An investigation on the Lasota-Wazewska model with a piecewise constant argument

Gizem Seyhan Öztepe

doi:10.15672/hujms.766819

Research Article

Year 2021, , 1500 - 1508, 15.10.2021

Gizem Seyhan Öztepe

https://doi.org/10.15672/hujms.766819

Abstract

References

[1] S. Abualrub and M. Aloqeili, Dynamics of the system of difference equations $x_{n+1}=A+\dfrac{y_{n-k}}{y_{n}}, y_{n+1}=B+\dfrac{x_{n-k}}{x_{n}},$ Qual. Theory Dyn. Syst. 19, 69, 2020.
[2] M. Akhmet, D.A. Çinçin and N. Cengiz, Exponential stability of periodic solutions of recurrent neural networks with functional dependence on piecewise constant argument, Turkish J. Math. 42 (1), 272–292, 2018.
[3] O. Arino and M. Kimmel, Stability analysis of models of cell production systems, Math. Modelling 7, 1269–1300, 1986.
[4] F. Bozkurt and A. Yousef, NeimarkSacker bifurcation of a chemotherapy treatment of glioblastoma multiform (GBM), Adv. Differ. Equ. 2019 (1), 397, 2019.
[5] E. Braverman and S.H. Saker, On a difference equation with exponentially decreasing nonlinearity, Discrete Dyn. Nat. Soc., 2011 1–17, 2011.
[6] L.A.V. Carvalho and K.L. Cooke, A nonlinear equation with piecewise continuous argument, Differential Integral Equations, 1 (3), 359–367, 1988.
[7] F. Cavalli and A. Naimzada, A multiscale time model with piecewise constant ar- gument for a boundedly rational monopolist, J. Diff. Eq. App. 22 (10), 1480–1489, 2016.
[8] G.E. Chatzarakis and T. Li, Oscillation criteria for delay and advanced differential equations with nonmonotone arguments, Complexity, 2018, Article ID 8237634, 1–18, 2018.
[9] A. Chávez, S. Castillo and M. Pinto, Discontinuous almost periodic type functions, almost automorphy of solutions of differential equations with discontinuous delay and applications, Electron. J. Qual. Theory Differ. Equ. 2014 (75), 1–17, 2014.
[10] L. Chen and F. Chen, Positive periodic solution of the discrete Lasota-Wazewska model with impulse, J. Difference Equ. Appl. 20 (3), 406–412, 2014.
[11] F. Cherif and M. Miraoui, New results for a LasotaWazewska model, Int. J. Biomath. 12 (2), 1950019, 2019.
[12] K.S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, Electron. J. Qual. Theory Differ. Equ. 46, 1–19, 2010.
[13] K.S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr. 292, (10), 2153–2164, 2019.
[14] L. Duan, L. Huang and Y. Chen, Global exponential stability of periodic solutions to a delay LasotaWazewska model with discontinuous harvesting, Proc. Amer. Math. Soc. 144, 561-573, 2016.
[15] J. Dudás and T. Krisztin, Global stability for the three-dimensional logistic map, Nonlinearity 34 (2), 894, 2021.
[16] M.M. El-Afifi, On the recursive sequence $x_{n+1}=\frac{\alpha+\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}$, Appl. Math. Comput. 147, 617–628, 2004.
[17] Z. Feng, X. Wu and L. Yang, Stability of a mathematical model with piecewise con- stant arguments for tumor-immune interaction under drug therapy, Internat. J. Bifur. Chaos, 29 (01), 1950009, 2019.
[18] K. Gopalsamy, M.R.S. Kulenovic and G. Ladas, On a logistic equation with piecewise constant arguments, Differ. Int. Eq. 4, 215–223, 1991.
[19] J. Graef, C. Qian and P. Spikes, Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull. 39 (3), 275–283, 1996.
[20] M. Gumus, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl. 24 (6), 976–991, 2018.
[21] F. Gurcan and F. Bozkurt, Global stability in a population model with piecewise con- stant arguments, J. Math. Anal. Appl. 360 (1), 334–342, 2009.
[22] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Ap- plications, Oxford University Press, 1991.
[23] I. Györi and S.I. Trofimchuk, Global attractivity in $ x' \left(t\right) =-\delta x\left(t\right)+pf\left( x\left( t-\tau \right)\right)$, Dynam. Systems Appl. 8 (2), 197–210, 1999.
[24] W.H. Joseph and J.S. Yu, Global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 24 (2), 269–286, 1995.
[25] F. Karakoç, Asymptotic behaviour of a population model with piecewise constant ar- gument, Appl. Math. Lett. 70, 7–13, 2017.
[26] F. Karakoç, Asymptotic behavior of a Lasota-Wazewska model under impulse effect, Dynam. Systems Appl. 29 (12), 3381–3394, 2020.
[27] S. Kartal and F. Gurcan, Stability and bifurcations analysis of a competition model with piecewise constant arguments, Math. Methods Appl. Sci. 38 (9), 1855–1866, 2015.
[28] V.L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications Vol. 256, Springer Science and Business Media, 1993.
[29] M.R.S. Kulenovi and G. Ladas, Linearized oscillations in population dynamics, Bull. Math. Biol. 49 (5), 615–627, 1987.
[30] M.R.S. Kulenovi and G. Ladas, Dynamics of Second Order Rational Difference Equa- tions, Chapman and Hall/CRC, Boca Raton, 2001.
[31] M.R.S. Kulenovi, G. Ladas and Y.G. Sficas, Global attractivity in population dynam- ics, Comput. Math. Appl. 18 (10-11), 925–928, 1989.
