos; The iterative cycle of calculating the population yields no discernible pattern and you get an effectively random number every successive year. Looking at it a bit closer, even in the chaotic region there are brief moments of clarity where the growth rate yields a ‘stable cycle’ — pretty cool.  

[Figure 1] The goal

My first pass (after correcting a dumb bug) looked like [figure 2] . What’s important here is that this is a plot of every iteration from the initial population of 0.05 to iteration 100. The goal image is slightly different because it only shows stable values between 0 and three. I figured 100 iterations would be enough to tell if things were converging or not. Overall, you can see the shapes you want, but the first few iterations really deviate from the intended image and it doesn’t really paint the right picture of hat is going on. 

[Figure 2] first pass

Filtering out the first 30 iterations, you get [figure 3]. This looks a lot better, but isn’t all the way there. ‘Hard Coding’ in the removal of the first 30 iterations gets you a much more refined overall shape, but it’s missing a few things. For one, the knee at a growth rate of 1 is not well defined, and the other things is that pulling out the first 30 iterations means they are not included in the chaotic region so it’s not a complete picture – we can do better.

[Figure 3] First 30 iterations filtered out

Next I simultaneously did two things: 1. I rounded all my data to three decimal places and 2. Wrote a function to check for convergence. Three decimal places seemed sufficient to check for convergence – After all I am an engineer, and not a mathematician. Much to my surprise, after some tooling around I found out that my initial assumption about 100 iterations being enough was not correct. Updating my file to 700 iterations, applying the convergence analysis, and filtering out all the non-stable data for stable growth rates yields [figure 4]. Pretty damn on the nose if you ask me!

[Figure 4] Final plot

One of the other things I was very interested in was how quickly the stable regions converged onto their values. [plot 5] Shows the number of iterations it took to find stability at a precision level of three decimal places; this includes growth rates that are bi/tri/quad/etc.-stable.

[Figure ] Wildly different convergence times.

If you’d like to check out the files I built, You can do so here: A few other things I would like to explore at a later date. I’m sure these are already answered questions, but this is about me going through the exercises of figuring it out, not just looking up the answer.
-Does the initial stating population significantly effect the behavior of anything? I chose 0.05 as the starting population, but didn’t exploration anything else.
-Can the Feigenbaum constant be analytically derived? I can put together a more detailed file to empirically derive it, but it would never be an exact value.
-The chaotic region has well defined boundaries. What are these boundaries, why do they exist, and can they be empirically calculated?

A bonus photo showing only the 10 first iterations. Purdy.

The Data

  • Total Miles hiked: 2653.2
  • Completion percentage: 100%
  • Hikingdays: 129
  • Full Hiking days (no time spent in town): 102
  • Marathon Days (>26.2 miles hiked): 40
  • Neros: 27
  • True zero days: 16
  • Days off trail due to injury and travel:15
  • Towns stopped in: 36
  • Average daily milage when hiking: 20.6
  • Days precipitated on: 37
  • Pairs of shoes used: 6
  • Lost items: 2 (massage stick and sun hat)
  • Total Ascent: 462,084 Ft.
  • Total descent: 461,555 Ft.
  • Notable injury count: 1 (left knee interfrateller fat pad swelling)
  • Lost toenails: 1
  • Longest stretch without a blister: 1140.4 (Tehachapi to Sied Valley)

Over the course of my hike I kept notes on my daily milage, as well as a few other metrics. If you are interested in looking at this data in more detail, it can be found here:

This plot tells the story of my hike the best. When analyzing the daily milage, there is just too much noise to look at each individual day. Observing a five day average yields much better results and is what is presented here. A note: when calculating the average, all non hiking days were removed from the data set. As expected, there is a strong overall upward trend, meaning I got a lot stronger while on the trail. Additionally, my milage dropped in the sierras at the end of my hike for two reasons. First, the sierras are more difficult due to the elevation (spending a lot of time hiking at over 10000 feet). Second, I wanted to slow down a bit and enjoy what is arguably the most beautiful part of the trail. 

It’s nearly impossible to determine which section of the trail is the hardest because there are so many factors. Trail conditions, water availability, weather, etc. all effect how difficult a section is are are difficult to quantify. Vertical movement per mile is a traceable metric that gives you a sense of how difficult that part of the trail is though. This is calculated by adding the ascent and descent and normalizing over the distance hiked that day. Again, a 5 day average is presented to reduce the noise. What’s interesting here is that the Sierras are the most difficult section according to conventional wisdom. The data shows that they are only a little above average per this metric though. Looking into it at a bit higher level we get the following numbers:

By this metric Washington is the clear front runner for difficulty, Oregon is the easiest (as expected) and the other three sections are clumped together in the middle. If the Sierras really are the most difficult to hike through, its due to other conditions like snow and altitude. 


