Convergence of Iterations

Convergence is a central problem in both computer science and in population biology. Will a program terminate? Will a population go to an equilibrium? In general these questions are quite difficult – even unsolvable. In this paper we will concentrate on very simple iterations of the form xt+1 = f(xt) where each xt is simply a real number and f(x) is a reasonable real function with a single fixed point. For such a system, we say that an iteration is “globally stable” if it approaches the fixed point for all starting points. We will show that there is a simple method which assures global stability. Our method uses bounding of f(x) by a self-inverse function. We call this bounding “enveloping” and we show that enveloping implies global stability. For a number of standard population models, we show that local stability implies enveloping by a self-inverse linear fractional function and hence global stability. We close with some remarks on extensions and limitations of our metho...

• Type: Conference
• Year: 2005
• Where: EUROCAST
• Authors: Paul Cull
• Keywords: EUROCAST 2005, Global Stability, Implies Global Stability, fixed Point