The model at the heart of the study tracks three populations over time: a prey species N, juvenile predators P1, and mature predators P2. The equations are non-linear ordinary differential equations in which the prey growth rate is modulated by a fear factor of the form 1/(1+kP2), where k quantifies the intensity of fear and P2 is the density of mature predators. This formulation traces back to a landmark 2011 experimental study showing that the mere perception of predation risk, independent of any direct killing, reduced prey reproduction by up to 40 percent in song sparrows. The model also incorporates intra-specific competition among juvenile predators, capturing sibling rivalry and competition for limited refuges, as well as a maturation process that connects the juvenile and adult predator classes.

At the default parameter values, the system settles into a stable coexistence equilibrium at approximately (16.02, 16.09, 32.19) for prey, juvenile predators, and mature predators respectively. The stability analysis relies on the eigenvalues of the Jacobian matrix, all of which have strictly negative real parts at these parameters, confirming that the equilibrium is a stable focus. The complex conjugate eigenvalue pair indicates damped oscillatory convergence with an oscillation period of roughly 20.7 time units and a damping timescale of about 244 time units. The damping ratio of approximately 0.013 is remarkably weak, meaning transient oscillations persist for roughly 12 full oscillation periods before decaying to 1/e of their initial amplitude. This weak damping has practical consequences: simulations integrated over insufficiently long horizons can misleadingly suggest persistent oscillations, which is why the authors integrated all simulations to at least 800 time units to guarantee convergence.

The pivotal discovery comes when the fear intensity parameter k is varied. As k increases, the equilibrium predator density drops steeply, from 74.8 at k = 0.001 down to 18.3 at k = 0.2. The real part of the complex conjugate eigenvalue pair crosses zero at a critical fear intensity of approximately k = 0.067, marking a supercritical Hopf bifurcation. Below this threshold, the system converges to the equilibrium; above it, the equilibrium becomes unstable and a stable limit cycle emerges, with oscillation amplitudes growing monotonically from 14.85 at k = 0.1 to 29.68 at k = 0.5. The biological interpretation is counterintuitive: higher fear reduces the prey growth rate, which in turn reduces the equilibrium predator population, weakening the negative feedback loop that ordinarily stabilizes the system. Eventually this weakened feedback triggers oscillatory instability. The finding extends theoretical results on fear-mediated dynamics to the stage-structured setting and carries implications for wildlife management, since conservation strategies that reduce perceived predation risk may paradoxically improve ecosystem stability.

Five machine learning methods were then deployed and benchmarked against this system, each addressing a fundamentally different task. The first, Sparse Identification of Non-linear Dynamics, or SINDy, discovers governing equations from time-series data by expressing the derivative vector as a sparse linear combination of candidate basis functions, selected through sequential thresholded least squares. Critically, the authors employed a generic polynomial library of degree two or less, containing ten candidate functions with no prior knowledge of the model structure, deliberately avoiding model-specific basis functions that would render the identification tautological. The sparsification threshold was set to 0.003 and derivatives were estimated from 4000 time points using central differencing.

The SINDy results constitute one of the most striking findings of the study. The predator equations were recovered exactly, with R² equal to 1.0 and coefficient errors below 0.01 percent, because the predator dynamics are polynomial in the state variables and all relevant monomials are present in the generic library. The prey equation, in contrast, was only approximated with R² of 0.9997, because the true growth term involves the rational function 1/(1+kP2), which cannot be exactly represented by a polynomial basis. This clean test of identifiability demonstrates that the polynomial structure of the predator equations is a discoverable feature of the data even without prior knowledge of the model, while simultaneously highlighting the fundamental limitation of polynomial SINDy when true dynamics involve rational non-linearities.

The second method, Dynamic Mode Decomposition, served as a deliberate negative control. DMD approximates the dynamics with a best-fit linear operator, and its linear assumption proved catastrophically inadequate for this strongly non-linear system. The reconstruction of the mature predator trajectory yielded an R² of −2.885, meaning the DMD prediction was worse than a constant prediction at the mean value. The authors emphasize that this is not a failure of the DMD implementation but a demonstration of the fundamental limitation of linear methods for systems with strongly non-linear oscillatory transients. In applications where the underlying dynamics are unknown, the failure of DMD itself serves as a quantitative diagnostic indicator of essential non-linearity, complementing the eigenvalue analysis.

Gaussian Process Regression emerged as the most practical tool for ecologists working with sparse, noisy field data. Each species was modeled independently as a Gaussian process with a composite kernel combining periodic and radial basis function components, designed to capture the oscillatory decay. The periodicity parameter matched the observed oscillation period of approximately 20 time units, and hyperparameters were optimized by maximizing the log marginal likelihood with five random restarts. Training data consisted of only 80 observations drawn at random time points, corrupted by additive Gaussian noise with standard deviations of 3.5, 1.8, and 1.5 for the three species. The resulting RMSE values of 1.92, 1.03, and 0.84 were consistently below the noise levels, demonstrating effective denoising, and the 95 percent confidence intervals were well calibrated, with the true trajectory falling within them at nearly all time points.

The remaining methods rounded out the framework. A physics-constrained function approximation, using smoothing splines with a physics-guided smoothing parameter calibrated to the noise variance, achieved RMSE values of 2.79, 1.78, and 0.86 with R² of 0.976, 0.954, and 0.989, along with physics residual diagnostics concentrated at early times where trajectories change most rapidly. A polynomial surrogate ODE model, trained with Ridge regression on degree-three monomials, achieved near-perfect derivative fitting and accurate trajectory reconstruction when forward-integrated. All numerical results were independently verified against high-precision Runge–Kutta solutions with relative tolerance of 10 to the −10 and absolute tolerance of 10 to the −12, and all performance metrics were computed directly from actual algorithmic outputs rather than hardcoded values.

The study also examined the effect of maturation delay on system stability through delay differential equation analysis, integrated with a fourth-order Runge–Kutta method at step size 0.01. The delay destabilizes the system in a manner qualitatively similar to the fear effect but through a mechanistically distinct pathway of information lag rather than reduced feedback. At zero delay the system follows the ordinary dynamics and converges to equilibrium; as delay increases, the damping rate decreases and oscillation amplitude grows. By a delay of 10 time units, oscillations persist indefinitely, and at 20 time units large-amplitude sustained oscillations dominate. Poincaré section analysis in the limit-cycle regime confirmed a stable period-1 cycle with no evidence of period-doubling or chaos in the explored parameter range. The authors acknowledge limitations, including the use of smoothing splines rather than full physics-informed neural networks, a simplified discrete-delay model, and reliance on simulated data rather than field observations. Future work will target genuine physics-informed neural networks, recurrent architectures, and neural ordinary differential equations, along with rational basis functions for improved prey equation recovery, extending this reproducible benchmark for data-driven ecology.

Subject of Research: A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis

Article Title: A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis

Article References: Ramraj, G., & Poornima, T. (2026). A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis. Frontiers in Artificial Intelligence, 9, Article 1887024. https://doi.org/10.3389/frai.2026.1887024

Image Credits: AI Generated

DOI: 10.3389/frai.2026.1887024

Keywords: machine, learning, framework, stage-structured, predator, prey, systems, fear, effect, Sparse, Identification, Gaussian