Asymptotic stability and diffusion-driven pattern formation in a predator-prey system with two chemicals

Abstract: This work analyzes a predator-prey cross-diffusion system coupled with two chemical substances under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R^n (n >= 2) with smooth boundary dOmega. Under appropriate conditions on the model parameters, the global existence of classical solutions is established. Furthermore, by constructing a suitable Lyapunov functional, the asymptotic stability of the spatially homogeneous steady state is proved. The emergence of spatial patterns induced by diffusion-driven instability is also investigated. Owing to the complexity of the resulting four-equation system, the criteria for Turing bifurcation are derived numerically rather than analytically. Numerical simulations are performed to generate Turing bifurcation diagrams, illustrating the dynamical responses of the system to variations in the predation rate. These results provide new insights into the role of predation intensity in the formation of spatial patterns in predator-prey systems mediated by two chemical substances.

• The paper demonstrates global well-posedness and boundedness, establishing conditions for coexistence or prey extinction using precise cross-diffusion constraints.
• It derives sharp criteria for diffusion-driven instability, showing that chemical cross-diffusion parameters critically shape Turing pattern morphology.
• Numerical simulations confirm theoretical predictions, illustrating transitions from labyrinthine stripes to hexagonal spots as predation response varies.

Asymptotic Stability and Pattern Formation in a Chemically Mediated Predator-Prey Cross-Diffusion System

Model Framework and Biological Motivation

This work addresses the mathematical analysis of a reaction-cross-diffusion system modeling predator–prey interactions modulated by two chemical substances. Unlike standard Lotka–Volterra or classical Turing-unstable reaction–diffusion models, this system incorporates cross-diffusive motilities that are chemically mediated: prey secrete attractants ($v_1$) that direct predator movement, whereas predators emit repellents ($v_2$) influencing prey avoidance fluxes. The resulting spatiotemporal dynamics capture both direct species interactions and the indirect regulation arising from chemical signaling. The coupled PDE system is posed in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \geq 2$, under homogeneous Neumann conditions, yielding a four-component model:

$$\begin{aligned} u_{1,t} &= \nabla \cdot (d_{11} \nabla u_1 + d_{12} \nabla v_1) + u_1(-\lambda_1 u_1 + \eta_1 u_2), \ u_{2,t} &= \nabla \cdot (d_{21} \nabla u_2 - d_{22} \nabla v_2) + u_2(-\lambda_2 u_2 - \eta_2 u_1), \ v_{1,t} &= d_3 \Delta v_1 + a_1 u_2 - b_1 v_1, \ v_{2,t} &= d_4 \Delta v_2 + a_2 u_1 - b_2 v_2. \ \end{aligned}$$

Here, $u_1$ (predator) and $u_2$ (prey) densities respond both to self-diffusion and to cross-diffusive taxis driven by the chemicals $v_1$ (prey attractant/chemoattractant) and $v_2$ (predator repellent/chemorepellent), with nontrivial cross-diffusion coefficients $d_{12}$ and $v_2$0. The model is motivated by numerous ecological and microbiological contexts (e.g., bacteria–phage and algae–zooplankton interactions) where motility, aggregation, and segregation are strongly regulated by autoinducers, kairomones, and other secondary metabolites.

Theoretical Results: Existence, Boundedness, Asymptotic Stability

Three main results are established.

(1) Global Well-posedness and Uniform Boundedness:

Using maximal regularity for non-degenerate parabolic systems subject to structural parabolicity constraints ($v_2$1, $v_2$2), classical solutions exist for all time and remain bounded, precluding finite-time blow-up in any dimension $v_2$3, given nonnegative initial data.

(2) Global Asymptotic Stability—Coexistence or Prey-Extinction:

The long-term fate is determined by the relative magnitudes of the predation response coefficient $v_2$4 and effective intraspecific self-limitation $v_2$5 in combination with chemical production/decay and cross-diffusion rates.

• If $v_2$6 and cross-diffusion is sufficiently weak, both species persist: the unique positive coexistence equilibrium is globally asymptotically stable. This is established rigorously using a Lyapunov functional of entropy-type and quadratic deviations, leading to exponential or algebraic convergence.
• If $v_2$7, only the predator survives asymptotically: the prey vanishes, and the system converges to the semi-trivial steady state with extinction in $v_2$8.

