### Abstract

This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational difference equations of order two (Formula Presented), with positive initial conditions x_{−1}, x_{0}, y_{−1} and y_{0} are nonzero real numbers. If we let u_{n} = x_{n}x_{n−1} and vn = y_{n}y_{n−1}, then these systems can be viewed as special cases of the system of the form un+1 = f (v_{n}), v_{n+1} = g(un). This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let w_{n} = u_{2n}, we have w_{n+1} = f (g(w_{n})) ≡ h(w_{n}). At a nonzero steady state w^{∗} of the last difference equation, we have |h′^{∗})| = |f′ (g(w^{∗}))g′ (w^{∗})| = 1, indicating that the system is degenerate at this steady state.

Original language | English (US) |
---|---|

Pages (from-to) | 321-333 |

Number of pages | 13 |

Journal | Journal of Computational Analysis and Applications |

Volume | 18 |

Issue number | 2 |

State | Published - Feb 1 2015 |

### Fingerprint

### Keywords

- Difference equations
- Periodic solution
- Recursive sequences
- Stability
- System of difference equations

### ASJC Scopus subject areas

- Computational Mathematics

### Cite this

*Journal of Computational Analysis and Applications*,

*18*(2), 321-333.

**Solutions and properties of some degenerate systems of difference equations.** / Alzahrani, E. O.; El-Dessoky, M. M.; Elsayed, E. M.; Kuang, Yang.

Research output: Contribution to journal › Article

*Journal of Computational Analysis and Applications*, vol. 18, no. 2, pp. 321-333.

}

TY - JOUR

T1 - Solutions and properties of some degenerate systems of difference equations

AU - Alzahrani, E. O.

AU - El-Dessoky, M. M.

AU - Elsayed, E. M.

AU - Kuang, Yang

PY - 2015/2/1

Y1 - 2015/2/1

N2 - This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational difference equations of order two (Formula Presented), with positive initial conditions x−1, x0, y−1 and y0 are nonzero real numbers. If we let un = xnxn−1 and vn = ynyn−1, then these systems can be viewed as special cases of the system of the form un+1 = f (vn), vn+1 = g(un). This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let wn = u2n, we have wn+1 = f (g(wn)) ≡ h(wn). At a nonzero steady state w∗ of the last difference equation, we have |h′∗)| = |f′ (g(w∗))g′ (w∗)| = 1, indicating that the system is degenerate at this steady state.

AB - This paper is devoted to obtain the form of the solution and the qualitative properties of the following systems of a rational difference equations of order two (Formula Presented), with positive initial conditions x−1, x0, y−1 and y0 are nonzero real numbers. If we let un = xnxn−1 and vn = ynyn−1, then these systems can be viewed as special cases of the system of the form un+1 = f (vn), vn+1 = g(un). This system has applications in modeling population growth with age structure or the dynamics of plant-herbivore interaction. Let wn = u2n, we have wn+1 = f (g(wn)) ≡ h(wn). At a nonzero steady state w∗ of the last difference equation, we have |h′∗)| = |f′ (g(w∗))g′ (w∗)| = 1, indicating that the system is degenerate at this steady state.

KW - Difference equations

KW - Periodic solution

KW - Recursive sequences

KW - Stability

KW - System of difference equations

UR - http://www.scopus.com/inward/record.url?scp=84940476654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84940476654&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84940476654

VL - 18

SP - 321

EP - 333

JO - Journal of Computational Analysis and Applications

JF - Journal of Computational Analysis and Applications

SN - 1521-1398

IS - 2

ER -