Physics-Informed Neural Networks for Incommensurate Fractional Volterra-Lotka Systems: Forward Solutions, Memory Asymmetry, and Inverse Parameter Recovery
Abstract
Abstract Fractional extensions of the classical Lotka-Volterra competition model invariablyimpose a single Caputo order on both species, an assumption that is mathematicallyconvenient yet ecologically unjustified: two organisms with different life historiesand habitat dependencies need not encode population memory at the same rate.We study the incommensurate coupled system in which species u and v are governedby Caputo operators of distinct orders α1 ̸= α2, respectively. Because the uniformpower-of-tα structure that underlies the power-series method of Yu and Feng [15] nolonger holds when the two orders differ, we turn instead to physics-informed neural networks(PINNs). Species-specific L1-scheme stencils are embedded as independent termsin the PINN training loss, producing a mesh-free solver valid across the whole time intervalof interest rather than only near t = 0. Three objectives drive the study. First, weverify that the PINN reproduces the commensurate power-series benchmark of [15] to amean absolute error of 2.3×10−3. Second, we show that memory asymmetry (α1 ̸= α2)generates dynamics absent from commensurate models: in particular, a nominally mutualisticparameter pair (C1,C2 > 0) can produce an apparent amensalistic outcomewhen the fractional orders differ substantially, a phenomenon we term a memory-inducedrole transition. Third, we demonstrate that all eight system unknowns—growth rates,intraspecific-limitation parameters, coupling coefficients, and both fractional…
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