Global Dynamics and Bifurcations Analysis of a Two‐Dimensional Discrete‐Time Lotka‐Volterra Model

Abstract

In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant R 2 . It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria (0, 0), (( 1 − 1)/ 3 , 0), (0, ( 4 −1)/ 6 ) and the unique positive equilibrium ((, by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria (( 1 − 1)/ 3 , 0), (0, ( 4 − 1)/ 6 ) and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium ((( 1 − 1) 6 − 2 ( 4 − 1))/( 3 6 − 2 5 ), ( 3 ( 4 − 1) + 5 (1 − 1 ))/( 3 6 − 2 5 )). Further it is shown that every positive solution of the discrete model is bounded and the set [0, 1 / 3 ] × [0, 4 / 6 ] is an invariant rectangle. It is proved that if 1 < 1 and 4 < 1, then equilibrium (0, 0) of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium ((( 1 − 1) 6 − 2 ( 4 − 1))/( 3 6 − 2 5 ), ( 3 ( 4 − 1) + 5 (1 − 1 ))/( 3 6 − 2 5 )) is a global attractor. Some numerical simulations are presented to illustrate theoretical results.

Bifurcation Analysis of a Discrete-Time Two-Species Model

“…is work deals with the study of local dynamics and bifurcation analysis of a discrete-time two-species model in ℝ 2 + . We In order for (43) to undergo Neimark-Sacker bifurcation, it is required that following discriminatory quantity, i.e., Ψ ̸ = 0 (see [6][7][8][9][10][11][12][13]).…”

Stability, Chaos, and Bifurcation Analysis of a Discrete Chemical System

“…Now, the nondegeneracy condition for hopf bifurcation is given by [19][20][21][22][23][24][25][26][27]…”

Neimark–Sacker Bifurcation of a Two-Dimensional Discrete-Time Chemical Model

“…It is noted that the following relation should be nonzero in order for (34) to undergo the Neimark-Sacker bifurcation (see [8][9][10][11][12][13][14][15][16]):…”