Two delays induce Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system

Abstract

In this paper, the dynamics of a modified Leslie-Gower predator-prey system with two delays and diffusion is considered. By calculating stability switching curves, the stability of positive equilibrium and the existence of Hopf bifurcation and double Hopf bifurcation are investigated on the parametric plane of two delays. Taking two time delays as bifurcation parameters, the normal form on the center manifold near the double Hopf bifurcation point is derived, and the unfoldings near the critical points are given. Finally, we obtain the complex dynamics near the double Hopf bifurcation point, including the existence of quasi-periodic solutions on a 2-torus, quasi-periodic solutions on a 3-torus, and strange attractors.Diffusive predator-prey models with delays have been investigated widely, and the delay induced Hopf bifurcation analysis has been well studied. However, the study about bifurcation analysis of predator-prey models with two simultaneously varying delays has not been well established. Neither the Hopf bifurcation theorem with two parameters nor the derivation process of normal form for two delays induced double Hopf bifurcation has been proposed in literatures. In this paper, we investigate a diffusive Leslie-Gower model with two delays, and carry out Hopf and double Hopf bifurcation analysis of the model. Applying the method of studying characteristic equation with two delays, we get the stability switching curves and the crossing direction, after which we give the Hopf bifurcation theorem in two-parameter plane for the first time. Under some condition, the intersections of two stability switching curves are double Hopf bifurcation points. To figure out the dynamics near the double Hopf bifurcation point, we calculate the normal form on the center manifold. The derivation process of normal form we use in this paper can be extended to other models with two delays, one delay, or without delay.

Permanence and uniform asymptotical stability of a ratio-dependent Leslie system with feedback controls on time scales

“…From (7) and (9), V (t) −ΞV (t), ∀t ∈ . Similar to the above argument, we can easily obtain uniform asymptotical stability of system (3). This completes the proof.…”

“…where x 1 and x 2 stand for the population (the density) of the preys and of the predators, respectively, p is the so-called predator functional response to predator and prey. In the last decades, the dynamical behaviors for the continuous-time Leslie predator-prey systems such as Hopf bifurcation [2,3], permanence [4], periodic solution [5,6], almost periodic solution [7,8], and stability [4], etc., have been widely investigated.…”

Dynamical analysis in a diffusive predator‐prey system with a delay and strong Allee effect

“…Assume that (H1) and (H2) are satisfied. Then there exists a positive integer N 0 such that equation (12) has a pair of purely imaginary roots ±i n when = n,j for n ∈ {0, 1, … , N 0 − 1}, where n,j is defined by (20).…”

“…From (20), it is obvious that { n, }| ∞ =0 is increasing on j for the fixed n. Then, n,0 = min ∈N 0 n, , where 0 ≤ n < N 0 . Let * = min{ n,0 , n = 0, 1, 2, … , N 0 − 1}.…”

“…8,17,18 In Wei and Wei, 19 they studied the delay feedback on the dynamics of chemical reaction models theoretically. In Du et al, 20 they investigated a Leslie-Gower predator-prey system with the feedback time delay in prey growth and a digestion time delay of the prey.…”