arXiv subjects

Gnanasekaran Shanmugasundaram

Publications and source records attributed to Gnanasekaran Shanmugasundaram.

5 recordsLinked to original sources

arXiv2026

Global existence, blow-up behavior and numerical simulations for a class of chemotaxis-driven fish-mussel systems

In this work, we investigate a chemotaxis-driven fish--mussel ecosystem model described by a coupled system of partial differential equations subject to homogeneous Neumann boundary conditions. Under suitable assumptions on the system parameters and initial data, we establish the global existence of classical solutions by employing semigroup methods together with a priori estimates. We also examine the possible blow-up behavior of solutions in a three-dimensional domain. To support the theoretical analysis, a finite element method is developed for the numerical approximation of the system, and convergence studies based on mesh refinement are carried out to verify the accuracy and stability of the proposed numerical scheme. Furthermore, numerical simulations illustrating the blow-up behavior of solutions in the computational domain are presented.

math.AP↗

arXiv2026

Global dynamics and diffusion-driven pattern formation in a predator-prey system with two chemicals

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.

math.AP↗

arXiv2026

Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production

This paper investigates a class of chemotaxis systems modeling lethal interactions in a smooth, bounded domain $Ω\subset \mathbb{R}^n$ with homogeneous Neumann boundary conditions. We examine two distinct cases: (i) a fully parabolic system where both equations exhibit parabolic dynamics, and (ii) a parabolic-elliptic system featuring a parabolic first equation coupled with an elliptic second equation. Under appropriate parameter constraints, we establish the existence of unique globally bounded classical solutions for arbitrary spatial dimensions $n \geq 1$. Additionally, we employ carefully constructed Lyapunov functionals to analyze the long-term behavior of solutions, obtaining rigorous asymptotic stability results.

math.AP↗

arXiv2026

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

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.

math.AP↗

arXiv2025

Global existence and boundedness in an attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions

This paper deals with the following attraction-repulsion chemotaxis system with nonlocal logistic source and sublinear productions \[ \left\{ \begin{array}{rrll} &&u_t = d_1 Δu-χ\nabla\cdot(u^k \nabla v)+ξ\nabla\cdot(u^k \nabla w)+ μu^m \left(1-\int_Ωu(x,t){\rm d}x\right),\qquad &x\inΩ,\, t>0,\\ &&v_t = d_2 Δv-αv+f(u), &x\inΩ,\, t>0,\\ &&w_t = d_3 Δw-βw+f(u), &x\inΩ,\, t>0,\\ &&\frac{\partial u}{\partialν} = \frac{\partial v}{\partialν} = \frac{\partial w}{\partialν} = 0, &x\in\partialΩ,\, t>0,\\ &&u(x,0) = u_0, \quad v(x,0)=v_0, \quad w(x,0)=w_0,&x\inΩ, \end{array} \right. \] in an open, bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$ with smooth boundary $\partialΩ$. Assume the parameters $d_1$, $d_2$, $d_3$, $χ$, $ξ$, $α$, $β$ and $μ$ are positive constants, initial data $(u_0, v_0, w_0)$ are nonnegative and the function $f(u)\leq K u^l\in C^1([0, \infty))$ for some $K, l>0$. Under appropriate conditions on the parameter $k$, $l$ and $m$ we show that the above problem admits a unique globally bounded classical solution.

math.AP↗