Modelling Delay in Migration for Constant Predator and Predator-Density-Dependent Prey Migration

Abstract

Predator-prey models describe the dynamics of ecological systems in which two species, the predator and the prey, interact. The objective of this study is to formulate and analyze a predator-prey mathematical model, based on a system of delay differential equations that takes into consideration time delay in migration, with a prey migration rate that depends on the predator density and other factors like availability of its food. It is shown that the population density mainly depends on both barriers during migration and the migration rate. The rates of migration may be affected by factors such as infrastructure through natural habitat, destruction of the natural habitat through logging, natural disasters like re-outbreaks among others. In view of this, relevant agencies should employ measures which will deal with factors which slow down the rate of migration or cause barriers during migration for example reducing natural habitat land allocation to human settlement, agriculture or infrastructure.

A Predator-Prey Model with Logistic Growth for Constant Delayed Migration

“…Migration can either be constant or variable. If the number of species migrating per unit time is a constant portion of the species density in a given patch then the migration is constant, otherwise the migration is variable [8,9,13].…”

Density Dependent Delayed Migration for Rosenzweig-Macaurther Model with Holling Type II Predator Functional Response

“…Migration of these species from one patch to another occurs due to a number of factors like lack of food, intraspecific competition, climatic conditions, overpopulation in a patch, lack of security among others [11,12,13,14]. The rate of migration can either be constant or it variable [15,16,17]. For constant migration, the number of species moving per unit time is a constant fraction of the population of the species in a given patch [15].…”