Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect

Abstract

We investigate a diffusive predator-prey model by incorporating the fear effect into prey population, since the fear of predators could visibly reduce the reproduction of prey. By introducing the mature delay as bifurcation parameter, we find this makes the predator-prey system more complicated and usually induces Hopf and Hopf-Hopf bifurcations. The formulas determining the properties of Hopf and Hopf-Hopf bifurcations by computing the normal form on the center manifold are given. Near the Hopf-Hopf bifurcation point we give the detailed bifurcation set by investigating the universal unfoldings. Moreover, we show the existence of quasi-periodic orbits on three-torus near a Hopf-Hopf bifurcation point, leading to a strange attractor when further varying the parameter. We also find the existence of Bautin bifurcation numerically, then simulate the coexistence of stable constant stationary solution and periodic solution near this Bautin bifurcation point.

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

“…Previous studies have shown that in the predator-prey reaction-diffusion model, some equations have a double Hopf bifurcation, and some equations have a Turing-Hopf bifurcation [1,2,13]. However, in our paper, for this model, changing the values of time delay and diffusion would lead to the emergence of Turing-Hopf bifurcation of the system, while changing the values of time delay and growth rate would lead to the generation of double Hopf bifurcation.…”

“…Using the similar method in [11], we can easily show that the usual normal form of the original system (2) has the same form as (20), where ρ 1 > 0 and ρ 2 > 0. The normal form of the system (2) could be used to reflect the dynamic behavior near the bifurcation point, and the corresponding calculation steps can be referred to [11,13]. Numerical simulations show that the system exhibits periodic motions, quasi-periodic motions, and chaotic motions.…”

“…Lemma 5.1. For n 1 = 0, n 2 = 0, the coefficients A 11 11 , A 11 21 , A 13 12 , A 13 22 , B 11 210 , B 11 102 , B 13 111 , and B 13 003 of ( 19) are as follows. A 11 11 = 2ψ 1 (0) L 1 φ 1 (0) + L 2 φ 1 (−1) , A 11 21 = 0,…”

Further Study on Dynamics for a Fractional‐Order Competitor‐Competitor‐Mutualist Lotka–Volterra System

“…Zhang et al [7] revealed the effect of the fear factor on the periodic solution of a prey-predator model. As for concrete works, we refer the readers to [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Bifurcation and optimal control analysis of a delayed drinking model

“…From another perspective, in comparison with ordinary differential equations, delayed differential equations present more complex dynamics. In fact, delayed differential equations have been applied in various fields, for example, mathematical biology [27][28][29][30], epidemiology [31][32][33], and computer virus [34][35][36][37]. All the work above revealed that delay can influence dynamics of the systems significantly.…”