Mathematical model for herbivore/vegetation interaction in two-patch seasonal environment

In this paper we formulate a two-patch model for herbivore/vegetation interactions in seasonal environment. We assume the vegetation growth occurs during the raining season due to soil moisture. In this model animals are allowed to move between the patches searching for food. We show that when the migration propensity of leaving patch 1 is small the system has stable limit cycles. For large migration propensity we show that the system has stable limit cycle with 5 different frequencies, each of 2 years length, with a high total herbivore biomass and with a relatively high vegetation biomass. When we assume that the probability of leaving patch 1 is larger than the probability of leaving patch 2 the results show that the system has global attractive limit cycle with 5 different frequencies of 2 years length.