[32] A. Lasota, K. Loskot and M.C. Mackey, Stability properties of proliferatively coupled cell replication models, Acta Biotheor. 39, 1–14, 1991.
[33] A. Lasota and M.C. Mackey, Cell division and the stability of cellular replication, J. Math. Biol. 38, 241–261, 1999.
[34] T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys. 70 (3), Art. 86, 1–18, 2019.
[35] X. Li and Z. Wang, Global attractivity for a logistic equation with piecewise constant arguments. Differences and Differ. Eqs., in: Fields Inst. Commmun. 42, 215–222, 2004.
[36] X. Li and D. Zhu, Global asymptotic stability of a kind of nonlinear delay difference equations, Appl. Math.-JCU, Set. B, 17 (2), 178–183, 2002.
[37] X. Li and D. Zhu, Global asymptotic stability in a rational equation, J. Difference Equ. Appl. 9 (9), 833–839, 2003.
[38] X. Li and D. Zhu, Global asymptotic stability for two recursive difference equations, Appl. Math. Comput. 150 (2), 481–492, 2004.
[39] X. Li and D. Zhu, Two rational recursive sequences, Comput. Math. Appl. 47 (10-11), 1487–1494, 2004.
[40] Z. Li and D. Zhu, Global asymptotic stability of a higher order nonlinear difference equation, Appl. Math. Lett. 19 (9), 926–930, 2006.
[41] P. Liu and K. Gopalsamy, Global stability and chaos in a population model with piecewise constant arguments, Appl. Math. Comput. 101, 63–88, 1999.
[42] P. Liu and K. Gopalsamy, Global stability and chaos in a population model with piecewise constant arguments, Appl. Math. Comput. 101 (1), 63–88, 1999.
[43] G. Liu, A. Zhao and J. Yan, Existence and global attractivity of unique positive periodic solution for a LasotaWazewska model, Nonlinear Anal. 64, 1737–1746, 2006.
[44] E. Liz and C. Lois-Prados, A note on the Lasota discrete model for blood cell produc- tion, Discrete Contin. Dyn. Syst. Ser. B, 25 (2), 701–713, 2020.
[45] E. Liz, V. Tkachenko and S. Trofmchuk, Global stability in discrete population models with delayed-density dependence, Math. Biosci. 199 (1), 26–37, 2006.
[46] H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a logistic equation with piecewise constant argument, Nonlinear Differential Equations Appl. 8 (1), 45–52, 2001.
[47] Y. Muroya,Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270, 602–635, 2002.
[48] Y. Muroya, A sufficient condition on global stability in a logistic equation with piece- wise constant arguments, Hokkaido Math. J. 32, 75–83, 2003.
[49] Y. Muroya, Global attractivity for discrete models of nonautonomous logistic equa- tions, Comput. Math. Appl. 53 (7), 1059–1073, 2007.
[50] I. Ozturk and F. Bozkurt, Stability analysis of a population model with piecewise constant arguments, Nonlinear Anal. Real World Appl. 12 (3), 1532–1545, 2011.
[51] F. Qiuxiang and Y. Rong, On the Lasota-Wazewska model with piecewise constant argument, Acta Math. Sci. 26 (2), 371–378, 2006.
[52] S. Rihani, A. Kessab and F. Cherif, Pseudo-almost periodic solutions for a La- sotaWazewska model, Electron. J. Differential Equations, 2016, 1–17, 2016.
[53] S.H. Saker, Qualitative analysis of discrete nonlinear delay survival red blood cells model, Nonlinear Anal. Real World Appl. 9, 471–489, 2008.
[54] J. Shao, Pseudo-almost periodic solutions for a LasotaWazewska model with an oscil- lating death rate, Appl. Math. Lett. 43, 90–95, 2015.
[55] J.W.H. So and J.S. Yu, Global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 24 (2), 269–286, 1995.
[56] G. Stamov and I. Stamova, Impulsive Delayed LasotaWazewska Fractional Models: Global Stability of Integral Manifolds, Mathematics 7 (11), 10–25, 2019.
[57] S. Stevi and D. Tollu, Solvability and semi-cycle analysis of a class of nonlinear systems of difference equations, Math. Methods Appl. Sci. 42 (10), 3579–3615, 2019.
[58] V. Tkachenko and S. Trofimchuk, Global stability in difference equations satisfying the generalized Yorke condition, J. Math. Anal. Appl. 303 (1), 173–187, 2005.
[59] V. Tkachenko and S. Trofimchuk, A global attractivity criterion for nonlinear non- autonomous difference equations, J. Math. Anal. Appl. 322 (2), 901–912, 2006.
[60] K. Uesugi, Y. Muroya and E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2), 560–580, 2004.
[61] L. Wang, Qualitative analysis of a predatorprey model with rapid evolution and piece- wise constant arguments, Int. J. Biomath. 10 (07), 1750101, 2017.
[62] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of the red blood cells system, (Polish) Math. Stos. III, 6, 23–40, 1976.
[63] J. Wiener, Generalized Solution of Functional Differential Equations,World Scientific, Singapore, 1993.
[64] S. Xiao, Delay effect in the LasotaWazewska model with multiple time-varying delays. Int. J. Biomath. 11, 1850013, 2018.
[65] W. Xu and J. Li, Global attractivity of the model for the survival of red blood cells with several delays, Ann. Differential Equations, 14, 357–363, 1998.

