Analytical Study of a System of Difference Equation
Abdualrazaq Sanbo *
Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia and General studies department, Jeddah College of Telecom and Electronics, TVTC, B.P. 2816, Jeddah 21461, Saudi Arabia.
Elsayed M. Elsayed
Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia and Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt.
*Author to whom correspondence should be addressed.
Abstract
We study the qualitative behavior of a predator-prey model, where the carrying capacity of the predators environment is proportional to the number of prey. The considered system is given by the following rational difference equations:
$$x_{n+1}=\dfrac{x_{n} y_{n-2}}{y_{n-1} + y_{n-2}},\quad y_{n+1}=\dfrac{y_{n} x_{n-2}}{\pm x_{n-1} \pm x_{n-2}},\quad
n=0,1,\cdots,$$
where the initial conditions x-2; x-1; x0; y-2; y-1; y0 are arbitrary positive real numbers. Also, we give specic form of the solutions of some special cases of this equation. Some numerical examples are given to verify our theoretical results.
Keywords: Dierence equations, Recursive sequences, Stability, Boundedness