A Fractional-order mathematical model  to analyze the stability and develop a sterilization strategy for the habitat of stray dogs

Zafer Öztürk; Ali Yousef; Halis Bilgil; Sezer Sorgun

doi:10.11121/ijocta.1418

Authors

Zafer Öztürk Institute of Science, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Turkiye https://orcid.org/0000-0001-5662-4670
Ali Yousef Department of Natural Sciences and Mathematics, College of Engineering, International University of Science and Technology in Kuwait, 92400 Al-Ardiya, Kuwait https://orcid.org/0000-0002-8824-5947
Halis Bilgil Faculty of Engineering, Architecture and Design, Kayseri University, 38280, Kayseri, Turkiye https://orcid.org/0000-0002-8329-5806
Sezer Sorgun Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Türkiye https://orcid.org/0000-0001-8708-1226

DOI:

https://doi.org/10.11121/ijocta.1418

Keywords:

Fractional-order Differential Equation, Euler Method, Stability Analysis, Stray Dog Population and Sterilization, Strategy, Semi-cycle Solution

Abstract

Today, the socio-cultural lack of some countries with increased urbanization has led to the unconscious breeding of stray dogs. The failure to care for the offspring of possessive dogs or ignoring the responsibility to find a suitable family for the offspring increased the dog population on the streets and in the shelters. In this study, our main target is to analyze the habitat of stray dogs and the strategy of how to control the population without damaging the ecosystem of the species. For this aim, we establish a fractional-order differential equation system to investigate the fractal dimension with long-term memory that invovles two compartments; the non-sterilized dog population (x(t)) and the sterilized one (y(t)). Firstly, we analyze the stability of the equilibrium points using the Routh-Hurwitz criteria to discuss cases that should not affect the ecosystem of the dog population, but control the stray dog population in the habitat. Since the intervention to the stray dog population occurs at discrete time impulses, we use the Euler method's discretization process to analyse the local and global stability around the equilibrium points. Besides this, we show that the solutions of the system represent semi-cycle behaviors. At the end of the study, we use accurate data to demonstrate the sterilization rate of stray dogs in their habitat.

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Author Biographies

Zafer Öztürk, Institute of Science, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Turkiye

Zafer Öztürk currently a doctorate student of mathematics in the Department of Mathematics, Nevsehir Haci Bektas Veli University. His interests include fractional modeling and bifurcations.

Ali Yousef, Department of Natural Sciences and Mathematics, College of Engineering, International University of Science and Technology in Kuwait, 92400 Al-Ardiya, Kuwait

Ali Yousef obtained his Ph.D. from the School of Mathematics, the University of Southampton, the UK. His main research area is sequential estimation and biological mathematics. He established the Department of Basic Sciences at Middle East University MEU in Jordan and was the statistical consultation center manager. In February 2016, he established and coordinated the mathematics program at Kuwait College of Science and Technology until September 2020. He published papers in sequential estimation and biomathematics related to HIV, breast cancer, Monoclonal Tumors, and Covid-19 in high-impact journals. Recently, he has been the chair of the Department of Natural Sciences and Mathematics at the International University of Science and Technology in Kuwait. In addition, a member of several committees in the university. He teaches mathematics, probability, and statistics to science and engineering students.

Halis Bilgil, Faculty of Engineering, Architecture and Design, Kayseri University, 38280, Kayseri, Turkiye

Halis Bilgil received Ph.D. in Mathematics from Erciyes University, Turkiye in 2010. He is a Professor with the Department of engineering basic sciences, Kayseri University, Turkiye. His current research interests include fractional differential systems, nonlinear evaluation equations, grey modeling, fluid mechanics and numerical analysis.

Sezer Sorgun, Department of Mathematics, Nevsehir Haci Bektas Veli University, 50300, Nevsehir, Türkiye

Sezer Sorgun is an associate professor at Department of Mathematics, Science and Letter Faculty, Nevsehir Haci Bektas Veli University. He received his PhD degree from the Erciyes University, Kayseri, Turkey. His research interests include ODEs, graf theory and matrix theory.

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