Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering two different prey behaviours’ transition
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 solutio...
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| Published in | Nonlinear dynamics Vol. 107; no. 1; pp. 1357 - 1381 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.01.2022
Springer Nature B.V |
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| Online Access | Get full text |
| ISSN | 0924-090X 1573-269X |
| DOI | 10.1007/s11071-021-07058-y |
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| 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. |
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| AbstractList | 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. |
| Author | Lv, Yehu |
| Author_xml | – sequence: 1 givenname: Yehu orcidid: 0000-0002-7824-9557 surname: Lv fullname: Lv, Yehu email: mathyehul@163.com organization: School of Mathematical Sciences, Beijing Normal University |
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| CitedBy_id | crossref_primary_10_1103_PhysRevE_109_024214 crossref_primary_10_1007_s00033_023_02167_7 crossref_primary_10_1016_j_chaos_2024_115375 crossref_primary_10_2139_ssrn_4199142 crossref_primary_10_2139_ssrn_4153589 crossref_primary_10_1007_s12190_024_02183_4 crossref_primary_10_3934_era_2023229 crossref_primary_10_1002_mma_10480 crossref_primary_10_1016_j_chaos_2022_112659 crossref_primary_10_1140_epjp_s13360_024_05862_0 crossref_primary_10_1063_5_0157354 |
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| Keywords | 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 |
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| SubjectTerms | 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 |
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| Title | Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering two different prey behaviours’ transition |
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