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 reasons

after a predator attack and the predator in search of food, from a patch to another

may not be instantaneous. This may be due to barriers such as a swollen river or

a busy infrastructure through the natural habitat. Recent predator-prey models

have either incorporated a logistic growth for the prey population or a time delay

in migration of the two species. Predator-prey models with logistic growth that

integrate time delays in migration of both species have been given little attention. In

this study, a logistic predator-prey model integrating a time delay in the migration of

both species is developed and analyzed. The developed model was solved using two

invariant manifolds; the symmetric manifold and the asymmetric manifold. Analysis

of the model shows that when the prey growth rate is less than or equal to the

prey migration rate, the two species coexist, otherwise both species become extinct.

Numerical simulations show that during migration of the species, a longer time delay

makes the model to stabilize at a slower rate compared to when the time delay is

shorter. It is also shown that the prey migration due to the predator density does not

greatly aect the prey density and existence compared to factors, such as logging,

bad climatic conditions and limited food resources in a patch, that cause the prey

to migrate. In the interest of species conservation, policies should be developed and

enacted which address factors which prolong time delays during migration of the

species by minimizing human activities and settlement in natural habitat.