An investigation on the Lasota-Wazewska model with a piecewise constant argument

Year 2021, , 1500 - 1508, 15.10.2021

Gizem Seyhan Öztepe

https://doi.org/10.15672/hujms.766819

Abstract

This paper is devoted to investigating the asymptotic stability of the equilibrium point of the Lasota-Wazewska model with a piecewise constant argument and it is proved that this point is an attractor. It is also shown that every oscillatory solution of the corresponding difference equation has semi-cycles of length at least two.

Keywords

Lasota-Wazewska model,, piecewise constant argument, asymptotic stability, oscillation

References

[1] S. Abualrub and M. Aloqeili, Dynamics of the system of difference equations $x_{n+1}=A+\dfrac{y_{n-k}}{y_{n}}, y_{n+1}=B+\dfrac{x_{n-k}}{x_{n}},$ Qual. Theory Dyn. Syst. 19, 69, 2020.
[2] M. Akhmet, D.A. Çinçin and N. Cengiz, Exponential stability of periodic solutions of recurrent neural networks with functional dependence on piecewise constant argument, Turkish J. Math. 42 (1), 272–292, 2018.
[3] O. Arino and M. Kimmel, Stability analysis of models of cell production systems, Math. Modelling 7, 1269–1300, 1986.
[4] F. Bozkurt and A. Yousef, NeimarkSacker bifurcation of a chemotherapy treatment of glioblastoma multiform (GBM), Adv. Differ. Equ. 2019 (1), 397, 2019.
[5] E. Braverman and S.H. Saker, On a difference equation with exponentially decreasing nonlinearity, Discrete Dyn. Nat. Soc., 2011 1–17, 2011.
[6] L.A.V. Carvalho and K.L. Cooke, A nonlinear equation with piecewise continuous argument, Differential Integral Equations, 1 (3), 359–367, 1988.
[7] F. Cavalli and A. Naimzada, A multiscale time model with piecewise constant ar- gument for a boundedly rational monopolist, J. Diff. Eq. App. 22 (10), 1480–1489, 2016.
[8] G.E. Chatzarakis and T. Li, Oscillation criteria for delay and advanced differential equations with nonmonotone arguments, Complexity, 2018, Article ID 8237634, 1–18, 2018.
[9] A. Chávez, S. Castillo and M. Pinto, Discontinuous almost periodic type functions, almost automorphy of solutions of differential equations with discontinuous delay and applications, Electron. J. Qual. Theory Differ. Equ. 2014 (75), 1–17, 2014.
[10] L. Chen and F. Chen, Positive periodic solution of the discrete Lasota-Wazewska model with impulse, J. Difference Equ. Appl. 20 (3), 406–412, 2014.
[11] F. Cherif and M. Miraoui, New results for a LasotaWazewska model, Int. J. Biomath. 12 (2), 1950019, 2019.
[12] K.S. Chiu and M. Pinto, Periodic solutions of differential equations with a general piecewise constant argument and applications, Electron. J. Qual. Theory Differ. Equ. 46, 1–19, 2010.
[13] K.S. Chiu and T. Li, Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr. 292, (10), 2153–2164, 2019.
[14] L. Duan, L. Huang and Y. Chen, Global exponential stability of periodic solutions to a delay LasotaWazewska model with discontinuous harvesting, Proc. Amer. Math. Soc. 144, 561-573, 2016.
[15] J. Dudás and T. Krisztin, Global stability for the three-dimensional logistic map, Nonlinearity 34 (2), 894, 2021.
[16] M.M. El-Afifi, On the recursive sequence $x_{n+1}=\frac{\alpha+\beta x_{n}+\gamma x_{n-1}}{B x_{n}+C x_{n-1}}$, Appl. Math. Comput. 147, 617–628, 2004.
[17] Z. Feng, X. Wu and L. Yang, Stability of a mathematical model with piecewise con- stant arguments for tumor-immune interaction under drug therapy, Internat. J. Bifur. Chaos, 29 (01), 1950009, 2019.
