Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

Abstract

In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant double-struckR+2. It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.

Bifurcations of a two-dimensional discrete-time predator–prey model

“…In order for (10) to undergo a Neimark-Sacker bifurcation, it is mandatory that the following discriminatory quantity, i.e., χ = 0 (see [22][23][24][25][26][27][28][29][30][31][32]),…”

Supercritical Neimark‐Sacker bifurcation of a discrete‐time Nicholson‐Bailey model

“…(see other studies [9][10][11][12] ) Let H( ) = 2 + B 1 + B 2 , where B 1 and B 2 are constants. Suppose H(1) > 0 and 1 and 2 are solutions of H( ) = 0.…”

“…In addition,ĤX nXn | (0,0) = 2l 11 ,ĤX nŶn | (0,0) = l 12 ,ĤŶ nŶn | (0,0) = 2l 13 ,ĤX nXnXn | (0,0) = 6l 14 , HX nXnŶn | (0,0) = 2l 15 ,ĤX nŶnŶn | (0,0) = 2l 16 ,ĤŶ nŶnŶn | (0,0) = 6l 17 , andKX nXn | (0,0) = 2l 21 ,KX nŶn | (0,0) = l 22 ,KŶ nŶn | (0,0) = 2l 23 ,KX nXnXn | (0,0) = 6l 24 , KX nXnŶn | (0,0) = 2l 25 ,KX nŶnŶn | (0,0) = 2l 26 ,KŶ nŶnŶn | (0,0) = 6l 27 . To guarantee the supercritical Neimark-Sacker bifurcation for (12), we require that following discriminatory quantity, ie, Ψ < 0 (see other studies [9][10][11][12][13][14][15][16][17][18][19][20].…”

Controlling chaos and Neimark-Sacker bifurcation in a discrete-time predator-prey system

“…In Section 3, choosing δ as the bifurcation parameter, Neimark-Sacker bifurcation analysis is studied. It is shown that the model (1.3) undergoes Neimark-Sacker bifurcation by using the bifurcation theory [23,33]. In Section 4, OGY control strategy is applied for chaos control due to occurrence of Neimark-Sacker bifurcation.…”