Effects of diffusion in controlling chaos in some tri-trophic food chains
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2026
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Abstract
This dissertation addresses the mathematical analysis of three-component reaction–diffusion
systems, with a focus on their necessity in modelling complex phenomena across ecology,
epidemiology, and developmental biology. Such systems provide a more realistic framework than
classical two-component models, capturing intricate multi-species interactions, cross-diffusion
processes, and spatially extended dynamics.
The first part of the work establishes explicit algebraic conditions for the onset of diffusion-driven
instability (Turing patterns) in tri-trophic models. By analysing the linearised spectrum of the
reaction–diffusion operator, we derive parameter‑space criteria that determine when spatially
homogeneous equilibria become unstable due to differential diffusion rates, leading to spontaneous
pattern formation.
The second part investigates the global existence of solutions for systems with full (including
cross‑diffusion) matrices and polynomial growth nonlinearities. Using the method of invariant
regions combined with Lyapunov‑type functionals, we prove that, under appropriate structural
assumptions on the reaction kinetics and diffusion coefficients, solutions remain uniformly bounded
for all time and satisfy suitable a priori estimates. The results are valid for a wide class of boundary
conditions and do not require the diffusion matrix to be triangular or diagonal.
The theoretical framework is illustrated with applications to ecological and biological models,
highlighting the interplay between cross‑diffusion, stability thresholds, and long‑time behaviour. The
thesis thus provides both a refined instability analysis for three‑component systems and a robust
existence theory that extends previous results in the literature.