[Paper Review] Qualitative analysis of strictly non-Volterra quadratic dynamical systems with continuous time

Rasulov Xaydar Raupovich|arXiv (Cornell University)|Nov 11, 2022

Differential Equations and Numerical Methods4 citations

TL;DR

This paper presents a qualitative analysis of a continuous-time strictly non-Volterra quadratic dynamical system, deriving its equilibrium points, phase portrait, and analytical solutions under specific symmetry conditions. It demonstrates that trajectories converge exponentially fast to the unique equilibrium point (1/3, 1/3, 1/3), with numerical and analytical solutions showing near-identical behavior for t ≥ 4, confirming the system's stability and hyperbolic saddle nature.

ABSTRACT

In this article, a continuous analogue of strictly non-Volterra quadratic dynamical systems with continuous time and points of equilibrium is investigated, a phase portrait of the system is constructed, numerical solutions are found, and a comparative analysis is carried out with a particular solution of the system.

Motivation & Objective

• To investigate the qualitative behavior of a continuous-time analogue of strictly non-Volterra quadratic stochastic operators.
• To identify equilibrium points and construct the phase portrait of the system.
• To derive analytical solutions under symmetric initial conditions (x₁ = x₂) and compare them with numerical solutions.
• To demonstrate the exponential convergence of trajectories to the equilibrium point.

Proposed method

• Formulate a system of three coupled nonlinear ODEs representing the continuous-time dynamics of a strictly non-Volterra quadratic operator.
• Use symmetry assumptions (x₁(t) = x₂(t)) to reduce the system to a solvable form, yielding analytical solutions (12) and (13).
• Apply the method of summing all equations to derive a scalar ODE for X = x₀ + x₁ + x₂, solving X′ = X² − X to analyze total population dynamics.
• Use MathCAD to compute numerical solutions for various initial conditions and compare them with analytical solutions.
• Construct phase portraits and trajectory plots to visualize system behavior, particularly convergence to equilibrium.
• Perform comparative analysis between numerical and analytical solutions, showing differences < 0.001 and near-coincidence for t ≥ 4.

Experimental results

Research questions

• RQ1 What are the equilibrium points of the continuous-time strictly non-Volterra quadratic dynamical system?
• RQ2 How do trajectories behave over time, and do they converge to a fixed point?
• RQ3 Under what conditions can analytical solutions be derived for this nonlinear system?
• RQ4 How do numerical solutions compare with analytical solutions in terms of accuracy and convergence?
• RQ5 What is the role of the constraint x₀ + x₁ + x₂ = 1 in shaping the system’s dynamics and phase portrait?

Key findings

• The system has a unique equilibrium point at (1/3, 1/3, 1/3), which is a hyperbolic saddle.
• The sum X = x₀ + x₁ + x₂ satisfies X′ = X² − X, with solutions X = 1/(1 − C eᵗ), showing X → 1 exponentially as t → ∞.
• For initial conditions with x₁ = x₂ and x₁ < 1/3, analytical solutions (12) are derived and shown to match numerical solutions with error < 0.001.
• For initial conditions with x₁ = x₂ and x₁ > 1/3, analytical solutions (13) are derived and similarly validated against numerical results.
• Numerical and analytical solutions coincide almost perfectly for t ≥ 4, confirming exponential convergence to the equilibrium point.
• The phase portrait consists of five curves: two equilibrium points (0 and 1), two rays (X < 0 and X > 1), and the interval (0,1), with X = 0 stable and X = 1 unstable.