Etna to Kennedy Meadows North

By the numbers:

Total miles hiked: 2338.4

Completion percentage: 88.17%

Expected completion date: 9/24

note: I’m writing this post from my phone as I do not have access to my laptop. As a result I’m not able to generate all the metrics I normally do.

The trail has a way of amplifying your emotional responses. Large events are commonplace and you tend to be more even keeled when they occur. Conversely, sometimes little things mean the world to you. This is a story of one of those small things.

As I hiked into the Burney Falls Guest Ranch I was excited because I knew I had a resupply box waiting for me that was sent from my college friend Brennan. As I hiked in I noticed a large cross that sat next to the main building as well as a few other religious statues. As it turns out, the retreat is run by a few devout Christians, and they have pretty strict rules for their guests, including a no-tolerance alcohol policy. After getting settled, I picked up my box and opened it to survey it’s contents.

Brennan’s box of goodies

As expected, the box contained all of the food I needed to make it to my next stop, but it also had a few other treats. Included was a fresh pair of socks, which I desperately needed, as well as a well protected beer. Not wanting to disrespect the rules of the retreat, I quickly packed the beer away without unwrapping it or even looking at what kind of beer it was.

The next day I set out to hike hat creek rim. It’s a notoriously arid section of the trail that only has pa single water cache over a ~30 mile stretch. On to of this there is very little shade and you spend most of the day hiking in direct sunlight. Naturally, it was over 90 degrees with clear blue skies when I hiked though. All of these factors make it a difficult section where you have to constantly be aware of your hydration level. Over the course of the day I drank nearly nine liters of water and only urinated twice. Basically all of the water I consumed that day exited my body as sweat.

After the sun started to set things cooled down a bit and I stopped for a break. While sitting in an ever-so-precious shady spot, I realized that I had not yet looked at what kind of beer I was carrying with me. Like a kid in Christmas I unwrapped the beer and after recognizing it, was completely overtaken by a wave of emotions. The beer was a Samuel Smith Oatmeal Stout; my favorite beer.

It’s not an exaggeration to say that Brennan has the most impeccable memory of any person I have ever met. His ability to store an recall memories at will is unparalleled. We have kept in touch since college, but I don’t believe beer has been a topic of conversation for the the better part of a decade. After all of those years he was able to remember what my favorite beer was and was generous enough to track it down (it isn’t readily available in the US) and provide one for me. The term grateful does not even begin to describe how I felt in that moment.

Hiking through hat creek rim made for an arduous day, but getting to enjoy my favorite beer at the end made it bearable.

A dinner fit for a king.


Ashland to Etna

By the numbers
Total Miles hiked: 1755.7
Completion percentage: 66.25%
Hiking days: 90
Zero days: 29
Total Ascent: 305520
Total descent: 305526
Notable injury count: 2
Toenails count: 9.5

There is no better feeling than putting an aggressive plan in place an executing on it. At the beginning or Oregon I committed to traversing the state in 17 days and am happy to say that I was able to honor that. I crossed over the Oregon/Calfornia border on the afternoon of my 17th day. As anticipated, the terrain was much softer allowing me in increase my milage. That being said, it was far from easy. The vast majority of my days were spent hiking at least 30 miles, with my longest day exceeding 35.

My last border crossing before the finish!

Oregon was an exercise in mental toughness rather than physical as most of the pressures to perform were internal rather than external. Up to this part of the trail the hiking was difficult because of extreme weather, tough terrain, and injuries; The trek across Oregon was a slog because the days were so long and repetitive. I had a very regular daily cadence of: wake up at 5:15am, start to hike at 6am, finish hiking at 8pm, and be asleep by 9pm. On top of that, save Crater lake and the odd mountain, there isn’t much of anything interesting to look at. Pushing myself in that environment was more difficult than I had anticipated. There wasn’t much of a reward involved and finding the motivation to keep going each day was a bit of a chore. One can only get excited to look at lush forest so much. This is in direct opposition to my experience in Washington, which had a fresh spectacular view around every corner and over every hill crest. Before I started to hike the trail, I would often hear from people that it was more of a mental challenge than a physical one. Oregon was the first time that felt like the truth to me.

