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Application of interval iterations to the entrainment problem in respiratory physiology

“…The cascade of doubling bifurcations we always observe dramatically changes for ε > 0.01 (only one bifurcation is observed) and for ε > 0.1 (no more bifurcations are observed). If I (0) becomes sufficiently wide we cannot obtain any cycle and there is always a unique stable interval equal to represents the 'Ricker model' for the growth of a single species in ecology literature (May & Oster 1976;Reich & Zaslavski 2002). Ricker introduced equation (6.4) to describe fish populations from the Pacific coast of Canada.…”

“…is then a stable fixed interval for the interval iterations dynamics. More generally, for proving the existence of fixed intervals for f •m (m = 1 in the previous example), we can use numerous fixed set theorems proven for the set-valued mappings (figure 11) (Nadler 1969;Smajdor 1985;Przytycki 1987;De Farias & van Roy 2000;Reich & Zaslavski 2002;Graczyk et al 2004;Kleptsyn & Nalskii 2004;Cheban & Mammana 2005;Desheng & Kloeden 2005;Kloeden & Valero 2005;Kamran 2007;Wlodarczyk et al 2007). These theorems involve in general a local Lipschitzian property of the set-valued mapping, for an adapted metric.…”

“…The equation defined by N t+1 = N t exp r 1 − N t K (6.4) represents the 'Ricker model' for the growth of a single species in ecology literature (May & Oster 1976;Reich & Zaslavski 2002). Ricker introduced equation (6.4) to describe fish populations from the Pacific coast of Canada.…”