Poincaré Map Construction for Some Classic Two Predators–One Prey Systems

Abstract

In this paper we consider a family of system with 2 predators feeding on one prey. We show how to construct a positively invariant set in which it is possible to define a Poincaré map for examining the behaviour of the system, mainly in the case when both predators survive. We relate it to examples from earlier works.

Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

“…The structure of Poincaré maps corresponding to the condition s = const, ṡ < 0 near fixed points is studied in [21]. It is shown in [8] and [9] that for a broad range of parameter values the considered system exhibits a strong contraction in the (y 1 + y 2 )-direction in which case its dynamics can be approximated by the one-dimensional map given by…”

“…Remark 1. Note that for parameter values which do not lead to a strong contraction in the (y 1 + y 2 )-direction, system (3) cannot be described by a onedimensional map and presumably may exhibit such phenomena as Shil'nikov's spiral chaos and coexistence of at least three attractors (see [3,21,27]). As discussed below, this cannot occur in the one-dimensional model (4).…”

Model Map and Multistability for a Two Predator-One Prey System

“…We consider a known system of n predators and one prey originally represented in [3,4,5] and generalized in [9]. In [11] it is shown how to rewrite it into the form…”

“…It is proved ( [9,2]), that in this case the set defined by V < 1, where V = x q 1 + y q 2 + s, q i = 1 + m i + a i − m i λ i , is positively invariant and absorbes all solutions, where the system is defined.…”

“…Conditions for extinction of one predator and coexistence of both can be found in [9,11]. A numerical overview of the behaviour of the system for different parameters can be found in [11].…”