Dynamics of the Attractor in the Lotka–Volterra Equations

Abstract

We show that for stable dissipative Lotka Volterra systems the dynamics on the attractor are hamiltonian and we argue that complex dynamics can occur. 1998Academic Press

Competitive Interaction Model for Online Social Networks’ Users’ Data Forwarding at a Subnet

“…Then we tested the predictability of the model when population's densities vary over time. In the numerical simulations interesting results were found about the sensitivity of the two species' growth rate in the case of limited network supply identified by the carrying capacity in the model under a routing protocol that treats equally all data packets [22][23][24][25][26][27].…”

Hamiltonian evolutionary games

“…A Poisson structure on R n defined by the bivector π A (x) = D x A D x was introduced in [2]. System (3.9) is Hamiltonian in the interior of R n + w.r.t.π A having H as Hamiltonian function.…”

“…More recently, in the 1980's, Redheffer et al developed further the teory of dissipative LV systems, introducing and studying the class of stably dissipative systems [14][15][16][17][18]. In [2] a re-interpretation was given for the Hamiltonian character of the dynamics of any conservative LV system: there is a Poisson structure on R n + which makes the system Hamiltonian. Another interesting fact from [2], which stresses the importance of studying Hamiltonian LV systems, is that the limit dynamics of any stably dissipative LV system is described by a conservative LV system.…”

A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations

“…This example shows that the authors in [15] decided to keep themselves with a Volterra RFDEs of the type (36) in the case of n = 3 species because is still possible to give a geometric description of a set of global bounded solutions, due, mainly, to the Poisson integrability (see [3]) of two associated 3-dimensional Lotka-Volterra ODE systems.…”

“…But for higher dimensional systems the topology of orbits in phase space is much more complex, and understanding this topology is a challenging problem. The following theorem (see [3]) is perhaps the first result in this direction. Theorem 1.2.…”