Dynamical analysis of seasonal migrating population; the effect of regular hunting to the coexistence

Seasonal migration among wild populations is commonly seen especially in the wild life region. The migration takes place during a certain season where logistical condition and the existing territory can no longer support the life of the whole population. In this case portion of the population migrate to the better place as part of their survival, and returning back to the home place when the logistical condition is improved. Here we model the dynamic of North-South annual migration of Impala population in Zimbabwe, where portion of population in the Southern part move to the North in the beginning of the dry season and portion of them return back to the South in the wet season. Here the North area has a better environmental carrying capacity than the South. Different processes take place during the year, partial migration to the south (during the month of December and January), partial migration to the north (during the month of June and July), and birth process (during the month of November and December). We construct a discrete dynamical model for simulating the annual migrating process. It is found that a stable co-existence always occurs when no hunting takes place in all season. When hunting is allowed, the co-existence could be severely affected. We obtain here a threshold condition for co-existence and show numerical simulations for different hunting scenarios.