Desynchronization of strongly nonlinear oscillations by coupling strengthening

TL;DR

The paper analyzes how strong nonlinear oscillations in ecological networks desynchronize when coupling strength increases. By linking spatial modulational instabilities in reaction-diffusion systems to network desynchronization through a spectral mapping and by developing a master-stability function, it offers a unified framework to predict synchronization regimes. Using a three-species Lotka–Volterra model, it exposes finite-interval instability bands in both RD and network settings and derives practical criteria via Metzler spectral properties and MSF analysis. The work yields actionable insights into when synchrony can be maintained or lost in complex ecological networks and provides tools applicable to broader multi-species dynamical systems.

Abstract

We investigate cyclic dominance models and their extensions to both network systems and reaction-diffusion frameworks. Using linear stability analysis, we establish the relationship between the stability of synchronized states in network systems and the response of homogeneous solutions subjected to spatially periodic perturbations. Furthermore, we explore the mathematical properties of networks characterized by strong nonlinear oscillations in an ecological context. Finally, we present numerical results for the master stability function of a competitive three-species Lotka-Volterra model, highlighting its role in understanding the dynamics of cyclic competition.

Figures (9)

• Figure 1: The parameter space $(\alpha,\gamma)$. In region (A) the coexistence point is the attractor, in region (B) the attractor is a limit cycle, and in region (C) the attractor is a heteroclinic cycle.
• Figure 2: Period doubling at a finite wavenumber for $d_u=1,d_v=d_w=0$. In (A),(B),(C) the Real, Imaginary, and Modulus of the two leading eigenvalues are shown respectively at $\alpha=2.3427$.
• Figure 3: Period doubling at finite wavenumber for $d_u=1,d_v=d_w=0$. (A) The disturbance with $k=k_*$ at $\alpha=2.3427$. Its period is twice the period of the base solution (B).
• Figure 4: Period doubling bifurcation in the network with the two coupled subsystems (A) and its corresponding Floquet multiplier (B). Similarly to Fig.\ref{['PD1']}, $\alpha=2.3427, \gamma =0.5,D=k_*^2/2=0.1922$.
• Figure 5: Synchronization between the two subsystems. $u_1(t),u_2(t)$ as a function of time (A) and the differences $u_1-u_2,v_1-v_2,,w_1-w_2,$ as a function of time (B). $\alpha=2.4327,\gamma=0.5,D=0.15,(u_1(0),v_1(0),w_1(0))=(0.1,0.15,0.05)$$,(u_2(0),v_2(0),w_2(0))=(0.3,0.15,0.05)$.