Dynamics of a host-parasite model connected with immigration

Abstract

The growth of the populations depending on interspecific interactions may exhibit drastic changes. The stability of populations change not only interspecific interactions but also under external effects such as immigration effect. Thus, population
models can show complex dynamics. Immigration effect often simplify the population dynamics; and tends to supress chaotic behavior. This situation which can allow the local stability analysis of population is important for control of a two species interacting system. In this paper, we investigated dynamics of a host-parasite model with the immigration parameter added to the host population under the constant searching efficiency. We conclude that immigration parameter produces certain interesting results.

Keywords

• Host-parasite model,stability analysis,equilibrium point,discrete dynamical systems,Misra-Mitra model

Kaynakça

1. L.J.S. Allen.: An introduction to mathematical biology, Pearson, New Jersey, (2007).
2. J.R. Beddington, C. A. Free and J. H. Lawton.: Dynamic complexity in predator-prey models framed in difference equation, Nature 255 (1975), 58-60.
3. M.P. Hassell, H.N. Comins.: Discrete-time models for two species competition. Theor. Popul. Biol. 9 (1976), 202-221.
4. S. Kapc¸ak, U. Ufuktepe and S. Elaydi.: Stability and invariant manifolds of a generalized Beddington host-parasitoid model, Journal of Biological Dynamics 7 (2013), 233-253.
5. M.R.S Kulenovic, G. Ladas.: Dynamics of second order rational difference equations: with open problems and conjectures, America, (2002).
6. J.C. Misra, A. Mitra.: Instabilities in single-species and host-parasite systems:period-doubling bifurcations and chaos, Comput. Math. Appl. 52 (2006),525-538.
7. J. D. Murray.: Mathematical Biology, New York , (2002).
8. A. Nicholson and V. Bailey.: The balance of animal population, 3, Proc. Zool. Soc. Lond., (1935).

9. U. Ufuktepe, S. Kapc¸ak.: Stability analysis of a host parasite model, Advances of Difference Equations 79 (2013) 2-7.
10. Ö. Ak Gümüs¸, F. Kangalgil.: Allee effect and stability in discrete-time host-parasitoid model, Journal of Advanced Research in Applied Mathematics, 7 (2015), 94-99.
11. Ö. Ak Gümüs¸.: Dynamical conseqences and stability analysis of a new host parasitoid model, General Mathematics Notes 27(2015), 9-15.
12. Q. Din, Ö. Ak Gümüs¸ and H. Khalil.: Neimark-Sacker bifurcation and chaotic behavior of a modified host-parasitoid model, Zeitschrift fur Naturforschung, 72 (2017), 25-37.
13. L Stone, D. Hart, Effects of immigration on the dynamics of simple population models: Theoretical Population Biology 55 (1999), 227-234.
14. H.I. McCallum.: Effects of immigration on chaotic population dynamics, Journal Theoretical Biology, 154 (1992) 277-284.
15. R. D. Holt.: Immigration and the dynamics of peripheral populations, in “Advances in Herpetology and Evolutionary Biology”, Museum of Comparative Zoology, Harvard University,Cambridge, MA, (1983).