---

Journal of Applied Nonlinear Dynamics 10(4) (2021) 791--800 | DOI:10.5890/JAND.2021.12.015

Z.S. Boxonov$^1$ , U.A. Rozikov$^{1,2,3}$

$^1$ V.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences

$^2$ AKFA University, 1st Deadlock 10, Kukcha Darvoza, 100095, Tashkent, Uzbekistan

$^3$ Faculty of Mathematics, National University of Uzbekistan

Download Full Text PDF

 

Abstract

We study the discrete-time dynamical systems of a model of wild mosquito population with distinct birth (denoted by $\beta$) and death (denoted by $\mu$) rates. The case $\beta=\mu$ was considered in our previous work. In this paper we prove that for $\beta<\mu$ the mosquito population will die and for $\beta>\mu$ the population will survive, namely, the number of the larvaes goes to infinite and the number of adults has finite limit ${\alpha\over \mu}$, where $\alpha>0$ is the maximum emergence rete.

Acknowledgments

We thank both referees for their useful comments.

References

1. [1]	Alphey, L., Benedict, M., Bellini, R., Clark, G.G., Dame, D.A., Service, M.W., and Dobson, S.L. (2010), Sterile-insect methods for control of mosquito-borne diseases: An analysis, Vector Borne Zoonotic Dis., {\bf 10} 295--311.

2. [2]	Bartlettand, A.C. and Staten, R.T. (1996), Sterile Insect Release Method and Other Genetic Control Strategies, Radcliffes IPM World Textbook.

3. [3]	Becker, N. (2003), Mosquitoes and Their Control, Kluwer Academic/Plenum, New York.

4. [4]	Rozikov, U.A. (2020), Population dynamics: algebraic and probabilistic approach. World Sci. Publ.: Singapore, 460 pp.

5. [5]	Li, J. (2011), {Malaria model with stage-structured mosquitoes}, Math. Biol. Eng., 8, 753--768.

6. [6]	Rozikov, U.A. and Velasco, M.V. (2019), {A discrete-time dynamical system and an evolution algebra of mosquito population}, { Jour. Math. Biology}, 78(4) 1225--1244.

7. [7]	Rozikov, U.A. and Boxonov, Z.S. (2020), A discrete-time dynamical system of stage-structured wild and sterile mosquito population. arXiv:2002.11995.

8. [8]	Li, J., Cai, L., and Li, Y. (2017), {Stage-structured wild and sterile mosquito population models and their dynamics}, Journal of Biological Dynamics, {\bf 11}(2), 79--101.

9. [9]	Devaney, R.L. (2003), An Introduction to Chaotic Dynamical System Westview Press.

10. [10]	Mukhamedov, F., Saburov, M., and Qaralleh, I. (2013), {On $\xi^{(s)}$-quadratic stochastic operators on two dimensional simplex and their behavior}, Abst. Appl. Anal., Article ID 942038, 12 p.

11. [11]	Mukhamedov, F. and Saburov, M. (2014), {On dynamics of Lotka-Volterra type operators}, Bull. Malay. Math. Sci. Soc., 37 59--64.

12. [12]	Rozikov, U.A. (2019), An Introduction to Mathematical billiards, World Sci. Publ.: Singapore, 224 pp.