I’m writing this post in while in Etna California. I’m now 100 miles into NorCal and firmly in mountainous terrain. Crossing over the state border brought with it an abrupt change in the trail and the return of external pressures. The day I crossed over the border I was met with heavy rain and steep, rock covered terrain. Except for a flat stretch coming up in 100 miles or so, the rest of my hike will be like this. I have less than 900 miles to go till competition and am looking forward to returning to pushing my physical limits.

The elevation profile of the trail coming into and out of Sied Valley. Note: not to scale

Cascade locks to Bend

By the numbers
Total Miles hiked: 1354.5
Completion percentage: 51.11%
Hiking days: 75
Zero days: 27
Total Ascent: 246370 ft.
Total descent: 246836 ft.
Notable injury count: 1
Toenails count: 9.5

The stint from Cascade locks to Bend was largely uneventful, which is actually is a good thing. It did include a significant milestone though; I am officially over halfway done with the trail. It is energizing to know that each step I take is closer to the finish than to the start. Things are going according to plan and I am humming along at a pace that feels good. Optimism about completing the trail is still high.

Mentally I am in a great place, but on the physical side of things I have started to degrade a bit. For the last few hundred miles or so I have been developing neuropathy in both of my feet. It’s an issue that has been building over time, but it feels like it has leveled off. Neuropathy seems to be pretty common on the trail which is reassuring because I know you can hike through it, but it is still quite painful and frustrating to deal with. It manifests in a slightly different way each day, but for the most part it follows a general pattern: In the morning when I take my first steps out of my tent I am in extreme pain and every “step” is a hobble at best. After a few minutes of walking around doing chores most of the pain subsides and I feel ready to hike. The first mile or so comes with a lot of discomfort in the form of shooting pain that runs from the balls of my feet to the heels. After I am sufficiently warm and loose the pain will go away completely and I am able to hike pain free for about 20 miles. Towards the end of the day the pain will return and hiking becomes challenging. I can manage the pain at the end of the day with nsaids, but it does not deal with the pain completely.

On top of this, the nerves in the balls of my feet and toes are not registering touch properly. Instead of the normal sensations I should feel when they are touched I experience intense tingling, much like when your leg will fall asleep. Luckily this only happens when my feet are bare, and it doesn’t happen when I am hiking. I have been keeping the neuropathy at bay with multiple foot massages each day as well as keeping my feet elevated when I sleep. It seems to be helping and am hoping that it will stifle the progression of it and, if I am lucky, reduce the symptoms. All things considered, it’s not too bad and is something I know I can deal with; If this is the worst thing I have to deal with on the trail than I consider myself quite lucky.

my very professional halfway marker.

Packwood to Cascade Locks

By the numbers
Total Miles hiked: 1207.8
Completion percentage: 45.58%
Hiking days: 70
Zero days: 27
Total Ascent: 218,928 ft.
Total descent: 224,000 ft.
Notable injury count: 0
Toenails count: 9

I’m happy to report that I have completed all of Washington state. It’s a nice morale boost after the difficulties faced in Washington. The state took me 28 days to complete, and I was rained on for 23 of those days. It was challenging both mentally and physically, but I am coming out on the other side with enormous confidence and optimism about the rest of the trail. When you step foot on the PCT the odds are against you for finishing, but for the first time it feels like I have a real shot at completing the trail.

Arriving at the famous ‘Bridge of the gods’, which spans Hood river, dividing Oregon and Washington

The initial learning stage for through hiking is over for me and I feel like I have totally acclimated to life on the trail. At this point, everything I have to do to be successful feels natural and I’m not being met with any unexpected obstacles; For the first time I feel ‘on top of it’. This is ushering in the second phase of my hike. The first phase was defined by me learning how to be out here, and the second is defined by me learning to deal the monotony of each day. A lot of mental energy and time were devoted to dealing with life on the trail. Now that I don’t have to spend nearly as much mental energy on things like that, a void has been left in how I spend my time. Oregon and northern California are flat, and do not offer nearly as many spectacular views as Washington. I’m going to continue hiking solo, and each day is going to feel very similar. It’s going to be a challenge to keep my brain occupied, but I am looking forward to falling into a rhythm.

My daily milage count continues to grow. Coming into Cascade locks I was able to complete my two longest days on the trail so far (29.7 and 32.2 miles) in addition to hiking a 20 mile day at a pace that would be needed for 30. This was a stress test on my body to see If I could handle the Oregon challenge. I’m happy to report that I feel great after that short stint, but am still weary of pushing that hard for an extended period of time. For the time being, I’m planning on getting through Oregon in 17 days, which will allow for a more reasonable daily mileage, plus a zero.

