PREDATOR-PREY MODEL WITH LOGISTIC GROWTH FOR CONSTANTAND DENSITY-DEPENDENT DELAYED MIGRATION

Abstract

Predator-prey models describe the interaction between two species, the prey which serves as a food source to the predator. The migration of the prey for safety reasonsafter a predator attack and the predator in search of food, from a patch to anothermay not be instantaneous. This may be due to barriers such as a swollen river ora busy infrastructure through the natural habitat. Recent predator-prey modelshave either incorporated a logistic growth for the prey population or a time delayin migration of the two species. Predator-prey models with logistic growth thatintegrate time delays in migration of both species have been given little attention. Inthis study, a logistic predator-prey model integrating a time delay in the migration ofboth species is developed and analyzed. The developed model was solved using twoinvariant manifolds; the symmetric manifold and the asymmetric manifold. Analysisof the model shows that when the prey growth rate is less than or equal to theprey migration rate, the two species coexist, otherwise both species become extinct.Numerical simulations show that during migration of the species, a longer time delaymakes the model to stabilize at a slower rate compared to when the time delay isshorter. It is also shown that the prey migration due to the predator density does notgreatly affect the prey density and existence compared to factors, such as logging,bad climatic conditions and limited food resources in a patch, that cause the preyto migrate. In the interest of species conservation, policies should be developed andenacted which address factors which prolong time delays during migration of thespecies by minimizing human activities and settlement in natural habitat.