Dynamical Systems Analysis in Competitive Ecosystems

Competitive ecosystems are often modelled as dynamical systems in which interacting species or functional groups obey rules of growth and inhibition. Analysis of these models seeks to uncover the long-term behaviour emerging from nonlinear interactions, including stable coexistence, oscillations and chaotic fluctuations. Continuous-time formulations, such as Lotka–Volterra and Kolmogorov systems, capture smooth population dynamics, whereas discrete-time maps model seasonal or generational effects. Central concepts include invariant manifolds that structure global phase space, and the carrying simplex, an attracting hypersurface on which competitive dynamics reduce in dimension. Bifurcation theory reveals how gradual changes in parameters trigger qualitative shifts, from fixed points to periodic orbits and chaotic attractors. Recent advances have extended classical monotone theory into non-monotone settings, exploited cosymmetry to trace families of periodic solutions, and developed topological tools to locate fixed points under weak regularity assumptions. Computational experiments now routinely supplement analytical criteria, enabling the exploration of multistability and transient phenomena across a range of ecological scenarios. These insights inform biodiversity management, invasive species control and ecosystem restoration by identifying parameter regimes that promote resilience or precipitate collapse.

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A mathematical study of three competing species in a spatially homogeneous domain employed the theory of cosymmetry to link the destruction of one-parameter equilibria to the emergence of continuous families of limit cycles. Computational experiments in MATLAB uncovered extreme multistability, with coexisting isolated cycles and stationary solutions, illustrating how oscillatory regimes may drive overpopulation followed by collapse.

In the discrete-time Atkinson–Allen model of four competing species, a cascade of quasi-period-doubling bifurcations was shown to arise from a Neimark–Sacker transition at the unique positive fixed point. The resulting chaotic attractor lies on a globally attracting invariant manifold, implying that even simple invasion attempts into a trimorphic community can induce unpredictability in species abundances.

A recent extension of nullcline-based criteria into non-monotone Kolmogorov systems demonstrated that phase-plane partitioning and translation-arc theory can secure global attractivity and repulsion results. Applications to models with weak Allee effects and pioneer–climax interactions confirmed that folklore stability criteria also hold in broader classes of discrete and continuous competitive systems.

Dynamical Systems Analysis in Competitive Ecosystems publication trend

The graph below shows the total number of articles in dynamical systems analysis in competitive ecosystems across all publications each year (not limited to Nature Index journals).

Technical terms

Carrying simplex: A globally attracting invariant hypersurface of codimension one within which competitive trajectories evolve.

Bifurcation: A qualitative change in system dynamics—such as the birth of a periodic orbit or transition to chaos—induced by variation of a parameter.

Limit cycle: A closed trajectory in phase space representing a stable or unstable periodic solution of a dynamical system.

Multistability: The coexistence of two or more distinct stable attractors under the same set of parameters.

References

1. Mathematical model of three competing populations and multistability of periodic regimes. Izvestiya VUZ Applied Nonlinear Dynamics (2023).
2. Chaotic attractors in Atkinson–Allen model of four competing species. Journal of Biological Dynamics (2020).
3. Simple dynamics in non-monotone Kolmogorov systems. Proceedings of the Royal Society of Edinburgh Section A Mathematics (2021).
4. A Strategy to Locate Fixed Points and Global Perturbations of ODE’s: Mixing Topology with Metric Conditions. Journal of Dynamics and Differential Equations (2013).