Dynamics and optimal harvesting of a fractional predator–prey system with fear and anti-predator mechanisms

A mathematical modeling study reports that stronger ecological memory, represented by a smaller fractional derivative order, can stabilize coexistence in a predator–prey system. Larger fractional orders can lead to sustained population oscillations in the analyzed model. The researchers developed a Caputo fractional-order model combining fear-induced reproductive suppression, proportional prey refuge, active anti-predator retaliation, a Holling type II functional response and nonlinear saturation harvesting. Their analytical and numerical results indicate that the fractional memory order can act as a regulator of ecological stability, although several conclusions remain model-based or numerically supported rather than demonstrated in a real ecosystem.

The model represents prey and predator population densities with fractional differential equations. Fear reduces the effective prey birth rate as predator density rises, while refuge reduces the fraction of prey exposed to predation. Anti-predator behavior adds a cost to predators through a prey–predator encounter term. Harvesting is represented by a saturating Michaelis–Menten function, and the Caputo derivative incorporates the history of past population dynamics. The classical integer-order system is recovered as a special case.

The study first established mathematical properties of the model, including non-negativity of solutions from non-negative initial conditions, uniform boundedness, and existence and uniqueness of solutions. The equilibrium analysis considered an extinction state, a prey-only state and a coexistence state. The local stability analysis used Matignon’s criterion for fractional systems. It also derived sufficient Lyapunov conditions for global asymptotic stability. These conditions are conservative and do not hold for the baseline parameter set used in some simulations, so global convergence in that setting was assessed numerically rather than guaranteed by the global-stability theorem.

At the coexistence equilibrium, the analysis identified a critical fractional order at which the equilibrium can lose stability through a Hopf bifurcation. Below the threshold, the coexistence state is locally asymptotically stable; above it, the model predicts sustained oscillations. The critical value is determined from the eigenvalues of the Jacobian matrix and therefore depends on the ecological parameters and the equilibrium state. The numerically located threshold agreed with the analytical formula to within 0.5% in the reported simulations.

Numerical evaluation of the first Lyapunov coefficient provided evidence that the Hopf bifurcation is supercritical, meaning the model predicts an orbitally stable limit cycle emerging as the fractional order passes above the threshold. The authors note that this calculation used an integer-order center-manifold formula as an approximation for the fractional system, so a fully rigorous proof of supercriticality was not provided. Phase portraits, bifurcation diagrams and basin classifications were consistent with the predicted transition from a stable equilibrium to a stable periodic orbit.

Parameter explorations produced distinct model predictions for the included mechanisms. In the numerical simulations, increasing fear reduced oscillation amplitude and shifted the stability boundary. Increasing refuge produced a similar pattern in the model. Stronger anti-predator retaliation lowered predator abundance by imposing a direct cost and indirectly allowing prey recovery. A derived parameter boundary marked combinations of refuge and retaliation at which the coexistence equilibrium ceases to exist and predator persistence is no longer possible. Sensitivity analyses indicated that the critical fractional order was most strongly influenced by parameters governing the strength and saturation of predator–prey interactions and was comparatively robust to moderate variation in the behavioral and harvesting parameters.

For resource management, the authors formulated a finite-horizon optimal-control problem to maximize discounted net revenue from harvesting, with quadratic costs of harvesting effort. The harvesting-control derivation used Pontryagin’s Maximum Principle. It described maximum-effort, interior-control and zero-effort regimes. The authors presented the resulting pattern as bang–singular–bang harvesting. Numerical solutions showed this time-varying policy producing approximately 20% to 21.5% greater discounted net revenue than a constant-effort benchmark across the reported stable fractional-order range. The authors also reported higher time-averaged prey and predator densities under the optimal policy than under the comparison scenarios.

The singular portion of the control was supported by numerical evidence that the switching function remained close to zero over an extended interval. A rigorous proof that this interval is non-degenerate was not supplied; the authors state that verifying the required generalized Legendre–Clebsch conditions for the fractional optimal-control problem remains open. In addition, the economic results use illustrative rather than field-calibrated values for price and harvesting costs, so the reported revenue advantage is a property of the modeled scenarios rather than evidence of a financial outcome in a particular fishery or wildlife system.

Overall, the study presents a theoretical framework in which fractional memory, fear, refuge, anti-predator behavior and saturating harvesting can be analyzed together. Its principal model prediction is that stronger memory can stabilize coexistence and narrow the conditions under which persistent cycles arise, while a time-dependent harvest policy can increase modeled economic returns. The authors acknowledged fixed memory order and approximate evidence for supercriticality as limitations. They also noted conservative global-stability conditions and an unverified singular control arc. Proposed extensions include spatial processes, time delays and variable-order memory. Other proposed directions include environmental forcing and field-data parameter estimation.

Subject of Research: Mathematical analysis and numerical simulation of a fractional predator–prey model with fear, refuge, anti-predator behavior and optimal harvesting

Article Title: Dynamics and optimal harvesting of a fractional predator–prey system with fear and anti-predator mechanisms

Article References: Govindan, M. K., & Singh, A. K. (2026). Dynamics and optimal harvesting of a fractional predator–prey system with fear and anti-predator mechanisms. PLOS One, 21(10), e0360316. https://doi.org/10.1371/journal.pone.0360316

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DOI: 10.1371/journal.pone.0360316

Keywords: fractional predator–prey model, Caputo derivatives, ecological memory, fear effects, prey refuge, anti-predator behavior, Hopf bifurcation, optimal harvesting, population dynamics, mathematical ecology, Dynamics, optimal