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In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a(8,0)-mode Turing–Hopf bifurcationand unstable for a(3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a(0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.
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Stability and bifurcation of dynamic contact lines in two dimensions
The moving contact line between a fluid, liquid and solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models, previous studies have shown that the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number, $$Ca_{{crit}}$$ , above which no steady-state solution can be found. Below $$Ca_{{crit}}$$ , both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against $Ca$ , a fold bifurcation appears where the stable and unstable branches meet. Interestingly, the significance of this bifurcation structure to the transient dynamics has yet to be explored. This article develops a computational model and uses ideas from dynamical systems theory to show the profound importance of the unstable solutions on the transient behaviour. By perturbing the stable state by the eigenmodes calculated from a linear stability analysis it is shown that the unstable branch is an ‘edge’ state that is responsible for the eventual dynamical outcomes and that the system can become transient when $$Ca< Ca_{{crit}}$$ due to finite-amplitude perturbations. Furthermore, when $$Ca>Ca_{{crit}}$$ , we show that the trajectories in phase space closely follow the unstable branch.
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- Award ID(s):
- 1935968
- PAR ID:
- 10412211
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 945
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2022.526
