Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering two different prey behaviours’ transition

• Published in: Nonlinear dynamics Vol. 107; no. 1; pp. 1357 - 1381
• Main Author: Lv, Yehu
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.01.2022; Springer Nature B.V
• Subjects: Automotive Engineering; Behavior; Boundary conditions; Canonical forms; Classical Mechanics; Control; Defense; Diffusion; Dynamical Systems; Eigenvalues; Eigenvectors; Engineering; Equilibrium; Hopf bifurcation; Mathematical models; Mechanical Engineering; Original Paper; Partial differential equations; Population; Predator-prey simulation; Predators; Stability analysis; Vibration; Cross-diffusion; Individual behaviour; Turing–Hopf bifurcation; analysis on manifolds; 92 Biology and other natural sciences; Herd behaviour; 35 Partial differential equations; 58 Global analysis; Predator–prey model; Spatially inhomogeneous periodic solution; Self-diffusion
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-021-07058-y

In this paper, we study the Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analysing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by calculating and analysing the normal form on the centre manifold associated with the Turing–Hopf bifurcation, we investigate the dynamical classification near the Turing–Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.