There is a theorem in the Intelligent Design literature called Basener's ceiling. It states that all evolutionary algorithms that attempt to optimize some fitness function will eventually reach a plateau where complexity no longer increases. This is seen in practice when engineers and computer scientists try to use algorithmic forms of evolution.
There are a number of standard models for the evolutionary process of mutation and selection as a mathematical dynamical system on a fitness space. We apply basic topology and dynamical systems results to prove that every such evolutionary dynamical system with a finite spatial domain is asymp- totic to a recurrent orbit; to an observer the system will appear to repeat a known state infinitely often. In a mathematical evolutionary dynamical system driven by increasing fitness, the system will reach a point after which there is not observable increase in fitness.
Biological evolution is the same as an evolutionary algorithm in the relevant aspects. It is a search to optimize reproductive fitness, where the fitness function is just the fact of survival or lack thereof.
How then is evolution able to make organisms continually increase in complexity, as history shows, despite Basener's ceiling?