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.

Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

“…This technique was used successfully in [13–16] 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.…”

“…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]. …”

Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

“…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]:…”

“…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.…”

“…The fourth section uses the symmetries for the map T showing that this map is conjugate to its inverse through an involution. Then, we use time reversal symmetry method [13,20] based on the symmetries to effectively find some feasible periods and corresponding orbits of the map T.…”