[18] K. Gopalsamy, M.R.S. Kulenovic and G. Ladas, On a logistic equation with piecewise constant arguments, Differ. Int. Eq. 4, 215–223, 1991.
[19] J. Graef, C. Qian and P. Spikes, Oscillation and global attractivity in a periodic delay equation, Canad. Math. Bull. 39 (3), 275–283, 1996.
[20] M. Gumus, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl. 24 (6), 976–991, 2018.
[21] F. Gurcan and F. Bozkurt, Global stability in a population model with piecewise con- stant arguments, J. Math. Anal. Appl. 360 (1), 334–342, 2009.
[22] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Ap- plications, Oxford University Press, 1991.
[23] I. Györi and S.I. Trofimchuk, Global attractivity in $ x' \left(t\right) =-\delta x\left(t\right)+pf\left( x\left( t-\tau \right)\right)$, Dynam. Systems Appl. 8 (2), 197–210, 1999.
[24] W.H. Joseph and J.S. Yu, Global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 24 (2), 269–286, 1995.
[25] F. Karakoç, Asymptotic behaviour of a population model with piecewise constant ar- gument, Appl. Math. Lett. 70, 7–13, 2017.
[26] F. Karakoç, Asymptotic behavior of a Lasota-Wazewska model under impulse effect, Dynam. Systems Appl. 29 (12), 3381–3394, 2020.
[27] S. Kartal and F. Gurcan, Stability and bifurcations analysis of a competition model with piecewise constant arguments, Math. Methods Appl. Sci. 38 (9), 1855–1866, 2015.
[28] V.L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications Vol. 256, Springer Science and Business Media, 1993.
[29] M.R.S. Kulenovi and G. Ladas, Linearized oscillations in population dynamics, Bull. Math. Biol. 49 (5), 615–627, 1987.
[30] M.R.S. Kulenovi and G. Ladas, Dynamics of Second Order Rational Difference Equa- tions, Chapman and Hall/CRC, Boca Raton, 2001.
[31] M.R.S. Kulenovi, G. Ladas and Y.G. Sficas, Global attractivity in population dynam- ics, Comput. Math. Appl. 18 (10-11), 925–928, 1989.
[32] A. Lasota, K. Loskot and M.C. Mackey, Stability properties of proliferatively coupled cell replication models, Acta Biotheor. 39, 1–14, 1991.
[33] A. Lasota and M.C. Mackey, Cell division and the stability of cellular replication, J. Math. Biol. 38, 241–261, 1999.
[34] T. Li, N. Pintus and G. Viglialoro, Properties of solutions to porous medium problems with different sources and boundary conditions, Z. Angew. Math. Phys. 70 (3), Art. 86, 1–18, 2019.
[35] X. Li and Z. Wang, Global attractivity for a logistic equation with piecewise constant arguments. Differences and Differ. Eqs., in: Fields Inst. Commmun. 42, 215–222, 2004.
[36] X. Li and D. Zhu, Global asymptotic stability of a kind of nonlinear delay difference equations, Appl. Math.-JCU, Set. B, 17 (2), 178–183, 2002.
[37] X. Li and D. Zhu, Global asymptotic stability in a rational equation, J. Difference Equ. Appl. 9 (9), 833–839, 2003.
[38] X. Li and D. Zhu, Global asymptotic stability for two recursive difference equations, Appl. Math. Comput. 150 (2), 481–492, 2004.
[39] X. Li and D. Zhu, Two rational recursive sequences, Comput. Math. Appl. 47 (10-11), 1487–1494, 2004.
[40] Z. Li and D. Zhu, Global asymptotic stability of a higher order nonlinear difference equation, Appl. Math. Lett. 19 (9), 926–930, 2006.
[41] P. Liu and K. Gopalsamy, Global stability and chaos in a population model with piecewise constant arguments, Appl. Math. Comput. 101, 63–88, 1999.
[42] P. Liu and K. Gopalsamy, Global stability and chaos in a population model with piecewise constant arguments, Appl. Math. Comput. 101 (1), 63–88, 1999.
