"Long spatio-temporal transients in slow-fast Bazykin's model"
The presence of multiple timescales in complex biological or ecological systems is ubiquitous in nature. Mathematically, these systems are known as 'slow-fast' systems or singularly perturbed systems. Recently, there has been a rising interest among researchers to study the ecological implications of slow-fast systems. The presence of multiple timescales in a biological system inevitably gives rise to the study of long transients. Here, we consider a slow-fast predator-prey model with Bazykin-type reaction kinetics to study the spatio-temporal long transients. The temporal counterpart of the system shows the existence of peculiar periodic solutions: canard and relaxation oscillation. However, a parametric domain is identified where the system shows the existence of two canard cycles, stable and unstable. Even in the spatially extended system, a spatio-temporal canard explosion is observed. This implies that the system dynamics change abruptly from small to large amplitude oscillations within an exponentially small parameter interval. The system dynamics become much more complex near a bifurcation threshold. We argue that the spatial average of the species density over time is not enough to capture the spatial heterogeneity of the distribution of the species. Hence, we introduce two additional metrics to identify the rich spatio-temporal dynamics, which include a variety of long transient regimes.
Additional authors: Sergei Petrovskii, School of Mathematics & Actuarial Science, University of Leicester, Leicester LE1 7RH, UK; Vitaly Volpert, Institut Camille Jordan, UMR 5208 CNRS, University Lyon 1, 69622 Villeurbanne, France; Malay Banerjee, Indian Institute of Technology Kanpur, Kanpur - 208016, India