🧬 biology

Physics-Informed Neural Networks for Incommensurate Fractional Volterra-Lotka Systems: Forward Solutions, Memory Asymmetry, and Inverse Parameter Recovery

This paper introduces a physics-informed neural network framework with species-specific L1-scheme stencils to solve incommensurate fractional Lotka-Volterra systems, demonstrating that memory asymmetry can induce ecological role transitions and enabling the accurate joint recovery of all system parameters from noisy data.

Original authors: Humaira Zulfiqar

Published 2026-09-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Humaira Zulfiqar

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

For decades, ecologists have relied on a classic mathematical model to understand how two species interact in the wild. This model, known as the Lotka-Volterra system, tracks how populations grow, shrink, and influence one another through competition, predation, or mutual benefit. The equations are simple enough to be written on a napkin, yet they capture the essential drama of nature: the rise and fall of predators and prey, or the struggle for resources between rivals. However, these traditional equations assume that time flows the same way for every creature involved. They treat the past as a single, uniform memory that affects all species equally. In reality, a long-lived elephant and a short-lived insect do not experience time or history in the same way. An elephant's population might be shaped by memories of droughts from decades ago, while an insect's numbers might react only to the conditions of the last few weeks. This difference in how organisms "remember" their past is a crucial piece of the ecological puzzle that standard models have long ignored.

A new study by researcher Humaira Zulfiqar tackles this oversight by building a more flexible version of the classic model. Instead of forcing both species to share the same memory of the past, the study allows each species to have its own unique rate of remembering. The researcher used a modern computational tool called a physics-informed neural network, which is essentially a computer program trained to solve complex equations by learning the laws of physics rather than just crunching numbers. By teaching this program to respect the specific, different ways each species remembers its history, the study uncovered behaviors that were previously invisible. The findings suggest that when two species have very different memories, their relationship can change in surprising ways. A pair of animals that should be helping each other might, under the right conditions of memory difference, end up acting as if one is harming the other. This discovery challenges the idea that we can always predict ecological outcomes based solely on the type of interaction, showing that the speed at which a species forgets or remembers the past is just as important as the interaction itself.

The core of this work lies in addressing a mathematical limitation that had stumped researchers for some time. Previous attempts to add "memory" to these population models worked well only when both species were assumed to have identical memory rates. When the rates differed, the standard mathematical tools failed because the equations no longer followed a neat, predictable pattern. Zulfiqar turned to a different approach, using a neural network that could handle these messy, unequal conditions. The network was designed with two separate "memory" settings, one for each species, allowing the computer to simulate how populations evolve when one species looks back further into the past than the other. The researchers first tested this new method against known solutions for the simpler, equal-memory case to ensure it was accurate. The computer's predictions matched the established mathematical benchmarks with an average error of less than three-thousandths, proving the method was reliable.

Once the method was verified, the study explored what happens when the two species have different memory rates. The simulations covered all six classic types of ecological relationships, from pure competition to mutualism. In every case, the unequal memory rates changed the details of the population curves. For instance, in a competition scenario, the species with the longer memory (the one that remembers past conditions for a longer time) tended to decline more slowly when faced with pressure. It was as if that species was more stubborn, holding onto the advantages of a better past for longer. In predator-prey scenarios, the size of the population swings depended on the specific memory rate of each animal, rather than a single shared factor. These results showed that the "memory" of a species is not just a background detail; it actively shapes the rhythm and intensity of their survival.

The most striking discovery emerged when the researchers looked at a scenario where two species were theoretically helping each other. In the standard model, if both species benefit from the other's presence, they will always grow together. However, in this new model, when one species had a very short memory and the other had a very long memory, the outcome shifted dramatically. The species with the short memory grew rapidly, while the species with the long memory was suppressed and declined. To an observer looking only at the final numbers, it would appear that one species was harming the other, even though the underlying rules said they were helping each other. The researchers call this a "memory-induced role transition." It suggests that in the real world, what looks like a harmful relationship might actually be a helpful one that is distorted by the different ways the two species experience time. This phenomenon does not happen when both species have the same memory rate, highlighting that the difference in their internal clocks is the key driver.

