Rosenzweig-Macaurther Model with Holling type II Predator Functional Response for Constant Delayed Migration

Main Article Content

Apima B. Samuel
Lawi O. George
Nthiiri J. Kagendo

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. In this paper, a Rosenzweig-MacAurther model with a Holling-type II predator functional response and time delay in the migration of both species is developed and analysed. Stability analysis of the system shows that depending on the prey growth and prey migration rates either both species go to extinction or co-exist. Numerical simulations show that a longer delay in the migration of the species leads makes the model to stabilize at a slower rate compared to when the delay is shorter. Relevant agencies like
the Kenya Wildlife Service should address factors that slow down migration of species, for example, destruction of natural habitats for human settlement and activities, which may cause delay in migration.

Keywords:
Rosenzweig-Mac Aurther model, delay, migration.

Article Details

How to Cite
Samuel, A. B., George, L. O., & Kagendo, N. J. (2019). Rosenzweig-Macaurther Model with Holling type II Predator Functional Response for Constant Delayed Migration. Asian Research Journal of Mathematics, 15(1), 1-14. https://doi.org/10.9734/arjom/2019/v15i130140
Section
Original Research Article

References

Brauer F, Chaves CC. Mathematical models in population biology and epidemiology. Second Edition, Springer, New York; 2001.

Murray JD. Mathematical Biology: I. An Introduction Third Edition. Springer Verlag, Berlin; 2002.

Abdllaoui AE, Auger PM, Kooi BW, Parra RB, Mchich R. Eects of Density -dependent migrations on stability of a two-patch predator-prey model. Mathematical Bioscience. 2007;210:335-354.

Mchich R, Auger PM, Poggiale JC. Eect of predator density dependent dispersal of prey on stability of a predator-prey system. Mathematical Biosciences. 2007;206:343-356.

Wasike AM, Bong'ang'a AS, Lawi GO, Nyukuri MO. A predator-prey model with a time lag in the migration. Applied Mathematical Science. 2014;8-75:3721-3732.

Apima BS. A predator-prey model incorporating delay in migration. MSc. Thesis, Masinde Muliro University of Science and Technology, Kakamega, Kenya; 2014.

Pillai P, Gonzalez A, Loreau M. Evolution of dispersal in a predator-prey meta- community. The American Naturalist. 2012;179(2):204-216.

Abadi, Dian S, Choirotul U. Stability analysis of Lotka-Volterra model with Holling type II functional response. Scientic Research Journal. 2013;I(V):22-26.

Huang Y. Predator migration in response to prey density. What are the consequences? J. Math. Biol. 2001;43:561-581.

Rosenzweig M, MacArthur R. Graphical representation and stability conditions of predator-prey interaction. American Naturalist. 1963;97:209-223.

Xu C, Li P. Oscillations for a delayed predator-prey model with Hassell- Varley type functional responses. Comptes Rendus Biologies. 2015;338(4):227-240.

Xu C, Li P. Bifurcation behaviors analysis on a predator-prey model with nonlinear diffusion and delay. Journal of Dynamical and Control Systems. 2014;20(1):105-122.

Xu C, Li P. On the periodicity and global stability for a discrete delayed predatorprey model. International Journal of Mathematics 2013;24(10):1350086.

Xu C, Liao M. Bifurcation analysis of an autonomous epidemic predator- prey model with delay. Annali di Matematica Pura ed Applicata. 2014;193(1):23-28.

Xu C, Liao M. Bifurcation behaviors in a delayed three-species food-chain model with Holling type-II functional response. Applicable Analysis. 2013;92(12):2468-2486.

Xu C, Liao M, He X. Stability and Hopf bifurcation analysis for a Lokta-Volterra predatorprey model with two delays. International Journal of Applied Mathematics & Computer Science. 2011;21(1): 97C107.

Comins HN, Blatt DWE. Predator-prey models in spatially heterogeneous Environments. Journal of Theoretical Biology. 1974;48:75-83.

Neubert MG, Klepac P, Van Den Driessche P. Stabilizing Dispersal Delays in Predator-Prey Meta-population Models Theoretical Population Biology. 2002;61:339-347.

Hale JK, Lunel SV, Introduction to functional dierential equations. Springer-Verlag, New York; 1993.