Friday Fractal LVIII: A Poison Weed

You might recall a young girl named Alice. Alice liked to contemplate things, such as what would happen if sage advice were ignored. "If you drink much from a bottle marked "poison", it is almost certain to disagree with you, sooner or later," she wisely noted. You don’t have to look into children’s literature, or even for a corrupt pharmacist, in order to find a "bottle" marked "poison". In nature, markings which warn of danger, similar to the crossbones pictured on old-fashioned bottles of toxic substances, are not altogether rare. In animals, and sometimes plants, this trait is known as "aposematism". Consider a skunk’s stripes, the bright scarlet hourglass on the belly of the black widow spider, or the brilliant orange hues on the wings of a monarch butterfly. These animals aren’t exactly flaunting their flashy designs; they’re giving a warning to any potential predators. The bright contrast says "don’t eat me; I taste bad." Birds who prey on insects learn quickly to avoid brightly colored patterns of an aposematic insect, or suffer for the mistake.

Many poisonous or unpalatable insects derive their toxins from the plants they hatch in and subsequently consume. In the case of the Monarch butterfly, and a number of other bright, aposematic insects, the host plant is the common milkweed. In meadows and ditches a select variety of insects lay their eggs on the poisonous leaves of the milkweed plant--or in some cases, inside the stems and roots. The patterned beetle mimicked in today’s fractal does just this--lays its eggs inside the milkweed plant leaving the young nymphs to feed on the leaves.

In order to imitate the milkweed blossom, I used Newton’s method, which calculates the placement of pixels based on their attraction to various basins. (Imagine rolling a ball along a bumpy landscape; the ball will rest in the valleys, or basins, even if it starts on a hill.) The basins in this fractal were determined by the trigonometric functions, exp(z) = log(z). Then, just to make things more complex, I set the various colors of the pixels based on which ones were trapped within an ellipse. The "beetle" is a Julia set, colored in a similar fashion, but trapping the pixels according to the calculations of the orbit of the fractal set:

It is rather easy to spot my "fractal beetle" ....and it was equally easy to spot this beetle on a milkweed plant near Big Dry Creek in Colorado:

Red Milkweed Beetle (Tetraopes tetraophthalmus) on Showy Milkweed (Asclepias speciosa)

The bright colors on the back of this beetle warn predators of its unpleasant taste. The colors are similar to another insect hosted by the milkweed plant, the milkweed bug. A "true bug", the milkweed bug has black and red stripes, in nearly the same shades as the red milkweed beetle.

You might think the milkweed plant is providing its own mini-habitat by hosting these specific insects. But the red milkweed beetle depends on the presence of grass species as much as it does the milkweed species. The beetle clings to the stalks of the grass during ovipositioning. Anurag A. Agrawal, a member of the department of botany at the University of Toronto, discovered the beetle’s preference caused a competitive, facilitating relationship between the milkweed plant and the milkweed beetles:

"[T]he associational effect of grass on milkweed resulted in milkweed suffering the non-additive effects of competition and herbivory, whereas grass enjoyed competitive release by facilitating its neighbor’s herbivore. Many traits of milkweed (e.g., growth, reproduction, and several resistance traits) showed variation among 23 full-sibling families, indicating that competitive ability and resistance may be subject to natural selection. A multiple regression analysis on family means revealed that leaf trichome density and nitrogen content were negatively genetically correlated with abundance of Tetraopes adults, but probability of flowering and plant height were positively associated. Leaf miners were most strongly negatively affected by latex and trichomes. Thus, complex interactions among competition, root herbivory, and plant genetic variation affect the herbivore and plant community and may result in diffuse coevolution between milkweed and its herbivores."

In order for the habitat to succeed, many complex parts interact with one another, constantly adapting to small changes, in order to survive. In this complex depth, however, we find strange and wonderful beauty. Of course, this is true not only for milkweed, grasses, and beetles, but for many complex, fractal patterns.

Fractal by the author using ChaosPro.