Time-Periodic Carrying Simplex for a Competitive System of Carathéodory ODEs

We establish the existence of a time-periodic carrying simplex for competitive systems of Kolmogorov-type ordinary differential equations of the form ẋᵢ = xᵢ gᵢ(t,x) x ˙ i = x i g i ( t , x ) , i=1, ,d i = 1 , ⋯ , d , on Rᵈ₊ R + d , where the vector field g is T -periodic in time and satisfies only Carathéodory conditions rather than the classical C¹ C 1 regularity assumed in earlier work. This relaxed regularity requires a careful reworking of global existence, uniqueness, and continuous dependence results for the associated initial value problem on the closed first orthant. We show that the time- T Poincaré map P generated by the flow is retrotone, and apply recent results on the carrying simplex for retrotone maps to obtain for P a carrying simplex Σ Σ that is an unordered, Lipschitz, invariant manifold homeomorphic to the standard probability simplex, and that attracts all nonzero trajectories of the discrete dynamical system generated by P . We then extend this construction to the full nonautonomous flow by introducing an auxiliary autonomous system on the extended phase space T × Rᵈ₊ T × R + d , where T T is the circle of length T , generated via a skew-product formulation. In the context of this extended phase space we then construct a compact global attractor and identify, as its relative boundary, a time-periodic family of carrying simplices \Σ ₛ ∈ T { Σ s } s ∈ T that depend continuously on the phase s , each homeomorphic to the standard simplex and each attracting, uniformly in s , all nonzero orbits of the corresponding time-shifted flow. We further show that the restriction of the extended flow to the time-periodic carrying simplex is topologically conjugate, via orthogonal projection, to a flow generated by a reduced time-periodic Carathéodory system on a $$(d-1)$$ ( d - 1 ) -dimensional linear subspace, extending prior results in the autonomous setting to the nonautonomous, low-regularity case. These results provide a rigorous foundation for the qualitative analysis of seasonally forced competitive population models in which growth rates need only be measurable and locally Lipschitz in time, broadening the applicability of carrying simplex theory to more realistic ecological settings.