Modelling delay in migration for constant predator And predator-density-dependent prey migration

Abstract

ABSTRACT Predator-prey models describe the dynamics of ecological systems in which two species, the predator and the prey, interact. The classical Lotka-Volterra model is the simplest predator-prey model. Migration of species, due to predation and other factors, in predator-prey systems has been studied. In reality the prey do not migrate instantly upon being preyed on neither do the predators migrate immediately they lack food. Delay in the migration of species in such systems for constant rates has been investigated. However, if the migration of the prey is largely due to predation, then it is expected that the prey migration rate will depend on the density of the predators. 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.Analysis of the formulated model shows that for both the Symmetric and Asymmetric manifolds, the system is unstable when the prey migration rate is less than the prey growth rate. This means that the predator and prey species will become extinct in either patches. On the other hand, when the prey migration rate is greater than the prey growth rate, then the resulting system is stable. This implies that the two species in both patches will coexist. Furthermore, when the prey migration rate is equal to the prey growth rate, a periodic solution occurs. This means that the two populations will fluctuate by rising or falling almost in equal measure. Numerical analysis show that delay has a stabilizing effect on the system and in the presence of delay, the species populations decrease at a faster rate compared to the case without delay.It is shown that the population density mainly depends on the migration rate which may be affected by factors such as infrastructure through natural habitat, destruction of the natural habitat through logging, natural disasters like fire-outbreaks among others. In view of this, relevant agencies like the Kenya Wildlife Service and the government should employ measures which will deal with factors which cause barriers during migration for example reducing natural habitat land allocation to human settlement, agriculture or infrastructure.