Local stability analysis on the predator-prey model with intraguild predation

We consider the predator-prey system with a common consuming resource that was proposed by Holt and Polis in 1997 to introduce the the effects of intraguild predation in modelling community ecology. Some of the results suggest that strong intraguild predation can even foster the coexistence of species. In 2018, the spatiotemporal dynamics of the model proposed was further analyzed to illustrate the theoretical findings previously mentioned in 1997. In this thesis, we perform transformations to the system, in order  to study a simplified equivalent system. The number of parameters is re- duced without altering the biological meaning of the system or the dynamic  behaviour. The local stability of the model is studied at each of the two pos- itive boundary equilibria and at the positive interior equilibrium by finding  the intervals of the parameters involved. The behaviour of the system will depend on which intervals the parameters fall. The emphasis is put on the  ranges of the predation rate assuming, there is less that can be done to in- fluence the parameters representing the natural birth and death rates of the prey and predator. By using the qualitative theory for autonomous planar systems, we show under which conditions each positive boundary equilibria can be a saddle, saddle node, or stable, and the interior positive equilibrium  is locally asymptotically stable. Under certain conditions the positive in- terior equilibrium is a stable node. It is interesting to note that when the  consumption of the common resources are equal for the predator and prey species then we would be dealing with a symmetric system.