It was a hot day in the foothills north of Del Valle Reservoir. I had left school at noon to watch wildlife in one of my favorite areas. I rode my bicycle out to the hospital where my mother worked and climbed the hill to the south. Over the crest, a two track road led to a chaparral and oak covered draw containing a dry creek bed. Here I had spent hours watching deer, ground squirrels, lizards and snakes. Once I had even seen a striped skunk.
At the time, I was enamored of food webs; diagrams of energy and nutrient flow within ecosystems. I spent countless hours researching the connections of each new species I would find hoping to enhance our high school’s diagram.
On this day I spotted a common kingsnake making its way through the sparse grass toward the path about eight feet away. I carefully lay down to watch the snake; wishing it would come close on the path in front of me.
A motion seen from the corner of my eye I drew my attention to a large Pinacate beetle lumbering onto the path. It was only a couple feet from my face so I shifted my gaze to it.
Near the center of the path the beetle halted standing on its head in a defensive posture that allowed for the emission of a protective odor. Almost before I could focus a white-footed mouse sprang from its path side cover and seized it in its forepaws. The mouse swiftly flipped it around and stuffed the stink emitter into the dust; then proceeded to eat it. A mouse eating a beetle, why that’s carnivorous!
I was so excited I almost jumped up. This little interaction would allow me to draw a link between two species already on the school’s food web. This was very rare given the diagram was established before I had come to the school. In my time at the school I had never seen a contribution made.
As a university biology student I was discouraged from pursuing food webs by the distain in which my professors held them. They were viewed as child’s play, not true science. Eventually I left food webs, and even Biology, to study physics; or so I thought.
Years later in graduate school, the food web interest caught me again; this time in the form of directed graphs. A directed graph is a series of linked nodes where the connections have a direction and perhaps a magnitude. For example the link I had added connecting the stink beetle and the mouse. The link is directional in that energy and nutrients flow from the beetle to the mouse, but not the other way. Well that is, unless other stink beetles eat mice.
This started me on a study of the Mathematics of graph theory, with a particular focus on stability criteria and equilibrium. At the time, the concept of the balance of nature was just losing its prevalence in ecosystem dynamics. Qualitative models based on signed directed graphs (digraph) that didn’t require equilibrium for analysis were starting to take hold.
I made a signed digraph for the food web I had copied from the classroom wall and tried to decipher the embedded feedback mechanisms and their magnitudes with loop analysis. I never got reasonable results, but attributed this to my imperfect understanding of the process, or perhaps missing elements.
Then I met the work of theoretical physicist Per Bak. He wrote about the dynamics of sand piles, and described their motions as self-organized criticality. Professor Bak suggested that complex systems, of which ecosystems are an example, do not live in equilibrium – ever. Rather they organize themselves into a “poised, critical state”; way out of balance. Self-organized criticality theory implies that because graph theory focuses on equilibrium to make inference it isn’t a particularly wonderful tool for studying food webs. Ecosystems only appear to be balanced because we lacked the imagination to envision their true nature. They are constantly reacting to perturbations large and small and cannot be understood solely in terms of their parts.
As an example, in a forested ecosystem trees grow in competition with each other, gaining and losing advantage due to external events and internal growth allocation. A tree may overtop its neighbor by allocating growth to a particular branch thereby gaining additional sunlight. Being slender and long, the overtopping branch, and perhaps the entire top of the tree, may break in a heavy snowfall. Strong winds may strengthen the stems of some trees while toppling others. These perturbations are not evenly distributed, as Professor Bak noted, they occur in a power distribution; meaning that small events will vastly out number large ones. In this case, branch damage is much more common than a tree destroying event, which is more common than a forest fire.
Stewart Kauffman developed this idea further suggesting a more general natural law: all systems of sufficient complexity evolve to the edge of chaos. This is one of the most exciting and chilling concepts to come from Complexity theory. Exciting for what it leads to; chilling for what it leads away from.
Probably the most chilling deduction is, the future behavior of a system of sufficient complexity can never be predicted with any certainty. This is true even if you know the state of all of its components to any arbitrary precision. Our world has not yet come to grips with this mathematical artifact. It is so contrary to what everyone knows.
Interestingly, my knowledge of digraphs has proven helpful in other realms; I use them frequently in circuit analysis and business analytics. There has been a great interest, rooted in graph theory, on understanding on social networks. You just never know where ideas will connect or pop up – complexity in action.
Oh, I never did find out where the kingsnake went off to. It disappeared while I was watching the mouse devour the beetle – another lesson in awareness. Never get so close to a track that you lose sight of your world.