Delay-Induced Triple-Zero Bifurcation in a Delayed Leslie-Type Predator–Prey Model with Additive Allee Effect

Abstract

In this paper, a Leslie-type predator–prey model with ratio-dependent functional response and Allee effect on prey is considered. We first study the existence of the multiple positive equilibria and their stability. Then we investigate the effect of delay on the distribution of the roots of characteristic equation and obtain the conditions for the occurrence of simple-zero, double-zero and triple-zero singularities. The formulations for calculating the normal form of the triple-zero bifurcation of the delay differential equations are derived. We show that, under certain conditions on the parameters, the system exhibits homoclinic orbit, heteroclinic orbit and periodic orbit.

A New Chaotic System with Only Nonhyperbolic Equilibrium Points: Dynamics and Its Engineering Application

“…Also, the corresponding eigenspace gives rise to a slow manifold. In Jiang et al [50], Gamero et al [51], and Freire et al [52], some methods have been suggested to derive the normal forms on the center manifold, which can be used to study the dynamics near triple zero eigenvalues. e stability of this equilibrium point cannot be determined under the Shilnikov criteria.…”

Dynamics of a Predator–Prey Model with the Additive Predation in Prey

“…Here we only consider real solutions. For the solution z t ∈ C 0 of Equation (28), when µ = 0, we have Ż(t) = iw 0 τZ + q * (θ), f (0, W(Z, Z, θ) + 2 {Zq(θ)}) = iw 0 τZ + q * (0) f (0, W(Z, Z, 0) + 2 {Zq(0)}) = iw 0 τZ + q * (0) f 0 (Z, Z) = iw 0 τZ + g(Z, Z), where g(Z, Z) = q * (0 (32), we have z t (θ) = (z 1t (θ), z 2t (θ)) T = W(t, θ) + Zq(θ) + Z q(θ). Combined with the definitions of W and q(θ), the following expressions can be obtained…”

“…See, for example [2,12,[17][18][19][20][21][22][23]. The models with a component Allee effect, of a type similar to (1), was first mentioned by Kostitizin [24] and applied in [2,11,[25][26][27][28][29][30][31][32] .…”

Existence of positive periodic solutions for a predator-prey model

“…By using the iterative technique and further precise analysis, sufficient conditions on the global attractivity of a positive equilibrium for a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge were obtained [4]. On the other hand, many researchers have considered delayed prey and predator models [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. For example, Nindjin et al have discussed the following delayed predator-prey model [5]:…”