[43] G. Liu, A. Zhao and J. Yan, Existence and global attractivity of unique positive periodic solution for a LasotaWazewska model, Nonlinear Anal. 64, 1737–1746, 2006.
[44] E. Liz and C. Lois-Prados, A note on the Lasota discrete model for blood cell produc- tion, Discrete Contin. Dyn. Syst. Ser. B, 25 (2), 701–713, 2020.
[45] E. Liz, V. Tkachenko and S. Trofmchuk, Global stability in discrete population models with delayed-density dependence, Math. Biosci. 199 (1), 26–37, 2006.
[46] H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a logistic equation with piecewise constant argument, Nonlinear Differential Equations Appl. 8 (1), 45–52, 2001.
[47] Y. Muroya,Persistence, contractivity and global stability in logistic equations with piecewise constant delays, J. Math. Anal. Appl. 270, 602–635, 2002.
[48] Y. Muroya, A sufficient condition on global stability in a logistic equation with piece- wise constant arguments, Hokkaido Math. J. 32, 75–83, 2003.
[49] Y. Muroya, Global attractivity for discrete models of nonautonomous logistic equa- tions, Comput. Math. Appl. 53 (7), 1059–1073, 2007.
[50] I. Ozturk and F. Bozkurt, Stability analysis of a population model with piecewise constant arguments, Nonlinear Anal. Real World Appl. 12 (3), 1532–1545, 2011.
[51] F. Qiuxiang and Y. Rong, On the Lasota-Wazewska model with piecewise constant argument, Acta Math. Sci. 26 (2), 371–378, 2006.
[52] S. Rihani, A. Kessab and F. Cherif, Pseudo-almost periodic solutions for a La- sotaWazewska model, Electron. J. Differential Equations, 2016, 1–17, 2016.
[53] S.H. Saker, Qualitative analysis of discrete nonlinear delay survival red blood cells model, Nonlinear Anal. Real World Appl. 9, 471–489, 2008.
[54] J. Shao, Pseudo-almost periodic solutions for a LasotaWazewska model with an oscil- lating death rate, Appl. Math. Lett. 43, 90–95, 2015.
[55] J.W.H. So and J.S. Yu, Global stability in a logistic equation with piecewise constant arguments, Hokkaido Math. J. 24 (2), 269–286, 1995.
[56] G. Stamov and I. Stamova, Impulsive Delayed LasotaWazewska Fractional Models: Global Stability of Integral Manifolds, Mathematics 7 (11), 10–25, 2019.
[57] S. Stevi and D. Tollu, Solvability and semi-cycle analysis of a class of nonlinear systems of difference equations, Math. Methods Appl. Sci. 42 (10), 3579–3615, 2019.
[58] V. Tkachenko and S. Trofimchuk, Global stability in difference equations satisfying the generalized Yorke condition, J. Math. Anal. Appl. 303 (1), 173–187, 2005.
[59] V. Tkachenko and S. Trofimchuk, A global attractivity criterion for nonlinear non- autonomous difference equations, J. Math. Anal. Appl. 322 (2), 901–912, 2006.
[60] K. Uesugi, Y. Muroya and E. Ishiwata, On the global attractivity for a logistic equation with piecewise constant arguments, J. Math. Anal. Appl. 294 (2), 560–580, 2004.
[61] L. Wang, Qualitative analysis of a predatorprey model with rapid evolution and piece- wise constant arguments, Int. J. Biomath. 10 (07), 1750101, 2017.
[62] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of the red blood cells system, (Polish) Math. Stos. III, 6, 23–40, 1976.
[63] J. Wiener, Generalized Solution of Functional Differential Equations,World Scientific, Singapore, 1993.
[64] S. Xiao, Delay effect in the LasotaWazewska model with multiple time-varying delays. Int. J. Biomath. 11, 1850013, 2018.
[65] W. Xu and J. Li, Global attractivity of the model for the survival of red blood cells with several delays, Ann. Differential Equations, 14, 357–363, 1998.

