Lie symmetry analysis and similarity reductions for the tempered-fractional Keller Segel system

Technical Summary: Lie Symmetry Analysis and Similarity Reductions for the Tempered-Fractional Keller–Segel System

Problem Statement
The paper addresses the mathematical analysis of the Tempered-Fractional Keller–Segel (TFKS) model, which describes chemotaxis with anomalous diffusion. While the classical Keller–Segel (KS) model assumes normal diffusion and often exhibits finite-time blow-up (singularity formation), and the fractional KS (FKS) model captures long-range jumps but may still suffer from blow-up, the TFKS model introduces an exponential tempering mechanism. This mechanism cuts off extreme long jumps while preserving memory effects. The specific challenge addressed is the application of Lie symmetry analysis to this system. Standard Lie symmetry methods rely on local prolongation formulas, which are incompatible with the inherent non-locality of tempered fractional derivatives (both in time and space) and the exponential tempering factors that break standard time-translation invariance.

Methodology
To overcome the non-locality and tempering barriers, the authors employ a two-step reduction methodology:

1. Gauge Transformation: A transformation $v(x,t) = e^{\lambda t}u(x,t)$ and $w(x,t) = e^{\lambda t}c(x,t)$ is applied. Utilizing the identity $D^{\alpha,\lambda}_t u = e^{-\lambda t} {}^C D^\alpha_t (e^{\lambda t} u)$, the tempered Caputo time derivative is converted into a standard Caputo derivative. This restores compatibility with existing fractional Lie symmetry frameworks regarding the temporal operator.
2. Short-Jump Diffusion Approximation: The non-local tempered fractional Laplacian $L^{(\lambda)}_\alpha$ is approximated by a local second-order diffusion operator $c_2(\alpha, \lambda)\Delta$. This approximation is valid under the assumption that the characteristic jump length of the underlying Lévy process is small compared to the macroscopic spatial scale. This step converts the original integro-differential system into a local fractional partial differential equation (PDE), allowing the direct application of classical Lie point symmetry techniques.

Key Contributions and Results
The paper performs a complete Lie symmetry analysis on the reduced local TFKS system in one spatial dimension ($0 < \alpha < 1$).

• Symmetry Classification:

  • Generic Case ($\lambda > 0$, $\chi, r, \kappa > 0$): The explicit time-dependence introduced by the tempering factor $e^{-\lambda t}$ and the non-linear coupling terms restrict the symmetry group significantly. The only admitted continuous symmetry is spatial translation ($X_1 = \partial_x$).
  • Conditional Enlargements: Additional symmetries emerge only under specific parameter constraints:
    • If $\lambda = 0$ (no tempering), the system admits time translation ($\partial_t$) and spatial translation, forming an abelian algebra.
    • If chemotaxis ($\chi$) and logistic growth ($r$) are absent, a modified scaling-translation symmetry ($\partial_t - \frac{\lambda}{2}x\partial_x$) appears, compensating for the tempering effects in the local approximation.
  • Optimal Systems: The authors construct optimal systems of one-dimensional subalgebras for the generic case, the untempered case ($\lambda=0$), and the degenerate linear case ($\chi=r=0$).

• Similarity Reductions and Exact Solutions:
  Using the optimal systems, the reduced PDEs are transformed into Ordinary Differential Equations (ODEs) or fractional ODEs.

  • Spatially Homogeneous Solutions: For the generic case, the system reduces to a fractional ODE system. Using the Fractional Complex Transform (FCT), exact analytical solutions are derived in terms of quadratures.
    • Untempered ($\lambda=0$): The solution converges to the carrying capacity $K_0$ as $t \to \infty$.
    • Tempered ($\lambda>0$): Under the condition $0 < v_0 < K_0$, the solution exhibits a transient rise followed by a monotonic decay to zero (extinction). The tempering factor $e^{-\lambda t}$ acts as an effective time-dependent loss term that eventually overwhelms logistic growth.
  • Linear Degenerate Case ($\chi=r=0$): Exact solutions are found for both spatially uniform and scaling-invariant scenarios. The scaling solution yields a fractional tempered heat kernel (Gaussian in space with fractional operational time $s(t) = t^\alpha/\Gamma(1+\alpha)$ and an overall exponential decay factor).
  • Steady States: The analysis identifies constant equilibria $(0,0)$ and $(K_0, K_0/\kappa)$, as well as Green's function representations for the chemoattractant concentration $c$ given any cell density $u$.

• Biological Dynamics:
  The derived solutions highlight the critical role of the tempering parameter $\lambda$ and the logistic growth term. The logistic growth term regulates population dynamics by limiting cell proliferation, reducing the tendency toward unbounded aggregation, and under suitable conditions, preventing finite-time blow-up. In contrast, the tempering mechanism shifts the asymptotic behavior: in the absence of tempering, the model predicts survival at the carrying capacity; with tempering, the model predicts eventual extinction for initial densities satisfying $0 < v_0 < K_0$, due to the cumulative effect of the exponential decay. The fractional order $\alpha$ influences the speed of diffusion and the shape of the aggregation, with lower $\alpha$ values corresponding to subdiffusive, memory-dominated motion.

Significance and Claims
The paper claims to be the first study to apply Lie symmetry groups to the TFKS system. The primary significance lies in providing a rigorous mathematical framework to handle tempered fractional operators by converting them into a form amenable to classical symmetry analysis. The results offer analytical insights into the long-term behavior of chemotactic systems with tempered anomalous diffusion, specifically demonstrating how tempering mechanisms drive populations toward extinction rather than aggregation at a carrying capacity (under the condition $0 < v_0 < K_0$), while the logistic term serves to prevent finite-time blow-up. The methodology is presented as applicable to other tempered fractional differential equations, offering a bridge between classical and fractional modeling approaches. The authors note that while the short-jump approximation enables these analytical results, it may obscure natural scaling relationships between variables and the tempering parameter, and the analysis is currently restricted to the subdiffusive regime ($0 < \alpha < 1$).