Multimodal distribution of transient time of predator extinction in a three-species food chain.

The transient dynamics capture the time history in the behavior of a system before reaching an attractor. This paper deals with the statistics of transient dynamics in a classic tri-trophic food chain with bistability. The species of the food chain model either coexist or undergo a partial extinction with predator death after a transient time depending upon the initial population density. The distribution of transient time to predator extinction shows interesting patterns of inhomogeneity and anisotropy in the basin of the predator-free state. More precisely, the distribution shows a multimodal character when the initial points are located near a basin boundary and a unimodal character when chosen from a location far away from the boundary. The distribution is also anisotropic because the number of modes depends on the direction of the local of initial points. We define two new metrics, viz., homogeneity index and local isotropic index, to characterize the distinctive features of the distribution. We explain the origin of such multimodal distributions and try to present their ecological implications.

Bistability in a tri-trophic food chain model: Basin stability perspective.

The most important issue of concern in a food chain is the stability of species and their nature of persistence against system parameter changes. For understanding the stable dynamics and their response against parameter perturbation, the local stability analysis is an insufficient tool. A global stability analysis by the conventional techniques seems to supplement some of the shortcomings, however, it becomes more challenging for multistable ecosystems. Either of the techniques fails to provide a complete description of the complexity in dynamics that may evolve in the system, especially, when there is any transition between the stable states. A tri-trophic resource-consumer-predator food chain model has been revisited here that shows bistability and transition to monostability via a border collision that leads to a state of predator extinction. Although earlier studies have partially revealed the dynamics of such transitions, we would like to present additional and precise information by analyzing the system from the perspective of basin stability. By drawing different bifurcation diagrams against three important parameters, using different initial conditions, we identify the range of parameter values within which the stability of the states persists and changes to various complex dynamics. We emphasize the changes in the geometry of the basins of attraction and get a quantitative estimate of the nature of relative changes in the area of the basins (basin stability) during the transitions. Furthermore, we demonstrate the presence of a down-up control, in addition to the conventional bottom-up and top-down control phenomena in the food chain. The application of basin stability in food networks will go a long way for accurate analysis of their dynamics.