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

Abstract

The model describing the interaction between the predator and prey species is referred to as a predator-prey model. The migration of these species from one 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 density-dependent migration of both species have been given little attention. A Rosenzweig-MacAurther model with density-dependent migration and time delay in the migration of both species is developed and analyzed in this study. The Analysis of the model when the prey migration rate is greater than or equal to the prey growth rate, the two species will coexist, otherwise, at least one species will become extinct. A longer delay slows down the rate at which the predator and prey population increase or decrease, thus aecting the population density of these species. The prey migration due to the predator density does not greatly affect the prey density and existence compared to the other factors that cause the prey to migrate. These factors include human activities in the natural habitats like logging and natural causes like bad climatic conditions, limited food resources and overpopulation of the prey species in a patch among others.

A dynamic analysis of a prey–predator population model with a nonlinear harvesting rate

“…The persistence and extinction properties of a large marine ecosystem are not only dependent on the interspecific interaction between the marine species but it is also dependent on the external environment factors. To enrich model (1), many researchers modify the nonlinear functional response function, adding some other elements like pollution, toxicity, harvesting, age of the species, refuge, etc. [2,[5][6][7][8][9][10].…”

“…Many researchers are interested to the dynamic of predator-prey interactions models, and have investigated the processes that affect it [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The interaction between the predator and prey species can be modeled by the classical Lotka-Volterra model [15].…”

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

“…The Lotka-Volterra model is an example of a classical predator-prey model which represents the interaction of these species in a homogeneous (single patch) environment [2]. Most classical predator-prey models assume that the environment will always be homogeneous, this may not be realistic since the environment is heterogeneous (many patches) and its connected by a diffusion-like process referred to as migration [4,5,6,7,8,9]. Migration takes place when a species transfers from one patch to another due to some hostile situations in its initial patch, for example, ecological factors like floods, human settlement and activities in the natural habitat like logging, intraspecific competition, predation, among others [10,11,12].…”

“…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].…”

“…In reality, after a species encounters a hostile situations, the species may meet a barrier, like a busy infrastructure passing through the natural habitat or a swollen river, while trying to migrate. With these barriers, a delayed migration of the prey and predator species may occur [7,8,9,15].…”