[Paper Review] Chaotic dynamics in the Volterra predator-prey model via linked twist maps

Marina Pireddu, Fabio Zanolin|ArXiv.org|May 28, 2008

Mathematical Dynamics and Fractals23 references3 citations

TL;DR

This paper establishes the existence of chaotic dynamics in the periodically forced Volterra predator-prey model via linked twist maps, proving infinitely many periodic solutions and a topological horseshoe through geometric analysis of Poincaré return maps. The key contribution is the rigorous demonstration of chaotic-like behavior under periodic harvesting, extending classical Volterra dynamics with topological methods.

ABSTRACT

We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator--prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.

Motivation & Objective

• To investigate the existence of complex, chaotic dynamics in the periodically forced Volterra predator-prey system with harvesting.
• To extend classical Volterra dynamics—originally conservative and periodic—under periodic perturbations due to harvesting.
• To establish the presence of infinitely many periodic solutions and topological horseshoes via geometric and topological methods.
• To apply recent results on chaotic planar maps and linked twist map theory to a classical ecological model.

Proposed method

• Utilizes the Poincaré return map of the periodically forced Volterra system to analyze long-term dynamics.
• Applies the theory of linked twist maps to detect chaotic behavior through geometric twisting of annular regions.
• Constructs a compact invariant set Λ ⊆ 𝒫 that is semiconjugate to a two-sided m-shift via a continuous surjection g.
• Employs topological horseshoe theory to prove the existence of infinitely many periodic orbits and chaotic dynamics.
• Uses the surjection g: 𝒫 → Σ_m to map symbolic dynamics onto the phase space, ensuring that periodic sequences in the shift space lift to periodic points in the system.
• Relies on the injectivity and continuity of the Poincaré map and the structure of compact, disjoint invariant sets to define the symbolic dynamics.

Experimental results

Research questions

• RQ1 Does the periodically forced Volterra predator-prey model with harvesting exhibit chaotic dynamics despite its conservative origin?
• RQ2 Can the existence of infinitely many periodic solutions be rigorously proven in the presence of periodic harvesting?
• RQ3 To what extent do geometric features of linked twist maps enable the detection of topological horseshoes in planar ODEs?
• RQ4 How does the Poincaré map of the forced system reflect chaotic behavior through symbolic dynamics?
• RQ5 Can the semiconjugacy between the system's dynamics and a shift space be established to confirm chaotic-like dynamics?

Key findings

• The periodically forced Volterra model with harvesting exhibits infinitely many periodic solutions due to the presence of a topological horseshoe.
• A topological horseshoe is established via a semiconjugacy between the Poincaré map and a two-sided m-shift, confirming chaotic dynamics.
• The system's Poincaré map is shown to be semiconjugate to a Bernoulli shift on m symbols through a continuous surjection g.
• The set of periodic points of the Poincaré map is dense in the invariant compact set Λ, a hallmark of chaotic systems.
• For any k-periodic sequence in the shift space Σ_m, the preimage under g contains at least one k-periodic point of the system.
• The proof relies on the geometric twisting properties of linked twist maps and the existence of disjoint, invariant compact sets in the phase space.