Mechanisms leading to bursting oscillations in the system of predator-prey communities coupled by migrations

The purpose is to study the periodic regimes of the dynamics for two non-identical predator-prey communities coupled by migrations, associated with the partial synchronization of fluctuations in the abundance of communities. The combination of fluctuations in neighboring sites leads to the regimes that include both fast bursts (bursting oscillations) and slow oscillations (tonic spiking). These types of activity are characterized by a different ratio of synchronous and non-synchronous dynamics of communities in certain periods of time. In this paper, we describe scenarios of the transition between different types of burst activity. These types of dynamics differ from each other not so much in size, shape, and number of spikes in a burst, but in the order of these bursts relative to the slow-fast cycle. Methods. To study the proposed model, we use the bifurcation analysis methods of dynamic systems, as well as geometric methods based on the division of the full system into fast and slow equations (subsystems). Results. We showed that the dynamics of the first subsystem with a slow-fast limit cycle directly determines the dynamics of the second one with burst activity through a smooth dependence of regime on the number of predators and a non-smooth dependence on the number of prey. We constructed the invariant manifolds on which there are parts of dynamics with tonic (slow manifold) and burst (fast manifold) activity of the full system. Conclusion. We described the scenario for bursting with different waveforms, which are determined by the appearance of the fast invariant manifold and the location of its parts relative to the slow-fast cycle. The transitions between different types of burst are accompanied by a change in the oscillation period, the degree of synchronization, and, as a result, the dynamics becomes quasi-periodic when both communities are not synchronous with each other.