Cascade Locks also brought with it a few small treats. Because I ramped up my milage, I was able to catch my trail family for a day. I missed them immensely and was grateful for the time we got to spend together. Unfortunately the Trail family has shrunk even more. Casper has left the trail because of personal reasons and I was not able to say goodbye. Muscle also took time off for an injury, but he will be back. In an odd but very welcome twist of fate T-pain was in town as well. He changed his plans and is moving to a smaller town in northern Washington. His journey out to the PNW aligned with ours and we were able to see each other for an evening; beers and good times were had. When T pain left the next morning we promised each other that it would not be last time we saw each other.

The trail family left a day before I did, so I still have catching up to do. It’s likely that I will reach them somewhere in Oregon as I am planning to move a bit faster than them. Hopefully it will align with all of us being in Elk lake as I plan on taking my next zero there.

Brightside, Hot hands, myself, T pain, and 70/30


Leavenworth to Packwood

By the numbers
Total Miles hiked: 1060.0
Completion percentage: 40.0%
Hiking days: 64
Zero days: 26
Total Ascent: 195563
Total descent:195609
Notable injury count: 2
Toenails count: 9

I have spent the majority of my trail time hiking with other people. While I have loved the people I met along the way, I have decided to hike the remainder of the trail by myself. One of the original reasons I decided to do a long trail was because I wanted complete agency over my life and how I spent my time. Unfortunately, hiking with other people forces you to comply in ways that you might not want to, and removes some optionality in your decision making process. Hiking by myself is going to allow me to put in a few extra miles each day, and gives me an outside shot at finishing by mid September which means I would be able to attend my good friend’s wedding. For most of Washington, Carjack and I have been hiking together, but in Snoqualmie Pass I left early and am now roughly a day ahead.

During this stretch I feel like I have really hit my stride and am putting in great miles. Over the last 6 full days of hiking I have averaged 24.4 miles, which puts me in a great position. I finally have my hiker legs under me (it only took 1000 miles, grumble) and am prepping myself for Oregon. Because Oregon is very flat compared to the rest of the trail, it’s expected that you increase your daily milage by a significant amount. Many successful hikers average about 30 miles a day in Oregon, and I plan to do the same. In addition, there is a tradition on the PCT called ‘The Oregon Challenge’. It’s a pretty straightforward endeavor where you set the goal of going through all of the roughly 450 miles in Oregon in two weeks. I am toying with the idea of attempting it, but it requires a ~33 mile a day average, with no zero days. I’m not sure I’ll be capable of doing that and will make a game time decision when I reach the Oregon border in roughly 10 days.

The relatively minor increase in daily milage has also had an unexpected effect on me. Since I hit Washington, my weight has been plummeting. I have been tracking my daily calorie intake over the whole hike and managed to remain very close to the same weight over the first 700 miles. I don’t have access to a scale so I can’t find out my actual weight, but the weight loss is quite apparent when looking in the mirror. I’ll be increasing my daily calorie intake by about 500 calories for the remainder of Washington. The daily elevation gain in Washington is likely causing me to burn additional calories as well. My weight is something I am going to have to monitor closely when I get to Oregon because the hiking is going to be quite different. It’s possible that the increase in miles and decrease in elevation changes may end up being a wash. Only time will tell.

Calories. Calories. Calories.

I wish I could say that the last stretch of hiking was as beautiful and inviting as the rest of Washington, but I can’t. Save a single afternoon, the weather has been severely overcast and I have been rained on most days. Very few nice views were had and I have been unable to get a clear shot of Mt. Rainer, which I am hoping to see. In addition, the trail has been overrun by mosquitoes. When you are moving, the mosquitoes don’t bother you too much, but when you stop for longer than 30 seconds, a cloud of them will quickly be surrounding you. I have responded to this by wearing additional clothing that they can’t bite through and using bug spray, but it only does so much. One night I did the disservice to myself of counting the mosquito bites I had, and counted 38 (not including my back).

The next stretch of the trail is going to be a long one for me. Rather than stopping in Trout Lake for a full day, I’m only going to do a quick resupply and move on. Getting from Packwood to Stevenson will take me roughly a week and I will cover a little over 150 miles. It will be my longest stretch on the trail, and I am eager to tackle it and push myself a little bit further.

The view out of my tent one morning when I camped in a burn area. Most of the time It feels like I am hiking in a cloud.