Post-accumulation convergence in the symmetric model

Prove that solutions of the four-dimensional deterministic dynamical system for the symmetric matching-type predator–prey model, initialized at states of the form $(K\hat{n}, K^{1-\varepsilon}, K^m\hat{h}, K^{(1-\varepsilon)m})$, reach a neighborhood of the four-type coexistence equilibrium $z^*$ within a time of order $O(\varepsilon\log K)$, thereby establishing convergence after the accumulation time in the original symmetric model.

Background

In cases A and B, the successive prey and predator invasions accumulate at a finite time. At the accumulation time, all four logarithmic population exponents approach one, indicating that both prey types and both predator types should be macroscopic. The relevant limiting ecological state is the positive four-type coexistence equilibrium z∗z^* of the symmetric four-dimensional deterministic system.

The paper establishes convergence up to the accumulation point and proves post-accumulation convergence only after modifying the model so that intra-type and inter-type competition parameters differ. In the original symmetric model, the Lyapunov function is degenerate because its derivative has two null eigenvalues, and the authors therefore do not obtain the required O(εlog⁡K)O(\varepsilon\log K) bound.

References

However, we were unable to prove that solutions of the four dimensional dynamical system starting from initial conditions of the form $(K\hat{n}, K{1-\varepsilon},Km\hat{h}, K{(1-\varepsilon)m})$ reach a neighborhood of the coexisting equilibrium $z*$ in a time at most $O(\varepsilon\log(K))$. The main difficulty here comes from the symmetry of the system which makes the Lyapunov function degenerate in the sense that its derivative admits two null eigenvalues.

— Eco-evolutionary cycles in a matching type predator-prey interaction  (2609.04834 - Costa et al., 4 Sep 2026) in Section 4, subsection “Convergence of the exponents,” immediately before Section 5; revisited in Section 7, “Behaviour at the accumulation point,” and subsection “A slightly modified setting”