Investigation of the global dynamics of two exponential-form difference equations systems
<p>In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms:</p> <p class=&q...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-10-01
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| Series: | Electronic Research Archive |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/era.2023338?viewType=HTML |
| _version_ | 1797449840980918272 |
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| author | Merve Kara |
| author_facet | Merve Kara |
| author_sort | Merve Kara |
| collection | DOAJ |
| description | <p>In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms:</p>
<p class="disp_formula"> $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Psi_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Omega_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Upsilon_{n}}, \end{equation*} $ </p>
<p class="disp_formula">$ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Upsilon_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Psi_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Omega_{n}}, \end{equation*} $</p>
<p>for $ n\in \mathbb{N}_{0} $, where the initial conditions $ \Upsilon_{-j} $, $ \Psi_{-j} $, $ \Omega_{-j} $, for $ j\in\{0, 1\} $ and the parameters $ \Gamma_{i} $, $ \delta_{i} $, $ \Theta_{i} $ for $ i\in\{1, 2, 3\} $ are positive constants.</p> |
| first_indexed | 2024-03-09T14:31:49Z |
| format | Article |
| id | doaj.art-81f7feff9a9f4a2b8060ad542c6dc981 |
| institution | Directory Open Access Journal |
| issn | 2688-1594 |
| language | English |
| last_indexed | 2024-03-09T14:31:49Z |
| publishDate | 2023-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Electronic Research Archive |
| spelling | doaj.art-81f7feff9a9f4a2b8060ad542c6dc9812023-11-28T01:29:04ZengAIMS PressElectronic Research Archive2688-15942023-10-0131116697672410.3934/era.2023338Investigation of the global dynamics of two exponential-form difference equations systemsMerve Kara 0Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, Karaman 70100, Turkey<p>In this study, we investigate the boundedness, persistence of positive solutions, local and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the following difference equations systems of exponential forms:</p> <p class="disp_formula"> $ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Psi_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Omega_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Upsilon_{n}}, \end{equation*} $ </p> <p class="disp_formula">$ \begin{equation*} \Upsilon_{n+1} = \frac{\Gamma_{1}+\delta_{1}e^{-\Psi_{n-1}}}{\Theta_{1}+\Upsilon_{n}}, \ \Psi_{n+1} = \frac{\Gamma_{2}+\delta_{2}e^{-\Omega_{n-1}}}{\Theta_{2}+\Psi_{n}}, \ \Omega_{n+1} = \frac{\Gamma_{3}+\delta_{3}e^{-\Upsilon_{n-1}}}{\Theta_{3}+\Omega_{n}}, \end{equation*} $</p> <p>for $ n\in \mathbb{N}_{0} $, where the initial conditions $ \Upsilon_{-j} $, $ \Psi_{-j} $, $ \Omega_{-j} $, for $ j\in\{0, 1\} $ and the parameters $ \Gamma_{i} $, $ \delta_{i} $, $ \Theta_{i} $ for $ i\in\{1, 2, 3\} $ are positive constants.</p>https://www.aimspress.com/article/doi/10.3934/era.2023338?viewType=HTMLboundednesssystem of difference equationspersistencestability |
| spellingShingle | Merve Kara Investigation of the global dynamics of two exponential-form difference equations systems Electronic Research Archive boundedness system of difference equations persistence stability |
| title | Investigation of the global dynamics of two exponential-form difference equations systems |
| title_full | Investigation of the global dynamics of two exponential-form difference equations systems |
| title_fullStr | Investigation of the global dynamics of two exponential-form difference equations systems |
| title_full_unstemmed | Investigation of the global dynamics of two exponential-form difference equations systems |
| title_short | Investigation of the global dynamics of two exponential-form difference equations systems |
| title_sort | investigation of the global dynamics of two exponential form difference equations systems |
| topic | boundedness system of difference equations persistence stability |
| url | https://www.aimspress.com/article/doi/10.3934/era.2023338?viewType=HTML |
| work_keys_str_mv | AT mervekara investigationoftheglobaldynamicsoftwoexponentialformdifferenceequationssystems |