Dynamics of Modified predator-prey Models

Besides being structurally unstable, the Lotka-Volterra predator-prey model has another shortcoming due to the invalidity of the principle of mass action when the populations are very small. This leads to extremely large populations recovering from unrealistically small ones. The effects of linear modifications to structurally unstable continuous-time predator-prey models in a (small) neighbourhood of the origin are investigated here. In particular, it is shown that typically either a global attractor or repeller arises depending on the choice of coefficients. The analysis is based on Poincaré mappings, which allow an explicit representation for the classical Lotka-Volterra equations.