Lie Symmetry Classification and Invariant Solutions of Diffusive Lotka–Volterra Systems with Time-Dependent Coefficients

Diffusive Lotka–Volterra systems are widely used to describe interacting biological species, chemical reactions, and ecological processes. While Lie symmetry methods have been extensively applied to reaction–diffusion systems, complete symmetry classifications for diffusive Lotka–Volterra systems with time-dependent coefficients remain relatively scarce.

A class of two-component diffusive Lotka–Volterra systems with time-dependent coefficients is considered in the form

u_t=u_xx+u(b₁(t)u+c₁(t)v), v_t=d(t)v_xx+v(b₂(t)u+c₂(t)v),

where b_i(t), c_i(t) and d(t) are arbitrary smooth functions satisfying c₁²+b₂²≠0 and d(t)≠0, and at least one coefficient is non-constant.

The classical Lie symmetry approach is employed to perform a group classification of the considered class of systems. The equivalence group of the considered class is found, and a complete Lie symmetry classification of its inequivalent cases is obtained. The classification yields extended symmetry algebras, which lead to symmetry reductions to systems of ordinary differential equations. Corresponding invariant solutions are constructed.

The obtained results show that time-dependent coefficients significantly enrich the symmetry structure of diffusive Lotka–Volterra systems. The classification provides a systematic basis for constructing exact solutions and extends known results for constant-coefficient models.

Reference: Davydovych V. Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients, Symmetry, 10(2) (2018), 41.