Integrability and singularity structure of predator-prey system

Abstract

The ‘‘extended’’ ARS (Ablowitz, Ramani, and Segur) algorithm is introduced to characterize a dynamical system as Painlevé or otherwise; to that end, it is required that the formal series—the Laurent series, logarithmic, algebraic psi series about a movable singularity—are shown to converge in the deleted neighborhood of the singularity. The determinations thus obtained are compared with those following from the α method of Painlevé. An attempt is made to relate the structure of solutions about a movable singularity with that of first integrals (when they exist). All these ideas are illustrated by a comprehensive analysis of the general two-dimensional predator-prey system.

Necessary and Sufficient Conditions for Finite Time Singularities in Ordinary Differential Equations

“…without logarithmic terms), the convergence of these series has been proven in [9,10]. In the general case, recent general results on singular analysis for PDEs by Kichenassamy and co-workers [11,12,13] strongly suggest that that the Psi-series are convergent in general as has been successfully demonstrated on many specific examples [14,15,16,17]. However, in the absence of a well-defined rigorous proof, we leave here the convergence of the Psi-series as an assumption.…”

Singularity Structure of Third‐Order Dynamical Systems. I

“…In an earlier article [6], we obtained all the representations of the solutions of a two-dimensional normal quadratic system about a movable singularity, namely, Laurent series, logarithmic psi-series, or algebraic psi-series expansions, proving in each case the convergence of the relevant series in a deleted neighborhood of the movable singularity. In this article, we extend our study to general third-order systems.…”

“…Smith [7] studied a general second-order system where P and Q are polynomials of finite degree ν in x and y He transformed to a first-order Briot-Bouquet (B-B) equation in v and w and a third auxiliary equation, using the transformation v = 1/x z = y/x z = β + w He then applied the known results for the first-order B-B equation and obtained convergent series solutions about a movable singularity for the original second-order system by using the inverse transformation x = 1/v y = z/v (see also [6,8]).…”