Abstract
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.
Highlights
Since the groundbreaking works of Lotka [1] and Volterra [2], the predator-prey model is used to describe the dynamical interaction between two species and has been widely researched by many scholars in the fields of biology and mathematics [3, 4]
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
By computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail
Summary
Since the groundbreaking works of Lotka [1] and Volterra [2], the predator-prey model is used to describe the dynamical interaction between two species and has been widely researched by many scholars in the fields of biology and mathematics [3, 4]. Based on the above facts, the authors in [32] have proposed a modified predator-prey model with herd behavior described by the following ordinary differential equations ru u K. where r is the intrinsic growth rates of the prey, K is the carrying capacity of the prey, a is the maximum value of prey consumed by per predator per unit time, h is a threshold for the transition between herd grouping and solitary behaviour, m is the death rate of the predator in the absence of prey, e is the conversion or consumption rate of prey to predator. The existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion is researched.
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