These regimes are sharp: intermediate cases can yield both exponential and (exceptionally) algebraic convergence, with explicit rates dictated by Lyapunov differential inequalities.

Diffusion-Driven Instability: Criteria and Mechanisms

The work analyzes the linearized stability of the positive equilibrium in both the ODE system (spatially homogeneous) and the reaction–diffusion PDE system with and without cross-diffusion. In the absence of (cross-)diffusion, the ODE equilibrium is always locally stable under the structural conditions derived above. Pure self-diffusion ($v_2$9) is not sufficient to yield Turing bifurcation: all modes remain stable.

The central contribution is the characterization of diffusion-driven instability (Turing-type bifurcation) induced by cross-diffusion. The characteristic equation yields a quartic polynomial in the perturbation decay/growth rate; explicit analytical conditions for bifurcation were intractable. Instead, numerical computation of the full spectrum (as a function of wave number $\Omega \subset \mathbb{R}^n$0 and key parameters, especially $\Omega \subset \mathbb{R}^n$1) identifies parameter regimes in which one or more spatial modes exhibit positive real part, generating spatial symmetry breaking at finite wavelength.

Key findings:

• Cross-diffusion terms $\Omega \subset \mathbb{R}^n$2 are essential for destabilizing the equilibrium and generating non-homogeneous patterns: increasing these coefficients broadens the admissible unstable $\Omega \subset \mathbb{R}^n$3-range.
• The intensity of predatory response ($\Omega \subset \mathbb{R}^n$4) strongly governs the architecture and scale of emergent patterns: as $\Omega \subset \mathbb{R}^n$5 increases, patterns transition from labyrinthine stripes to isolated spots to regular hexagonal arrays.
• Variations in $\Omega \subset \mathbb{R}^n$6 (prey influence on predator movement) similarly modulate spatial complexity, with higher values favoring spot formation.

Numerical Simulations: Turing Patterns and Parameter Sensitivity

Extensive direct simulations (finite difference integration on large domains) confirm and extend the theoretical predictions. The system exhibits robust Turing patterns in cross-diffusive parameter regimes: co-located peaks in predator and prey densities are synchronized with local chemical maxima—the system generates "in-phase" spatially periodic aggregates rather than classical segregated stripes or anti-phase spot/stripe mosaics. The geometry and coherence of the patterns are highly sensitive to changes in $\Omega \subset \mathbb{R}^n$7, cross-diffusion magnitudes, and self-diffusion rates, as systematically explored.

Numerical results support the following nontrivial claims highlighted in the work:

• Spatially in-phase predator–prey patterning (contradicting classical Turing scenarios with out-of-phase maxima).
• Direct chemical mediation of cross-diffusion produces patterns with substantially richer morphologies, including labyrinths, hexagonal spots, and disordered microstructures, all observed within biologically realistic parameter sets.
• Sharp transitions between global stability and pattern-forming regimes as $\Omega \subset \mathbb{R}^n$8 crosses the stability threshold.

Implications and Future Directions

The theoretical framework and computational results demonstrate the utility of chemically mediated cross-diffusion models in ecological and microbiological pattern formation. The work both confirms and extends previous analyses by showing that even with uniform self-diffusion, cross-diffusion terms coupled to secreted chemicals can fundamentally alter community stability and spatial organization. The findings have direct relevance for the study of chemically regulated spatial heterogeneity in microbial biofilms, planktonic ecosystems, and general population biology.

Potential avenues for future research include:

• Analytical extension to cross-diffusive systems with explicit time delays, density-dependent sensitivity, or stochastic fluctuations.
• Application to real ecological data to estimate effective cross-diffusion and response coefficients.
• Multi-scale asymptotic analysis of pattern selection mechanisms near Turing thresholds.

Conclusion

This study provides rigorous existence, boundedness, and asymptotic stability results and a comprehensive bifurcation analysis for a chemically mediated, cross-diffusive predator–prey PDE system. It demonstrates, both analytically and numerically, that nontrivial cross-diffusion induced by chemical signaling can generate a spectrum of Turing-type spatial patterns, with distinct transitions governed by predation intensity and nonlinear feedback. These results will inform both theoretical studies of nonlinear PDEs in ecology and the modeling of biomolecular communication in spatially structured populations.

Reference:

"Asymptotic stability and diffusion-driven pattern formation in a predator-prey system with two chemicals" (2604.18129)