Abstract
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
x t + 1 = f(x t )
where each x t 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 method.
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Cull, P. (2005). Convergence of Iterations. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2005. EUROCAST 2005. Lecture Notes in Computer Science, vol 3643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556985_59
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DOI: https://doi.org/10.1007/11556985_59
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