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The simultaneous effect of chemotaxis and alarm-taxis on the global existence and stability of a predator-prey system

Gnanasekaran Shanmugasundaram, Jitraj Saha, Rafael Díaz Fuentes

TL;DR

The paper analyzes a fully parabolic predator–prey system incorporating both chemotaxis and alarm-taxis under Neumann boundary conditions. By combining rigorous parabolic theory with Lyapunov-based techniques, it proves global existence and uniform boundedness of classical solutions, and establishes exponential convergence to multiple equilibria dictated by interspecific interactions and parameter regimes. The authors derive explicit coexistence and semi-coexistence equilibria and provide sharp conditions under which each state is globally asymptotically stable, complemented by numerical simulations in 2D and 3D that corroborate the theoretical predictions. Overall, the work clarifies how chemotaxis and alarm-taxis coefficients govern long-time dynamics in predator–prey systems and offers a methodological template for analyzing similar chemo-taxis–alarm-taxis models.

Abstract

This study examines a fully parabolic predator-prey chemo-alarm-taxis system under homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n$ with a smooth boundary $\partialΩ$. Under specific parameter conditions, it is shown that the system admits a unique, globally bounded classical solution. The convergence of the solution is established through the construction of an appropriate Lyapunov functional. In addition, numerical simulations are presented to validate the asymptotic behaviour of the solution. The results highlight the significant role of chemotaxis and alarm-taxis coefficients in determining the existence and stability of predator-prey models, as discussed in the literature.

The simultaneous effect of chemotaxis and alarm-taxis on the global existence and stability of a predator-prey system

TL;DR

The paper analyzes a fully parabolic predator–prey system incorporating both chemotaxis and alarm-taxis under Neumann boundary conditions. By combining rigorous parabolic theory with Lyapunov-based techniques, it proves global existence and uniform boundedness of classical solutions, and establishes exponential convergence to multiple equilibria dictated by interspecific interactions and parameter regimes. The authors derive explicit coexistence and semi-coexistence equilibria and provide sharp conditions under which each state is globally asymptotically stable, complemented by numerical simulations in 2D and 3D that corroborate the theoretical predictions. Overall, the work clarifies how chemotaxis and alarm-taxis coefficients govern long-time dynamics in predator–prey systems and offers a methodological template for analyzing similar chemo-taxis–alarm-taxis models.

Abstract

This study examines a fully parabolic predator-prey chemo-alarm-taxis system under homogeneous Neumann boundary conditions in a bounded domain with a smooth boundary . Under specific parameter conditions, it is shown that the system admits a unique, globally bounded classical solution. The convergence of the solution is established through the construction of an appropriate Lyapunov functional. In addition, numerical simulations are presented to validate the asymptotic behaviour of the solution. The results highlight the significant role of chemotaxis and alarm-taxis coefficients in determining the existence and stability of predator-prey models, as discussed in the literature.
Paper Structure (12 sections, 21 theorems, 231 equations, 8 figures)

This paper contains 12 sections, 21 theorems, 231 equations, 8 figures.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries and local existence
  3. Global existence of solutions
  4. Global asymptotic stability
  5. Coexistence state of the species
  6. Secondary predator only existence state
  7. Semi-coexistence state of the species
  8. Prey vanishing state
  9. Primary predator vanishing state
  10. Proof of the main theorems
  11. Numerical examples
  12. Conclusion

Key Result

Theorem 1.1

Suppose that $\Omega \subset\mathbb{R}^n (n\geq 1)$, is a bounded domain with smooth boundary. Then there exists $\mu>0$, such that if $\min\{\mu_2, \mu_3\}>\mu$ and for any nonnegative initial data $(u_0, v_0, w_0, z_0)$ satisfying 1.2 for some $q>\max\{2, n\}$, the system 1.1 possesses a unique cl where the constant $C>0$.

Figures (8)

  • Figure 1: Populations at $t=30$ in 2D.
  • Figure 2: Populations at $t=20$ in 3D.
  • Figure 3: Populations at $t=20$ in 2D.
  • Figure 4: Populations at $t=20$ in 3D.
  • Figure 5: Populations at $t=20$ in 2D.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Theorem 1.1: Global existence of solutions
  • Remark 1.1
  • Theorem 1.2: Coexistence state of the species
  • Theorem 1.3: Secondary predator only existence state
  • Theorem 1.4: Semi-coexistence state of the species
  • Lemma 2.1: jzheng
  • Lemma 2.2: cstinner2014ytao2019
  • Lemma 2.3: Maximal Sobolev regularity xcaohieber
  • Lemma 2.4: Local Existence
  • proof
  • ...and 30 more