Attractors and long transients in a spatio-temporal slow–fast Bazykin's model

Spatio-temporal complexity of ecological dynamics has been a major focus of research for a few decades. Pattern formation, chaos, regime shifts and long transients are frequently observed in field data but specific factors and mechanisms responsible for the complex dynamics often remain obscure. An elementary building block of ecological population dynamics is a prey–predator system. In spite of its apparent simplicity, it has been demonstrated that a considerable part of ecological dynamical complexity may originate in this elementary system. A significant progress in understanding of the prey–predator system's potential complexity has been made over the last few years; however, there are yet many questions remaining. In this paper, we focus on the effect of intraspecific competition in the predator population; the prey–predator model accounting for such competition is known as Bazykin's model. We pay a particular attention to the case (often observed in real population communities) where the inherent prey and predator timescales are significantly different: the property known as a ‘slow–fast’ dynamics. Using an array of analytical methods along with numerical simulations, we provide a detailed analysis for the existence of Turing bifurcation and corresponding Turing pattern. We show how the Turing domain and corresponding spatial pattern changes in the presence of different (slow–fast) timescales. In doing that, we apply a novel approach to quantify the system solution by calculating its norm in two different metrics such as C0 and L2. We show that the slow–fast Bazykin's system exhibits a rich spatio-temporal dynamics, including a variety of long transient regimes that can last for hundreds and thousands of generations.