Nonlocal Models in Biology and Life Sciences: Sources, Developments, and Applications

TL;DR

Nonlocal models address limitations of local descriptions in biology by incorporating long-range interactions and memory effects across scales. The paper surveys diverse nonlocal formulations—ranging from reaction-diffusion and fractional models to peridynamics and network-based approaches—and their applications in ecology, cancer, neuroscience, and nanotechnology. It highlights methodological advances, including pattern formation analysis, data-driven kernel learning, and physics-informed neural networks, while outlining multiscale and multifidelity strategies to manage computational complexity. Overall, the work underscores the potential of nonlocal modeling to bridge scales, integrate data, and inform interventions in life and health sciences, while calling for closer collaboration between mathematicians, modelers, and experimental biologists to validate and operationalize these tools.

Abstract

Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances. This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including brain dynamics models that provide an important tool to understand neurodegenerative diseases better. In addition, we have discussed nonlocal modelling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales, including nanotechnology, where classical local models often fail to capture the complexities of nanoscale interactions, applied to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.