Beyond just simulating forward in time, the study also tested whether this new model could work backward to solve mysteries. The researchers created a set of fake population data with noise, simulating what a field scientist might collect in the wild. They then asked the computer to figure out the hidden rules that created that data, including the growth rates, the strength of the interactions, and the specific memory rates for both species. The neural network successfully recovered all eight hidden values from just thirty data points, with an error margin of less than 3.5 percent. This is a significant achievement because it shows that even with limited and imperfect data, it is possible to uncover the specific memory characteristics of different species. It opens the door for future studies to analyze real-world population data and determine if species are operating on different time scales, potentially explaining why some ecological interactions look different from what simple models predict.

The study concludes that the assumption of a single, shared memory for all interacting species is a simplification that misses important biological realities. By allowing for different memory rates, the model reveals a richer, more complex world of ecological dynamics. The findings do not overturn the classic models but rather expand them, showing that the history of a population matters differently for every creature. The research suggests that when ecologists see strange or unexpected patterns in nature, such as a mutualistic relationship that looks like a struggle, they should consider whether the two species simply have different ways of remembering their past. The work provides a new mathematical tool to explore these questions, moving beyond the idea that time is the same for everyone, and acknowledging that in the wild, the past is a different story for every species.

Technical Summary: Physics-Informed Neural Networks for Incommensurate Fractional Volterra-Lotka Systems

Problem Statement
The classical Lotka-Volterra model describes population dynamics through ordinary differential equations, while fractional extensions incorporate memory effects via Caputo derivatives. Existing literature on fractional Lotka-Volterra systems, such as the work by Yu and Feng [15], relies on the commensurate assumption, where both species share a single fractional order ( $\alpha_1 = \alpha_2 = \alpha$ ). This assumption is ecologically unjustified, as species with different life histories and habitat dependencies likely encode population memory at different rates.

The incommensurate system, where species $u$ and $v$ are governed by distinct Caputo orders ( $\alpha_1 \neq \alpha_2$ ), presents a significant mathematical challenge. The uniform power-of- $t^\alpha$ structure required for the power-series recursion method used in previous studies [15] collapses when orders differ. Consequently, there is a need for a computational approach capable of solving these coupled systems globally (across the entire time interval) and addressing inverse problems (parameter recovery) without relying on mesh-based discretization or series expansions valid only near $t=0$ .

Methodology
The paper proposes a Physics-Informed Neural Network (PINN) framework tailored for incommensurate fractional differential equations.

Architecture: A single fully-connected neural network $\phi_\theta: \mathbb{R}^+ \to \mathbb{R}^2$ maps time $t$ to population estimates $(\hat{u}_\theta(t), \hat{v}_\theta(t))$ . The network utilizes four hidden layers of width 20 with $tanh$ activation functions to ensure smoothness, which is critical for the finite-difference approximations required by the fractional derivative stencils.
Initial Conditions: To enforce initial conditions $u(0)=u_0$ and $v(0)=v_0$ exactly, the network outputs are structured as $\hat{u}_\theta(t) = u_0 + t^{\alpha_1} N_u(t)$ and $\hat{v}_\theta(t) = v_0 + t^{\alpha_2} N_v(t)$ , where $N_u$ and $N_v$ are scalar sub-networks.
Loss Function: The training minimizes a composite loss function $L(\theta) = L_{res} + \lambda_1 L_{IC} + \lambda_2 L_{data}$ $L (θ) = L_{r es} + λ_{1} L_{I C} + λ_{2} L_{d a t a}$ .
- Residual Loss ( $L_{res}$ ): This is the core innovation. It embeds the fractional ODE residuals directly into the loss. Crucially, the Caputo derivatives are approximated using species-specific L1-scheme stencils. The stencil for species $u$ uses order $\alpha_1$ , and the stencil for species $v$ uses order $\alpha_2$ . These stencils are applied independently to the network outputs, decoupling the fractional orders.
- Inverse Problem Formulation: For parameter recovery, the physical parameters (growth rates $A_i$ , limitation parameters $B_i$ , coupling coefficients $C_i$ , and fractional orders $\alpha_i$ ) are treated as trainable scalars alongside network weights. Positivity constraints on $A_i$ and $B_i$ are enforced via softplus reparameterization, while $\alpha_i$ are constrained to $(0, 1)$ via a sigmoid transformation.
Optimization: The system is trained using the Adam optimizer over 5,000 epochs with a decaying learning rate.

