An equation of biological windows and its applications

All biological events, phenomena and processes on Earth have a time of start and a time of end. An accurate and precise definition and determination of the window of a biological event is of both theoretical and practical value. In this paper, I define the window of a biological event and derive models for its calculation. Specifically, the window of a biological event is defined through the progression of its biological time. Let a biological event start at time tL and end at time tR. As time τ varies from tL to tR, its biological time varies from a(tL) at time tL to a(tR) at time tR. The equation of biological windows is defined as a ( t R ) − a ( t L ) = ∫ a d a = ∫ t L t R ( d a ( τ ) / d τ )   d τ , with a window width of tR − tL, a window height of da(τ)/dτ at time τ ∈ [tL, tR], and a window area of a(tR) − a(tL). The standard equation of biological windows is defined as 1 = ∫ t L t R ( 1 / ( a ( t R ) − a ( t L ) ) ) ( d a ( τ ) / d τ )   d τ , with a window width of tR − tL, a window height of (da(τ)/dτ)/[a(tR) − a(tL)] at time τ ∈ [tL, tR], and a window area of unity. A stochastic model is also given. Both types of model are related to multiple ecological factors through a first-order nonlinear partial differential equation. I use the special cases of both types of model to analyze data collected in the laboratory to demonstrate their utility.