A theoretical and computational study of heteroclinic cycles in Lotka–Volterra systems

In general, global attractors are composed of isolated invariant sets and the connections between them. This structure can possibly be highly complex, encompassing attraction basins, repeller sets and invariant sets that, collectively, form a dynamical landscape. Lotka-Volterra systems have long been pivotal as preliminary models for dynamics in complex networks exhibiting pairwise interactions. In scenarios involving Volterra-Lyapunov (VL) stable matrices, the dynamics is simplified in such a way that the positive solutions converge to a single, globally asymptotically stable stationary point as time tends to infinity, thereby excluding the existence of periodic solutions. In this work, we conduct a systematic study on the emergence of heteroclinic cycles within Lotka-Volterra systems characterized by Volterra-Lyapunov stable matrices. Although VL stability of the matrix implies that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ω</mml:mi></mml:math> -limit sets of solutions are always stationary points, our analysis of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -limit sets reveals finite sets of stationary points interconnected by global trajectories, forming structures referred to as heteroclinic cycles. Our findings indicate that even within the framework of VL stable matrices, such structures are more prevalent than previously thought in literature, driven by the interplay between the symmetric and antisymmetric components of the model matrix. This understanding also reinforces our comprehension of the classical three-dimensional May-Leonard model, which is known to be the unique case exhibiting heteroclinic cycle within the VL framework in dimension three, while also pointing to a surprising richness in the dynamics of these structures in higher dimensions.