#### Authors:

Asep K. Supriatna,Asep Sholahuddin,Hennie Husniah,#### DOI NO:

https://doi.org/10.26782/jmcms.2020.02.00006#### Keywords:

SI-terrorist model,logistic differential equation,fractional order,piecewise constant argument,#### Abstract

*The simplest model of terrorist growth model consists of two subpopulations, namely the susceptible subpopulation (S) and the militant or infected and infectious subpopulation (I). The model is governed by a coupled of differential equation reflecting the growth of the susceptible and infected subpopulations. Assuming a constant human population, the system can be reduced to a logistic differential equation. In this paper a fractional order delayed logistic equation is discussed and the discretization in the form of piecewise constant argument is used to find the solution. We use the first and the second order discretization method in the numerical scheme and investigate the effect of the fractional order in the growth of the underlying population modelled by the equation. We found that in general the discretization method can mimic the behavior of the original logistic equation for some parameters. However, destabilizing effect may occur depending on the combination of the values of related parameters, such as the fractional order, the intrinsic growth rate, and the piecewise constant argument parameter.*

#### Refference:

I. Atangana, “Application of Fractional Calculus to Epidemiology,” in Fractional dynamic (Eds. C. Cattani, H. M. Srivastava and X. Yang), De Gruyter Open, pp: 174-190, 2015.

II. K. Supriatna and H. P. Possingham, “Optimal harvesting for a predator–prey metapopulation”, Bulletin of Mathematical Biology, vol. 60, pp: 49–65, 1998.

III. K. Supriatna and H. P. Possingham, “Harvesting a two-patch predator-prey metapopulation”, Natural Resource Modeling, vol. 12, pp: 481-489, 1999.

IV. K. Supriatna, “Maximum sustainable yield for marine metapopulation governed by generalized coupled logistic equations”, J. Sustain. Sci. Manage., vol. 7, pp: 201-206, 2012.

V. K. Supriatna, A. P. Ramadhan and H. Husniah, “A decision support system for estimating growth parameters of commercial fish stock in fisheries industries”, Procedia Computer Science, vol. 59, pp: 331-339, 2015.

VI. K. Supriatna, A. Sholahuddin, A. P. Ramadhan and H. Husniah, “SOFish ver. 1.2 – a decision support system for fishery managers in managing complex fish stocks”, IOP Conf. Ser.: Earth Environ. Sci., vol. 2016, pp: 1-7, 2016.

VII. M. A. El-Sayed and S. M. Salman, “On a discretization process of fractional-order Riccati differential equation”, Journal of Fractional Calculus and Applications, vol. 4, pp: 251–259, 2013.

VIII. M. A. El-Sayed, S. M. Salman and N. A. Elabd, “On the fractional-order Nicholson equation, Applied Mathematical Sciences, vol. 10, pp: 503 – 518, 2016.

IX. M. A. El-Sayed, S. Z. Rida and A. A. M. Arafa, “Exact solutions of fractional-order biological population model”, Communications in Theoretical Physics, vol. 52, pp: 992-996, 2009.

X. N. Angstmann, B. I. Henry, B. A. Jacobs and A. V. McGann, “Discretization of fractional differential equations by a piecewise constant approximation, arXiv, vol. 1605.01815v1, pp: 1-8, 2016.

XI. Aru˘gaslan, “Dynamics of a harvested logistic type model with delay and piecewise constant argument”, J. Nonlinear Sci. Appl., vol. 8, pp: 507–517, 2015.

XII. F. Farokhi, M. Haeri and Tavazoei, “Comparing numerical methods for solving nonlinear fractional order differential equations”, In book: “New Trends in Nanotechnology and Fractional Calculus Applications”, pp: 171-179, 2010.

XIII. F. Udwadia, G. Leitmann and L. Lambertini, “A dynamical model of terrorism”, Discrete Dynamics in Nature and Society, vol. 2006, pp: 1–32, 2006.