There are 65 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Gizem Seyhan Öztepe 0000-0002-8170-7029
Publication Date	October 15, 2021
Published in Issue	Year 2021

Cite

APA	Seyhan Öztepe, G. (2021). An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics, 50(5), 1500-1508. https://doi.org/10.15672/hujms.766819
AMA	Seyhan Öztepe G. An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. October 2021;50(5):1500-1508. doi:10.15672/hujms.766819
Chicago	Seyhan Öztepe, Gizem. “An Investigation on the Lasota-Wazewska Model With a Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics 50, no. 5 (October 2021): 1500-1508. https://doi.org/10.15672/hujms.766819.
EndNote	Seyhan Öztepe G (October 1, 2021) An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics 50 5 1500–1508.
IEEE	G. Seyhan Öztepe, “An investigation on the Lasota-Wazewska model with a piecewise constant argument”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, pp. 1500–1508, 2021, doi: 10.15672/hujms.766819.
ISNAD	Seyhan Öztepe, Gizem. “An Investigation on the Lasota-Wazewska Model With a Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics 50/5 (October 2021), 1500-1508. https://doi.org/10.15672/hujms.766819.
JAMA	Seyhan Öztepe G. An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 2021;50:1500–1508.
MLA	Seyhan Öztepe, Gizem. “An Investigation on the Lasota-Wazewska Model With a Piecewise Constant Argument”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 5, 2021, pp. 1500-8, doi:10.15672/hujms.766819.
Vancouver	Seyhan Öztepe G. An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 2021;50(5):1500-8.

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