1. Logistic curve

Eco-systems show the complicate action such as cycling phenomena.

Plant growing or the increase of human population is shown by a logistic curve, which increase slightly at first and steeply at middle and gradually at last.

　y = k / { 1 + m * e ^ (-ct)} Eq. 1

Where, k, m, c are constants, and equation 1 satisfies next differential equation.

Constant k corresponds to final value of y.

dy / dt = c * y ( 1 - y / k ) Eq. 2

See logistic.xls

　2. Cycling phenomena of Weed and Grasshopper

Relationship of weed and grasshopper is a typical cycling phenomenon, and will be expressed as following equations.

dX / dt = ( k1 - a * Y) * X Eq. 3

dY / dt = ( b * X - k2) * Y Eq. 4

Where,

If there is no grasshopper, then equation 3 will be dX / dt = k1 * X, that is, X = e ^ (k1 * t), therefore weed will increase as like exponential function. But the breeding of grasshopper make weed fed and weed will decrease.

Finally the relationship of weed and grasshopper shows a cyclic phenomenon.

There are so-called pig cycle in breeding market of pig like as above phenomenon.