Key Contributions

Formulation of Species-Specific PINNs: The paper introduces a PINN architecture that handles incommensurate fractional orders by embedding independent L1-scheme stencils for each species within a single network loss function.
Validation against Benchmarks: The method is verified in the commensurate limit ( $\alpha_1 = \alpha_2$ ), demonstrating that it reproduces the power-series benchmark of Yu and Feng [15] with a mean absolute error of $2.3 \times 10^{-3}$ .
Discovery of Memory-Induced Role Transition: The study identifies a novel dynamical phenomenon where memory asymmetry ( $\alpha_1 \neq \alpha_2$ ) alters the qualitative nature of ecological interactions. Specifically, a parameter set nominally representing mutualism ( $C_1, C_2 > 0$ ) can produce an amensalistic outcome (one species harmed, the other unaffected) if the species have substantially different fractional orders.
Joint Inverse Parameter Recovery: The framework successfully recovers all eight unknown system parameters (including both fractional orders) from just thirty noisy population observations, achieving relative errors below 3.5%.
Theoretical Consistency: Proposition 6.1 establishes the theoretical link between the PINN solution and the power-series solution in the commensurate limit, proving that the PINN error is bounded by the residual tolerance, the L1 discretization error, and the power-series truncation error.

Results

Forward Solutions: Numerical experiments were conducted for all six classical ecological interaction types (pure competition, predator-prey, mutualism, commensalism, amensalism, neutralism). The results show that while the qualitative character of interactions is preserved, the quantitative behavior (e.g., oscillation amplitude, decay rates) depends independently on each species' fractional order.
Memory Asymmetry: Sensitivity analysis in predator-prey systems confirms that each species responds primarily to its own fractional order. The phase portraits for incommensurate orders exhibit geometrically distinct orbits that cannot be obtained by simply rescaling commensurate solutions.
Inverse Problem: In a mutualism scenario with added 2% Gaussian noise, the PINN recovered the true parameters $A_1, A_2, B_1, B_2, C_1, C_2, \alpha_1, \alpha_2$ with relative errors ranging from 0.38% to 2.00%. The fractional orders were identified with high precision despite being less directly observable than algebraic parameters.
Role Transition: The "memory-induced role transition" was quantified. When $\alpha_2$ is fixed at a high value (short memory, 0.95) and $\alpha_1$ is reduced (long memory), the population of species $u$ is progressively suppressed, leading to an apparent amensalistic dynamic despite positive coupling coefficients.

Significance and Claims
The paper claims to be the first work to:

Treat incommensurate orders in the fractional Lotka-Volterra system.
Solve the corresponding inverse problem (joint identification of all parameters and orders) for such systems.
Document the memory-induced role transition, suggesting that field observations of apparent amensalism in nominally symmetric systems may actually stem from differences in ecological memory timescales rather than interaction coefficients.

The author positions the PINN not as a replacement for analytical methods like power series (which remain superior near $t=0$ ), but as a complementary tool that provides global validity across the entire time interval and handles the incommensurate case where series methods fail. The study concludes that the incommensurate structure offers a more ecologically realistic modeling framework, though rigorous convergence proofs for the coupled training algorithm and applications to real-world data (e.g., lynx-hare cycles) remain open directions for future research.

Technical Summary: Physics-Informed Neural Networks for Incommensurate Fractional Volterra-Lotka Systems

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