Memory-Delay Stability Switching and Ecological Thresholds in a Dimensionally Balanced Fractional-Order Predator-Prey Model

Technical Summary: Memory-Delay Stability Switching and Ecological Thresholds in a Dimensionally Balanced Fractional-Order Predator-Prey Model

1. Problem Statement and Motivation

The paper addresses a critical methodological flaw in many existing fractional-order ecological models: dimensional inconsistency. In classical integer-order models, the rate of change $dx/dt$ has dimensions of $[Mass][Time]^{-1}$. However, the Caputo fractional derivative of order $0 < \alpha < 1$, denoted $D_t^\alpha$, inherently possesses dimensions of $[Mass][Time]^{-\alpha}$. Directly replacing the integer-order derivative with a fractional one without adjusting the system parameters results in a dimensional mismatch, rendering the model physically meaningless.

Furthermore, the authors argue that ecological systems are governed by multiple interacting mechanisms often studied in isolation:

• Memory effects: Biological populations respond to past history, not just current states.
• Time delays: Predator growth often lags behind prey consumption due to gestation or digestion.
• Behavioral factors: Fear of predation and prey refuge significantly alter population dynamics.
• Management: Nonlinear harvesting (e.g., fishing) is a realistic control mechanism.

The study aims to formulate a dimensionally balanced fractional-order predator-prey model that integrates these factors (fear, refuge, Holling type-III functional response, nonlinear harvesting, and time delay) while ensuring physical consistency.

2. Methodology

Model Formulation:
The authors construct a dimensionally balanced system by reconstructing time-dependent rate parameters. Instead of simply replacing $r$ with $r^\alpha$, they define fractional rate coefficients (e.g., $r_\alpha = r^\alpha$, $d_\alpha = d^\alpha$, $q_\alpha = q^\alpha$) to ensure every term in the differential equations has the dimension $[Mass][Time]^{-\alpha}$. Special care is taken with mixed-dimension parameters (like the self-limitation coefficient $\sigma$), redefining them as $\sigma_\alpha = \sigma d^{\alpha-1}$ to maintain homogeneity.

The resulting system (Equation 8) is:
$$\begin{aligned} D_t^\alpha X(t) &= \frac{r_\alpha X(t)}{1 + \kappa Y(t)} \left(1 - \frac{X(t)}{K}\right) - H_\alpha(X(t))Y(t) - \frac{q_\alpha X^2(t)}{c + X(t)} \\ D_t^\alpha Y(t) &= e H_\alpha(X(t-\tau))Y(t) - d_\alpha Y(t) - \sigma_\alpha Y^2(t) \end{aligned}$$
where $H_\alpha$ is the dimensionally balanced Holling type-III functional response, $\tau$ is the time delay, and $\alpha$ is the fractional order.

Analytical Framework:

• Well-posedness: The authors prove local existence, uniqueness, positivity, and ultimate boundedness of solutions using the Volterra integral formulation and fractional comparison principles.
• Equilibrium Analysis: They derive the prey-only equilibrium and the coexistence equilibrium. A predator invasion number ($R_0$) is defined to determine the threshold for predator persistence versus extinction.
• Stability Analysis:
  • Non-delayed ($\tau=0$): Stability is analyzed using the Matignon criterion, which requires eigenvalues $\mu$ of the Jacobian to satisfy $|\arg(\mu)| > \alpha\pi/2$.
  • Delayed ($\tau>0$): The characteristic equation is derived. The authors solve for critical frequencies and critical delay values ($\tau_j$) where roots cross the imaginary axis. Transversality conditions are established to determine the direction of stability switching.

Numerical Simulation:
Simulations are performed using a predictor-corrector method for Caputo fractional delay differential equations. The authors explicitly reconstruct dimensionally balanced coefficients for each simulation rather than using naive parameter substitution.

3. Key Contributions

1. Dimensional Consistency: The paper provides a rigorous framework for constructing fractional ecological models that are physically dimensionally consistent, correcting the common practice of direct derivative replacement.
2. Comprehensive Stability Classification: The study classifies and numerically verifies six distinct delay-induced stability scenarios for the coexistence equilibrium:
   • Remain-stable
   • Stable-to-unstable
   • Stable-unstable-stable (Stability bubble)
   • Remain-unstable
   • Unstable-to-stable
   • Unstable-stable-unstable (Instability bubble)
3. Memory-Delay Interaction: The analysis demonstrates how the fractional order $\alpha$ (memory) interacts with time delay $\tau$. The critical delay $\tau_0(\alpha)$ is shown to be a function of $\alpha$, indicating that memory effects can either stabilize or destabilize the system depending on the delay magnitude.
4. Ecological Management Thresholds: The paper links mathematical stability to ecological management by analyzing the impact of nonlinear harvesting ($q$) on the predator invasion number ($R_0$).

4. Results

• Existence and Persistence: The model ensures positive and bounded solutions. A threshold condition $R_0(q, \alpha) > 1$ is established for predator invasion. The authors prove that increasing harvesting intensity $q$ strictly decreases $R_0$, potentially driving the predator to extinction.
• Stability Switching:
  • For the non-delayed system, stability depends on the trace and determinant of the Jacobian and the fractional order $\alpha$.
  • For the delayed system, the introduction of $\tau$ can induce Hopf bifurcations. The study confirms that stability is not monotonic with respect to delay; the system can transition from stable to unstable and back to stable (or vice versa) as $\tau$ increases.
• Memory-Induced Stabilization: Numerical results show that for a fixed delay that causes oscillations in an integer-order model, reducing the fractional order $\alpha$ (increasing memory) can suppress oscillations and stabilize the coexistence equilibrium.
• Sensitivity: Sensitivity analysis indicates that the system's dynamics are highly responsive to variations in harvesting rates and initial population densities.

5. Significance and Claims

The paper claims that its primary significance lies in bridging the gap between physical consistency and ecological realism in fractional modeling. By ensuring dimensional balance, the model avoids the mathematical artifacts associated with naive fractionalization.

The authors emphasize that their work provides a unified framework to study how memory, delay, fear, refuge, and harvesting jointly influence population persistence. Specifically, they claim to have:

• Demonstrated that fractional memory can act as a stabilizing mechanism against delay-induced oscillations.
• Provided a complete classification of stability switching scenarios, showing that delay can have a dual role (destabilizing stable states and stabilizing unstable ones).
• Offered management insights, showing that harvesting intensity is a critical parameter that can be tuned to control predator invasion and coexistence.

The study concludes that future work could extend this framework to include spatial diffusion, stochastic perturbations, or optimal harvesting strategies, but the current contribution is focused on the rigorous formulation and stability analysis of the dimensionally balanced delayed system.