Complex dynamical behavior and control of a discrete ecological model

Cyclic types of ecological dynamics have been found in several biological species of forest Lepidoptera. There are several possible reasons for this, for example, interactions with consumers, predators, plant quality index, and interactions of density-dependent type. Furthermore, interaction with consumers and plant quality index is regarded as key ingredients to alter the dynamics for the parasitoid population. Consequently, the quality of food resources fluctuates due to the level of herbs. Such changes have been observed in different systems of forest pests. Lepidoptera (larch budmoth) is a destructive worm that affects high-altitude trees around the world and is rapidly declining and becoming extinct in large areas of the forest. Considering the interaction between the budmoth and plant quality index for larch trees in the mountain range in Switzerland (Swiss Alps), we discuss the dynamics of a discrete-time system. Ivlev type functional response regarding plant quality index is used for the formulation of discrete-time model concerning the interaction between the index of plant quality and budmoth. Moreover, the existence of steady-states, their local behaviors, and the boundedness of solutions are carried out for the discrete-time model under consideration. It is investigated that the model undergoes flip bifurcation about coexistence by applying the theory of normal forms and the center manifold theorem. Furthermore, the direction and the existence of Hopf bifurcation are explored for the model around its coexistence. Various methods of controlling chaos have been introduced to avoid system fluctuations and bifurcating attitudes. Validation of the analytical findings is illustrated through numerical simulations. Finally, the analytical results are validated by experimental and actual field data.