XIV. G. Seifert, “Second-order neutral delay-differential equations with piecewise constant time dependence”, J. Math. Anal. Appl. vol. 281, pp: 1–9, 2003.

XV. H. Husniah and A. K. Supriatna, “Marine biological metapopulation with coupled logistic growth functions: The MSY and quasi MSY”, AIP Conference Proceedings, vol. 2014, pp: 51-56, 2014.

XVI. H. Husniah, A. K. Supriatna and N. Anggriani, “System dynamics approach in managing complex biological resources”, ARPN Journal of Engineering and Applied Sciences, vol. 10, pp: 1685-1690, 2014.

XVII. H. Husniah, R. A. R. H. Anwar, D. Haspada and A. K. Supriatna, “Terrorist dynamics: a transient solution from theoretical point of view”, The International Conference on Policing and Society, Indonesia, pp: 255-262, 2018.

XVIII. H. Matsunaga, T. Hara and S. Sakata, “Global attractivity for a logistic equation with piecewise constant argument”, Nonlinear Differential Equations Appl. vol. 8, pp: 45–52, 2001.

XIX. K. Gopalsamy, “Stability and Oscillation in Delay Differential Equations of Population Dynamics”, Math. Appl., Kluwer Academic, Dordrecht, vol. 74, 1992.

XX. K. Krishnaveni, K. Kannan and S. R. Balachandar, “Approximate analytical solution for Fractional population growth model”, International Journal of Engineering and Technology, vol. 5, pp: 2832-2836, 2013.

XXI. K. L. Cooke and J. Wiener, “Retarded differential equations with piecewise constant delays”, J Math Anal Appl , vol. 99, pp: 265–297, 1984.

XXII. K. L. Cooke and J. Wiener, “An equation alternately of retarded and advanced type”, Proceedings of the American Mathematical Society , vol. 99, pp: 726-732, 1987.

XXIII. K. S. Chiu and M. Pinto, “Periodic solutions of differential equations with a general piecewise constant argument and applications”, Electronic Journal of Qualitative Theory of Differential Equations, vol. 46, pp: 1-19, 2010.

XXIV. L. Guerrini, “Analysis of an economic growth model with variable carrying capacity”, Int. Journal of Math. Analysis, vol. 7, pp: 1263 – 1270, 2013.

XXV. M. C. MacLean and A. W. Turner, “The logistic curve applied to Canada’s population”, Can. J. Econ. Political Sci., vol. 3, pp: 241–248, 1937.

XXVI. M. Caputo, “Linear model of dissipation whose Q is almost frequency independent-II”, Geophysical Journal International, vol. 13, pp: 529–539, 1967.

XXVII. M. U. Akhmet and D. Aru˘gaslan, “Lyapunov-Razumikhin method for differential equations with piecewise constant argument”, Discrete Contin. Dyn. Syst., vol. 25, pp: 457–466, 2009.

XXVIII. M. U. Akhmet, D. Aru˘gaslan and E. Yılmaz, “Method of Lyapunov functions for differential equations with piecewise constant delay”, J. Comput. Appl. Math. , vol. 235, pp: 4554–4560, 2011.

XXIX. N. Bacaër, “Verhulst and the logistic equation (1838)” in A Short History of Mathematical Population Dynamics, Springer, London, pp. 35-39, 2011.

XXX. N. Varalta, A. V. Gomes and R. F. Camargo, “A prelude to the fractional calculus applied to tumor dynamic”, TEMA (São Carlos), vol. 15, pp: 211-221, 2014.

XXXI. R. F. Al-Bar, “On the approximate solution of fractional logistic differential equation using operational matrices of Bernstein polynomials”, Applied Mathematics, vol. 6, pp: 2096-2103, 2015.

XXXII. R. P. Agarwal, A. M. El-Sayed and S. M. Salman, “Fractional-order Chua’s system: discretization, bifurcation and chaos”, Advances in Difference Equations, vol. 320, pp: 1-13, 2013.

