Spatiotemporal Oscillations Driven by Spatial Heterogeneity and Growth

Abstract

Temporal oscillations arise naturally in biological systems with feedback loops or explicit delays. When incorporating spatial effects, the travel times of propagating signals can also generate intrinsic delays, and this behaviour is complicated further in the presence of spatial heterogeneity. In this work, we investigate the mechanistic origin and dynamics of spatiotemporal oscillations driven by intrinsic diffusive delays in a representative activator-inhibitor system with spatial heterogeneity using a combination of mathematical modelling, simulations and numerical methods. In contrast to delay-differential equation models where delays are explicitly prescribed, in the partial differential equation system we consider here delays are an intrinsic consequence of signal travel time through space. We demonstrate that undamped oscillations can arise as a result of spatially heterogeneous feedback when the system size increases through a critical length. At this critical length the time for systems to reach steady-state diverges, and the systems first exhibit an eigenvalue with a positive real part and non-zero imaginary part, corresponding to a Hopf bifurcation. We infer that this critical length could define a theoretical upper limit for the size of systems where oscillations hinder function. The analysis pipeline derived in this work offers a novel route for understanding the onset of spatiotemporal oscillations and Hopf bifurcations in a diverse array of biological systems where interactions are mediated by diffusive species.