Predator-prey dynamics with refuge, alternate food, and harvesting strategies in a patchy habitat

A predator-prey dynamic reaction model is investigated in a two-layered water body where only the prey is subjected to harvesting. The surface layer (Layer-1) provides food for both species, while the prey migrates to deeper layer (Layer-2) as a refuge from predation. Although the prey is the preferred food for the predator, the predator can also consume alternative food resources that are abundantly available. The availability of alternative food resources plays a crucial role in species' coexistence by mitigating the risk of extinction. The main objective of the work was to explore the effect of different harvesting strategies (nonlinear and linear harvesting) on a predator-prey model with effort dynamics in a heterogeneous habitat. The analysis incorporates a dual timescale approach: the prey species migrate between the layers on a fast timescale, whereas the growth of resource biomass, prey-predator interactions, and harvesting dynamics evolve on a slow timescale. The complete model involving both slow and fast timescales has been investigated by using aggregated model. The reduced aggregated model is analyzed analytically as well as numerically. Moreover, it is demonstrated that the reduced system exhibits the bifurcations (transcritical and Hopf point) by setting the additional food parameter as a bifurcation parameter. A comparative study using different harvesting strategies found that there is chaos in the system when using linear harvesting in the predator-prey model. However, nonlinear harvesting gives only stable or periodic solutions. This concludes that nonlinear harvesting can control the chaos in the system. Additionally, a one-dimensional parametric bifurcation, phase portraits, and time series plots are also explored in the numerical simulation.