Akademik Veri Yönetim Sistemi
5th GLOBAL CONFERENCE on ENGINEERING RESEARCH, Balıkesir, Türkiye, 24 - 27 Eylül 2025, ss.168-169, (Özet Bildiri)
Reaction–diffusion systems are a central mathematical
framework for modeling spatially distributed processes in biology, ecology, and
chemistry. They describe the interplay between local nonlinear reactions and
spatial diffusion and have been used to explain phenomena ranging from pattern
formation in developmental biology to the dynamics of competing populations or
chemical concentrations. Classical models usually rely on logistic
self-limitation combined with simple bilinear cross-interactions of the Lotka–Volterra
type. While these formulations are mathematically tractable and biologically
insightful, they often fail to capture the fact that interspecific interactions
are constrained when the total population or concentration approaches a certain
carrying capacity. This motivates the introduction of a novel
reaction–diffusion system with a capacity-limited cross-interaction term.
In our work, we propose and analyze a two-species model
posed on a square spatial domain with homogeneous Neumann boundary conditions.
Here the cross-interaction between the two species is modulated by an Lp-type
capacity factor, which generalizes the idea of a carrying capacity beyond
simple linear constraints. When the total density is small, the interaction is
nearly bilinear, but as the combined density approaches the unit scale, the
interaction is suppressed. This functional form introduces a flexible mechanism
that interpolates between different competitive or cooperative regimes
depending on the value of p. By embedding this new interaction into a
reaction-diffusion framework, we extend the classical repertoire of
activator-inhibitor models and open the door qualitatively new dynamics.
The first part of our study is devoted to the
well-posedness of the system. Assuming smooth and positive initial data, and
positive diffusion and reaction parameters, we show that the system admits a
unique classical solution. The proof combines standard semi-linear parabolic
theory, Schauder estimates, and fixed-point arguments. We establish local
existence and uniqueness, then apply maximum principle techniques to show that
nonnegativity is preserved. Thus, the system is mathematically consistent: its
solutions exist, are unique, remain smooth, positive. This result ensures that
the model can be used reliably for both theoretical investigations and
numerical simulations.
The second part of our study focuses on numerical
simulations under different parameter regimes. We employ PDE (Partial
Differential Equation) Toolbox in MATLAB to solve the system on a
two-dimensional square domain with zero-flux boundary conditions. Our
simulations reveal a rich variety of dynamical behaviors. The role of the
capacity exponent p is especially interesting. Small p values produce smoother,
more gradual regulation of the cross-interaction, while larger p values impose
a sharper cutoff, leading to more pronounced and sharply localized spatial
structures. These computational experiments provide concrete evidence that the
Lp-capacity modification fundamentally alters the dynamics compared
to classical models.
Beyond the mathematical interest, the applications of this
framework are broad. In ecology, it can model two species competing for limited
resources when total density is constrained by an environmental carrying
capacity. In epidemiology, the model can capture interactions between two host
populations or pathogen strains when the overall population density is limited,
thereby incorporating saturation effects into disease spread. In chemical and
material sciences, the system could represent reactions in which the overall
concentration of reactants cannot exceed a maximum capacity, leading to slowed
reaction rates near saturation.
The combination of rigorous analysis, numerical evidence,
and interdisciplinary relevance underscores the value of this system as both a
mathematical object and an applied modeling tool.
Keywords—Reaction-diffusion systems; Partial differential equations; Lp-capacity interaction