An occasional excursion from topics photographic into the realm of mathematics and statistical modeling. I will cover some modeling mathematics for the interactions of predatory and prey in a series of three posts and include a program for the reader to use for further exploration.
Introduce a herd of deer into a forest. Let the population grow. Now add a predator, like a family of wolves. What will happen? It depends. A modeling program can be useful to explore the possibilities.
According to some basic principles of ecology, populations grow following some fairly simple mathematics. Thomas Malthus (http://en.wikipedia.org/wiki/Thomas_Robert_Malthus) was familiar with these principles and the basics have not changed much since then. A given population of mice, squirrels, deer, or humans has an intrinsic rate of growth that is limited by the carrying capacity of the environment.
Stating this in mathematical terms:
Population at time T+1 = Population at time T + Growth between T and T+1
Growth = intrinsic rate of growth x carrying capacity of the environment
Of course the carrying capacity is not a constant. For example, advances in food production technology can increase the food availability for growth of human populations. Conversely, climatic variation could result in vegetative variation causing swings in the carrying capacity for a theoretical population of deer.
Shown in the first chart is the plot of a deer herd population that grows under ideal conditions: the carrying capacity is reasonably fixed and there are no predators to eat the deer. They live a happy, idilic life. At the start there is a herd of 5000 deer growing at a rate of 50%/year (they are VERY randy deer). The carrying capacity (blue line) is 20000 and the relative variability in the carrying capacity is less than 1%. The deer herd (green line and green dots) grows rapidly until it reaches the carrying capacity and stabilizes without further variation. The wolf family (red) has not been introduced yet so the red line is flat with a constant value of zero.
Chart 1: Constant carrying capacity and no predators
Of course, if the environment is not constant, neither will the carrying capacity be constant. If variation is added to the equation for carrying capacity, the deer population will also fluctuate as in the next chart where the carrying capacity has been assigned a variability of 40%. The variation in the deer herd (green) matches the variation in the carrying capacity (blue). Over the short term, the deer herd survives, with a population size that sometimes exceeds and other times is lower than the average carrying capacity.
Chart 2: Variable carrying capacity and no predators
When the carrying capacity variation is increased to 50%, the deer herd still shows rapid growth initially from a starting value of 5000 until it reaches the carrying capacity of 20000, then drops rapidly to 10000, rebounds, but then crashes to zero. Of course, once the herd size reaches zero, it is difficult to rebound and the population remains at zero after just 20 generations.
Chart 3: Highly variable carrying capacity and no predators
In the next posting I will introduce a family of wolves and see how this changes the dynamics of the herd's population growth. In the last posting I will provide you with a programmed spreadsheet where you can experiment with different starting conditions and see for yourself how the environment, prey, and predator interact.
Zap! Pull your clothes from the dryer and unless you've used one of those funny smelling anti-static papers you just might get a shock of electricity. When the clothes tumble in the dryer and the water is removed, electrons are removed from their bonding atoms and static charges build on the clothing the same way walking across a rug on a dry day builds charge which is then released with a zap when you touch a conducting surface, like a door knob. Zap!
When I tossed my clothes into a pile for sorting, I watched as a loose strand from one sock reached upward as to defy gravity but stayed tethered to the sock. Even knowing why this is so (or at least one explanation) it fascinates. Over the next few minutes I held off sorting the clothes and took a few photographs.
Fig 1: Back lit sock with strand balanced against gravity by the repulsive force of static electricity
Even more amazing (to me), was the fine structure in the hair-like appendages that stood in repulsion to the main strand due to the repulsive forces of like charges. When electrons are removed from neutrally charged atoms, a net positive charge remains. The force between like charges is a repulsive one so the individual strands are forced apart.
Fig 2: fine structure detail from Fig 1 showing small fibers of the sock in mutual repulsion.
A few minutes later, the long strand was clearly losing the battle against gravity as the collected charges slowly dissipated as nitrogen and oxygen molecules took them away.
Fig 3: Strand 5 minutes later
Eventually, the rebellious strand lost its charge and rejoined the sock.
Fig 4: Composite showing positions of strand over 5 minutes as it loses charge and settles under the force of gravity
West Oakland's 16th Train Station was closed following Loma Prieta in 1989. The local train service has been relocated, but the 16th Street Train Station is gaining a new life as a center for community activities and special performances.
For more information, see http://www.indiegogo.com/16thStreetStation
Wildcat Creek did not run dry this year and still has a respectable flow. As I watched some leaves floating slowly by, I thought ahead when heavy rains will fill the creek above the rocks I was standing on. I also wondered how fast the leaves would be moving with the increased flow. One way to visualize this would be to take a time-lapse sequence!
Using the iPad I shot two sequences of 200 frames each. The first sequence was shot at 10 fps and the second at 1 fps. The two sequences were combined into a 40 second video using iMovie for the iPad and a soundtrack was added from a recording made a few years ago using a Sony MD Walkman. The first 20 seconds of the video shows the flow at normal speed and the second 20 seconds shows the flow at approximately 10 times normal speed. When the rains come, I can take a standard sequence and compare them. Perhaps not the most direct way to determine the comparative speed of flow in a creek, but it offers an alternative approach.
Located near Napa, the di Rosa Art Preserve is 200 acres of the selected works of Northern California artists and includes metal work, ceramics, photography, paintings, and other categories that I cannot identify (e.g., 50's vintage cars adorned with a collage of collected items). For more information and tours see http://www.dirosaart.org/.