Bogdanov–Takens bifurcation in a predator–prey model

Abstract

Abstract. In this paper, we investigate a class of predator-prey model with age structure and discuss whether the model can undergo Bogdanov-Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator-prey model has an unique positive equilibrium which is Bogdanov-Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov-Takens bifurcation in a small neighborhood of this positive equilibrium.

Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge

“…Similar conclusions were also obtained in the virus model, epidemic model, consumer-resource symbiosis model, alcoholism model and other models with age structure [2,5,7,8,20,22,38,46,47]. We also refer to [17,21,23,43] for the studies on the Bogdanov-Takens bifurcation and Zero-Hopf bifurcation problems of relevant models.…”

Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition

“…Via following the algorithm, they also derived the third-order normal form of Bogdanov-Takens bifurcation for Van der Pol's oscillator with delayed feedback, and then investigated the corresponding dynamics and found homoclinic orbits, heteroclinic orbits and large stable limit cycle. However, there are only a few related researches on Bogdanov-Takens bifurcation of partial (functional) differential equations [16,32,42]. For example, Faria [16] analogously developed the recursive algorithm for calculating the normal form of semilinear functional differential equation, and also discussed the calculations for normal form of Bogdanov-Takens bifurcation.…”

“…For example, Faria [16] analogously developed the recursive algorithm for calculating the normal form of semilinear functional differential equation, and also discussed the calculations for normal form of Bogdanov-Takens bifurcation. Lately, Liu, Magal and Xiao [32] also analyzed Bogdanov-Takens bifurcation of a class of predator-prey model with age structure, by utilizing normal form method and center manifold theory. Generally speaking, one of the possible reasons for the lack of discusses on Bogdanov-Takens bifurcation of partial (functional) differential equations might be that the associated bifurcation analysis and the calculations of normal form for partial (functional) differential equations are much more complicated, and another major one may be that spatiotemporal solutions arising from Bogdanov-Takens bifurcation of reaction-diffusion models involving only local interactions and Neumann boundary conditions are generally unstable [32,42].…”

“…Lately, Liu, Magal and Xiao [32] also analyzed Bogdanov-Takens bifurcation of a class of predator-prey model with age structure, by utilizing normal form method and center manifold theory. Generally speaking, one of the possible reasons for the lack of discusses on Bogdanov-Takens bifurcation of partial (functional) differential equations might be that the associated bifurcation analysis and the calculations of normal form for partial (functional) differential equations are much more complicated, and another major one may be that spatiotemporal solutions arising from Bogdanov-Takens bifurcation of reaction-diffusion models involving only local interactions and Neumann boundary conditions are generally unstable [32,42]. On the other hand, some recent researches [8,11,33,39] have already shown that nonlocal interaction can stabilize spatiotemporal solutions to a certain extent.…”

Formulation of the normal forms of Turing-Hopf bifurcation in reaction-diffusion systems with time delay

“…For example, the existence of Hopf bifurcations and associated stability switches have been considered in many recent work [6, 10, 22-24, 44, 53, 54, 66, 68, 69, 71]. More recently with the integrated semigroup theory, the center manifold and normal form theory for semilinear equations with non-dense domain have also been developed [35][36][37][38].…”