Instability Cascades, Lotka-Volterra Population Equations, and Hamiltonian Chaos

Abstract

When an instability mode which is driven unstable can have its growth reversed because it is depleted by driving (pumping) another similar mode via a three-mode interaction, and when that process can be repeated over several pump modes, one can speak of an instability cascade. If the modes that are not acting as pumps are heavily damped (as in nonlinear Landau damping for plasma waves), the relative phases have no interesting behavior and the equations for the* mode actions are a particular case of the general Lotka-Volterra interaction system, that of a nearest-neighbor cascade. 1 (This can be thought of as a system of predation where each population species preys only on the population next nearest the ultimate source and is being preyed on only by the population next further from the source.)The general Lotka-Volterra equation system 2 is as follows, for the variables n i9 their natural linear growth rates k i9 and interaction parameters a u and 0,:Here n i (which must be positive to be acceptable) can represent the ith concentration of, for instance, species, chemical component, or wave action. The a u is antisymmetric, reflecting the zero-sum nature of the ij pair interactions, while the j3, coefficients (which are positive) are the Volterra species equivalence numbers. For chemistry the p { coefficients are such that ^i/fi i is the ratio of the number of molecules involved in the ij interaction, and for three-wave interactions all the P t coefficients are 1 (from the Manley-Rowe relations). The original two-speciesLett. 6>, 411 (1981); M. LeBerre, E. Ressayre, and A. Tallet, Phys. Rev. A'25, 1604 (1982), and references therein.version of these equations was invoked by Lotka 2 for autocatalytic reactions and by Volterra 2 to explain fluctuations in fish species in the Adriatic sea, while several authors have used them to consider plasma-wave instability competition. 1 On numerical investigation, it became evident that the time behavior could be remarkably irregular. Using Poincarg surfaces of section we found the behavior to resemble that of Hamilto-7 nian chaos, 3 and then realized that the system was indeed Hamiltonian. We recapitulate for convenience an earlier treatment 2 of an even number of equations, with a slight (but important) variation, because this leads naturally to the Hamiltonian. We definewhere y { is the formal equilibrium value of n { . The y { hence must satisfy Eq. (1) with h i set equal to zero.For an odd number of equations the determinant of the antisymmetric a u is necessarily zero, which is why we consider only even systems with IT members and invertible a { /a. Note that, unlike n/s, some y/s may be negative. (We return to this point later when discussing nearestneighbor cascades.) It has been shown 2 that there is a general Lotka-Volterra invariant, which we call H because it is in truth a Hamiltonian, given by ^EiMwi-^lnwj+yftal-yJ).(2)We have used w i here (rather than the earlier 2 w.i/Pi) because this allows the following attractive form for the equations when H is expressed Pa...

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

“…Nonetheless, also in this situation, for an even number of species with n ≥ 4 it was demonstrated that Eq. (1) can display chaotic behavior resembling Hamiltonian chaos [10]. We also mention that when three or more species are in cyclic competition according to a dynamics described by Eqs.…”

“…Since Lotka and Volterra's seminal and pioneering works [1,2], many decades ago, modeling of interacting, competing species has received considerable attention in the fields of biology, ecology, mathematics [3,4,5,6,7,8,9,10,11,12,13,14,15], and, more recently, in the physics literature as well [16,17,18,19,20,21,22,23,24,25,26,27]. In their remarkably simple deterministic model, Lotka and Volterra considered two coupled nonlinear differential equations that mimic the temporal evolution of a two-species system of competing predator and prey populations.…”

One solution of the 3D Jacobi identities allows determining an infinity of them

“…and find two relationships of the form (12). Clearly this is not possible in this case, as anticipated in Section 2.…”

“…As a final example we consider the Lotka-Volterra [4,10,12], [13]- [15] and Generalized Lotka-Volterra [6,7] structures of the form…”

New four-dimensional solutions of the Jacobi equations for Poisson structures

“…dynamics (for either Lotka-Volterra 11 or generalized Lotka-Volterra 8 systems), plasma models19 and systems such as the Toda and relativistic Toda lattices 15. The interested reader is referred to the primary reference for further examples and the full details regarding issues such as the determination of the Casimir invariants and the reduction to the Darboux canonical form for separable Poisson structures 31.…”