XXXIII. R. Pearl, L. J. Reed, “On the rate of growth of the population of the United States since 1790 and its mathematical representation”, Proc. Natl Acad. Sci., vol. 6, pp: 275–288, 1920.

XXXIV. S. Abbas, M. Banerjee and S. Momani, “Dynamical analysis of fractional-order modified logistic model”, Journal Computers & Mathematics with Applications, vol. 62, pp: 1098-1104, 2011.

XXXV. S. Busenberg and K. Cooke, “Vertically transmitted diseases: models and dynamics”, Biomathematics, Springer-Verlag, Berlin, vol. 23, 1993.

XXXVI. S. Das, P. K. Gupta and K. Vishal, “Approximate approach to the Das model of fractional logistic population growth”, Applications and Applied Mathematics: An International Journal, vol. 5, pp: 605 – 611, 2010.

XXXVII. S. Kazema, S. Abbas and S. Kumar, “Fractional-order Legendre functions for solving fractional-order differential equations”, Applied Mathematical Modelling, vol. 37, pp: 5498-5510, 2013.

XXXVIII. S. M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument deviations”, Internat. J. Math. & Math. Sci., vol. 6, pp: 671-703, 1983.

XXXIX. V. K. Srivastava, S. Kumar, M. K. Awasthi and B. K. Singh, “Two-dimensional time fractional-order biological population model and its analytical solution”, Egyptian Journal of Basic and Applied Sciences, vol. 1, pp: 71-76, 2014.

XL. W. Kermack and A. McKendrick, “Contributions to the mathematical theory of epidemics—I”, Bulletin of Mathematical Biology, vol. 53, pp: 33-55, 1991.

XLI. W. Kermack and A. McKendrick, “Contributions to the mathematical theory of epidemics—II. The problem of endemicity”, Bulletin of Mathematical Biology, vol. 53, pp: 57-87, 1991.

XLII. W. Kermack and A. McKendrick, “Contributions to the mathematical theory of epidemics—III. Further studies of the problem of endemicity”, Bulletin of Mathematical Biology, vol. 53, pp: 89-118, 1991.

XLIII. W. S. Gurney, S. P. Blythe and R. M. Nisbet, “Nicholson’s blowflies (revisited)”, Nature, vol. 287, pp: 17-21, 1980.

XLIV. Y. H. Xia “Positive periodic solutions for a neutral impulsive delayed Lotka-Volterra competition system with the effect of toxic substance”, Nonlinear Analysis: RWA, vol. 8, pp: 204-221, 2007.

XLV. Y. Kuang, “Delay Differential Equations with Applications in Population Dynamics”, New York, Academic Press, 1993.

XLVI. Y. Muroya, “Permanence, contractivity and global stability in logistic equations with general delays”, J. Math. Anal. Appl. vol. 302, pp: 389–401, 2005.

XLVII. Z. Dahmani and L. Tabharit, “Fractional order differential equations involving Caputo derivative”, Theory and Applications of Mathematics & Computer Science vol. 4, pp: 40-55, 2014.

XLVIII. Z. F. El-Raheem and S. M. Salman, “On a discretization process of fractional-order logistic differential equation”, Journal of the Egyptian Mathematical Society, vol 22, pp: 407–412, 2014.

XLIX. Z. Li, Z. Liang and Y. Yan “High-order numerical methods for solving time fractional partial differential equations”, Journal of Scientific Computing, vol. 71, pp: 785–803, 2017.

L. Z. Wang, “A numerical method for delayed fractional-order differential equations”, Journal of Applied Mathematics, vol. 2013, pp: 1-7, 2013.

LI. Z. Wang and J. Wu, “The stability in a logistic equation with piecewise constant arguments”, Differential Equations Dynam. Systems, vol. 14, pp: 179–193, 2006 .