Cooperation in a system with a shared renewable resource is shaped by a feedback loop: individual actions alter the resource, which in turn changes the payoff to cooperation. We study this loop in a spatial evolutionary game with a signed feedback strength λ. For an isolated interior fixed point on the differentiable, unclipped branch of the mutation-free mean-field model, negative feedback (λ < 0) makes the fixed point locally attracting, whereas positive feedback (λ > 0) makes it a saddle. This local classification provides a reference for the spatial dynamics. At the sampled negative-feedback operating points, demographic noise produces quasi-cycles: their amplitude scales as N−1/2, the rescaled spectra collapse with system size, and resource fluctuations lag cooperation by approximately 1 rad. Under positive feedback, runs initialized in the prosperous and desert states remain strongly separated over the observation window, and recovery slows near the loss of prosperity. Resource diffusion can reverse the motion of a depleted patch from retreat to invasion, with the two regimes separated by a zero-velocity boundary. In multiplex systems, weak mass-conserving exchange can rescue a damaged layer, whereas stronger exchange can spread the deficit and promote joint collapse. The same qualitative contrast persists on the tested square lattices, small-world networks, and scale-free networks under cumulative payoffs. The results show how the sign of a strategy–resource feedback organizes fluctuations, spatial invasion, and interdependent failure.
This paper systematically investigates the evolutionary law of the existence of subharmonic solutions for a two-degree-of-freedom piecewise-smooth vibro-impact vibration system across four resonance types. Via Hamiltonian normalization, the original system is transformed into a Hamiltonian form, and fractional-order memory damping and nonlinear stiffness are integrally incorporated into the analytical construction of the Melnikov function, which extends the applicability of the existing Melnikov theory to multi-factor coupled systems. Combined with the subharmonic solution bifurcation theorem, the criteria for the existence of solutions are obtained, and a unified analytical framework covering subharmonic resonance, double subharmonic resonance, super-harmonic resonance, and double super-harmonic resonance is established. Theoretical determination of the number of subharmonic solutions, the coexistence of multiple solutions, and the deterministic evolutionary characteristics of the solutions are realized through verification of the zero distribution of the Melnikov function and the Jacobian determinant. We quantify how excitation amplitude, fractional order, nonlinear stiffness, and linear damping regulate solution coexistence by altering Melnikov zeros. The results show that the system can exhibit multi-solution coexistence characteristics, corresponding to multiple coexisting periodic orbits. Double subharmonic resonance presents a nested trajectory structure, whose time-series responses satisfy fixed time-series frequency ratios. Under super-harmonic resonance and double super-harmonic resonance, the system violates the Jacobian non-vanishing criterion, admitting no nearby subharmonic solutions. In this paper, the variation law of subharmonic solutions under the regulation of excitation amplitude is verified by the visualization of phase trajectories and time-series responses, which provides a theoretical basis for the dynamic design and vibration control of piecewise-smooth mechanical systems.
Stochastic nonlinear systems, from gene circuits to financial markets, are ubiquitous. A key challenge is inferring their governing parameters from sparse event times rather than continuous trajectories. We introduce a Markov process inversion approach (MPIA) that recovers drift and diffusion coefficients of nonlinear diffusions solely from first-passage times to a fixed boundary. Using arithmetic and geometric Brownian motions—representing constant-coefficient and state-dependent (log-linear) diffusions, respectively—the latter a paradigmatic model of multiplicative noise capturing state-dependent fluctuations—we develop a two-step procedure: fitting the inverse Gaussian distribution to hitting times, then analytically recovering generator parameters. We provide a comprehensive analysis of distribution fitting strategies, including maximum likelihood, Bayesian, and robust methods, and establish their relative merits. Asymptotic normality and consistency are proven, and the method is validated through high-precision simulations involving both direct sampling and full-path simulation. Applied to a synthetic gene circuit, MPIA disentangles effective growth rate from noise strength using only threshold-crossing times—readily accessible in high-throughput microfluidic experiments but invisible to trajectory-based methods. This event-based framework directly addresses the data constraints of modern single-cell technologies, where photobleaching and storage limitations make continuous fluorescence tracking impractical for large populations. Practical guidelines for handling censored data, drift direction verification, and model diagnostics are provided. This work lays a foundational blueprint for inversion strategies tailored to diffusions with analytically tractable first-passage-time distributions—specifically, constant-coefficient, log-linear, and the broader class of Lamperti-reducible diffusions. Extensions to non-reducible models, such as Ornstein–Uhlenbeck, Cox–Ingersoll–Ross, and logistic diffusions, via spectral and moment-based techniques, are discussed as future directions.
While the so-called next-generation neural mass models provide exact descriptions of densely coupled neural populations in the thermodynamic limit, finite-size effects introduce stochastic fluctuations whose power spectrum we derived in our previous work. However, coupling between different populations in the brain is typically sparse, meaning that only a subset of neurons from one population actually projects connections to another. This necessitates the study of shot noise generated by subpopulations within densely coupled neural populations. In the present work, we perform such an analysis and show that the shot noise of subpopulations differs from that of the whole population in both intensity and spectral properties, depending non-trivially on subpopulation size. Generalizing the previously introduced nesting method, we derive an analytical expression for the power spectrum of subpopulation shot noise, which shows excellent agreement with direct numerical simulations. Importantly, our approach accounts for realistic, non-Lorentzian distributions of local neuron parameters using a previously developed reduction technique. Thus, our results constitute an important step toward a new class of stochastic mean-field models suited for hierarchical neural networks with dense local connectivity and sparse inter-population connectivity.
