2011
DOI: 10.1080/17513758.2010.488301
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Applications of KAM theory to population dynamics

Abstract: Computer simulations have shown that several classes of population models, including the May host-parasitoid model and the Ginzburg-Taneyhill 'maternal-quality' single species population model, exhibit extremely complicated orbit structures. These structures include islands-around-islands, ad infinitum, with the smaller islands containing stable periodic points of higher period. We identify the mechanism that generates this complexity and we discuss some biological implications.

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Cited by 12 publications
(20 citation statements)
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“…This technique was used successfully in [1316] in the case of difference equations while there exists vast literature in the case of differential equations; see [17, 18]. Computer simulations of the trajectories of (5) indicate the existence of an infinite nested family of invariant closed curves surrounding an elliptic fixed point, sequences of periodic islands in the regions between the invariant curves, and stochastic regions surrounding the periodic islands and between invariant closed curves.…”
Section: Introductionmentioning
confidence: 99%
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“…This technique was used successfully in [1316] in the case of difference equations while there exists vast literature in the case of differential equations; see [17, 18]. Computer simulations of the trajectories of (5) indicate the existence of an infinite nested family of invariant closed curves surrounding an elliptic fixed point, sequences of periodic islands in the regions between the invariant curves, and stochastic regions surrounding the periodic islands and between invariant closed curves.…”
Section: Introductionmentioning
confidence: 99%
“…The KAM theorem says that, under the addition of the remainder term, most of these invariant circles will survive as invariant closed curves under the full map [17, 18]. Precisely, the following result holds see [13, 18]. …”
Section: Introductionmentioning
confidence: 99%
“…Once the sets S 0,1 are known, the search for periodic orbits can be reduced to a one-dimensional root finding problem using the following result, see [13,20]:…”
Section: Symmetriesmentioning
confidence: 99%
“…A transformation R of the plane is said to be a time reversal symmetry for T if R −1 • T • R = T −1 , meaning that applying the transformation R to the map T is equivalent to iterating the map backwards in time, see [13,20]. If the time reversal symmetry R is an involution, i.e., R 2 = id, then the time reversal symmetry condition is equivalent to R • T • R = T −1 , and T can be written as the composition of two involutions T = I 1 • I 0 , with I 0 = R and I 1 = T • R. Note that if I 0 = R is a reversor, then so is I 1 = T • R. In addition, the jth involution, defined as I j := T j • R, is also a reversor.…”
Section: Symmetriesmentioning
confidence: 99%
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