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• W15022599 abstract "One dimensional difference equations are widely used in population biol- ogy. These seemingly simple models can show a variety of behaviors from stability to chaos. (see Cull, Yorke, May, Feigenbaum) We show how the enveloping technique can be used to demonstrate global and semi-global stability. We discuss the issue of whether local stability implies global stability. We give some examples of more com- plicated behavior which can co-exist with local stability. We show that local stability implies global stability even for models slightly more complicated than the usual mod- els. We address the issue of how complicated a model must be to have local without global stability, and we describe our candidates for the simplest such models. 1 Introduction Populations wax and wane. To understand these changes in population, we often use simple models. In this paper, we will study difference equation (10) models of population growth. Often these models have a single equilibrium point and for some values of the param- eters this equilibrium is stable. Biological modelers have often checked for stability with respect to small perturbations and then treated the equilibrium as being stable with re- spect to large perturbations. Remarkably, this logical jump never caused any difficulties. Eventually, a number of mathematical papers (4, 34, 32) showed that for some of the usual population models local stability implies global stability. These papers used a variety of methods including Lyapunov functions (23, 18, 19) Schwartzian derivative (34), and some ad hoc techniques (34, 1, 2, 4, 3, 5, 32). But, one uniform technique applicable to all usual population models was lacking. In particular, it was unclear whether the single hump of population models was sufficient to derive global stability from local stability. Finally, Cull and Chaffee (9, 8) were able to show that the usual population models were bounded by linear fractional functions and that this bounding was enough to show global stability from local stability. One aspect of the above approaches is the assumption that the models were three times continuously differentiable. Huang (22) pointed out that for some of these arguments, the continuously differentiable assumption was essential. It was also unclear whether biological modelers really wanted to make such an assumption about their models. Cull (13) showed that bounding (enveloping) by linear fractionals did not depend on differentiability, and, in fact, such enveloping could apply to discontinuous multi-functions. In contrast to such stability May (25, 26) and others have shown that without local stability, population models can show complicated behavior including chaos (24, 14). But, is more complicated behavior possible for population models when local stability is assumed? We investigate some generalizations of the usual population models and show that YES, more complicated behavior is possible even with local stability. How much more complicated does a population model have to be to allow more complicated behavior? We show that" @default.
• W15022599 created "2016-06-24" @default.
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• W15022599 date "2008-03-01" @default.
• W15022599 modified "2023-09-26" @default.
• W15022599 title "STABILITY AND INSTABILITY IN ONE DIMENSIONAL POPULATION MODELS" @default.
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• W15022599 doi "https://doi.org/10.32219/isms.67.2_105" @default.
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