| Summary: | In this thesis we constructed two mathematical models for herbivore/vegetation interactions in environment of two patches, using the metaphysiological approach and a density-dependent migrations. In the first model we considered the case when the environment is constant, and we constructed a system of four perturbed ordinary differential equations describing the dynamics when only herbivores allowed to move between the two patches searching for food. The model contain two different timescales, fast for migrations and slow for the other demographic changes in the system. We used the geometric singular perturbation theory in order to reduce the dimension of the system. Using the continuation software AUTO we provided bifurcation diagrams for the reduced systems and we also provided some numerical illustrations to show the dynamics of the system for different migrations propensities. We analyzed the bifurcation diagrams using Morse decompositions and Conley index theory, to confirm their correctness. We constructed a second mathematical model, by considering that the vegetation growth depends on seasonal rainfall and the soil moisture. We provided some numerical simulations to illustrate several variates of dynamics for different migration speed and, when the migration propensities and the vegetation quality are change.
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