Research Article Details

Abstract

Partial differential equations (PDEs) provide a powerful mathematical framework for understanding biological systems in which populations, chemical signals, cells, and resources vary across both space and time. Alan Turing's seminal reaction–diffusion theory demonstrated that spatial patterns can emerge spontaneously from initially homogeneous systems through interactions between local reactions and differential diffusion. Since then, reaction–diffusion models have developed into a broad family of nonlinear PDE frameworks used in developmental biology, population ecology, morphogenesis, and cellular organization. More recent approaches incorporate cross-diffusion, chemotaxis, nonlinear mobility, advection, density dependence, heterogeneous environments, and multiscale interactions. This review traces the conceptual development from classical Turing systems to modern nonlinear cross-diffusion models and examines their applications in spatial ecology and developmental biology. Particular attention is given to how cross-diffusion changes pattern-forming mechanisms by allowing the movement of one species or biological component to depend explicitly on the distribution of another. The paper compares classical reaction–diffusion and cross-diffusion formulations, discusses mathematical stability and pattern formation, and identifies opportunities for integrating nonlinear PDEs with computational simulation, experimental data, and multiscale biological modeling. The review argues that modern PDE models should be viewed not as replacements for Turing's framework but as extensions that incorporate increasingly realistic mechanisms of biological transport and interaction.

Keywords: reaction–diffusion, cross-diffusion, Turing patterns, nonlinear partial differential equations, spatial ecology, developmental biology, morphogenesis, mathematical biology