scholarly journals Patterns formed in a thin film with spatially homogeneous and non-homogeneous Derjaguin disjoining pressure

2021 ◽  
pp. 1-25
Author(s):  
ABDULWAHED S. ALSHAIKHI ◽  
MICHAEL GRINFELD ◽  
STEPHEN K. WILSON
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Bifurcation Theory ◽  
Disjoining Pressure ◽  
Thin Film Equations ◽  
Planar Substrate ◽  
Spatially Homogeneous ◽  
The One ◽  
Steady State Solutions ◽  
Average Film Thickness

We consider patterns formed in a two-dimensional thin film on a planar substrate with a Derjaguin disjoining pressure and periodic wettability stripes. We rigorously clarify some of the results obtained numerically by Honisch et al. [Langmuir 31: 10618–10631, 2015] and embed them in the general theory of thin-film equations. For the case of constant wettability, we elucidate the change in the global structure of branches of steady-state solutions as the average film thickness and the surface tension are varied. Specifically we find, by using methods of local bifurcation theory and the continuation software package AUTO, both nucleation and metastable regimes. We discuss admissible forms of spatially non-homogeneous disjoining pressure, arguing for a form that differs from the one used by Honisch et al., and study the dependence of the steady-state solutions on the wettability contrast in that case.

Bifurcation of steady-state solutions in predator-prey and competition systems

1984 ◽  
Vol 97 ◽  
pp. 21-34 ◽  
Author(s):  
J. Blat ◽  
K. J. Brown
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Existence Of Solutions ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Predator Prey ◽  
Interacting Species ◽  
Global Bifurcation Theory ◽  
Steady State Solutions

SynopsisWe discuss steady-state solutions of systems of semilinear reaction-diffusion equations which model situations in which two interacting species u and v inhabit the same bounded region. It is easy to find solutions to the systems such that either u or v is identically zero; such solutions correspond to the case where one of the species is extinct. By using decoupling and global bifurcation theory techniques, we prove the existence of solutions which are positive in both u and v corresponding to the case where the populations can co-exist.

SPATIAL DISORDER OF CNN — WITH ASYMMETRIC OUTPUT FUNCTION

2001 ◽  
Vol 11 (08) ◽  
pp. 2085-2095 ◽  
Author(s):  
JUNG-CHAO BAN ◽  
KAI-PING CHIEN ◽  
SONG-SUN LIN ◽  
CHENG-HSIUNG HSU
Keyword(s):  
Neural Networks ◽  
Steady State ◽  
Three Dimensional ◽  
Cellular Neural Networks ◽  
Output Function ◽  
One Dimensional ◽  
Spatial Entropy ◽  
The One ◽  
Steady State Solutions

This investigation will describe the spatial disorder of one-dimensional Cellular Neural Networks (CNN). The steady state solutions of the one-dimensional CNN can be replaced as an iteration map which is one dimensional under certain parameters. Then, the maps are chaotic and the spatial entropy of the steady state solutions is a three-dimensional devil-staircase like function.

scholarly journals Global Bifurcation Structure of a Predator-Prey System with a Spatial Degeneracy and B-D Functional Response

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Xiaozhou Feng ◽  
Changtong Li ◽  
Hao Sun ◽  
Yuzhen Wang
Keyword(s):  
Steady State ◽  
Functional Response ◽  
Bifurcation Theory ◽  
A Priori Estimates ◽  
Global Bifurcation ◽  
Local Bifurcation ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
Steady State Solutions

In this paper, we investigate a predator-prey system with Beddington–DeAngelis (B-D) functional response in a spatially degenerate heterogeneous environment. First, for the case of the weak growth rate on the prey ( λ 1 Ω < a < λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are established by the comparison principle; two local bifurcation solution branches depending on the bifurcation parameter are obtained by local bifurcation theory. Moreover, the demonstrated two local bifurcation solution branches can be extended to a bounded global bifurcation curve by the global bifurcation theory. Second, for the case of the strong growth rate on the prey ( a > λ 1 Ω 0 ), a priori estimates on any positive steady-state solutions are obtained by applying reduction to absurdity and the set of positive steady-state solutions forms an unbounded global bifurcation curve by the global bifurcation theory. In the end, discussions on the difference of the solution properties between the traditional predator-prey system and the predator-prey system with a spatial degeneracy and B-D functional response are addressed.

Transient Thin-Film Flow on a Moving Boundary of Arbitrary Topography

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
Roger E. Khayat ◽  
Tauqeer Muhammad
Keyword(s):  
Thin Film ◽  
Steady State ◽  
Moving Boundary ◽  
Initial Conditions ◽  
Flow Behavior ◽  
Galerkin Projection ◽  
Thin Film Flow ◽  
Flat Substrate ◽  
Thin Film Equations ◽  
Two Dimensional Flow

The transient two-dimensional flow of a thin Newtonian fluid film over a moving substrate of arbitrary shape is examined in this theoretical study. The interplay among inertia, initial conditions, substrate speed, and shape is examined for a fluid emerging from a channel, wherein Couette–Poiseuille conditions are assumed to prevail. The flow is dictated by the thin-film equations of the “boundary layer” type, which are solved by expanding the flow field in terms of orthonormal modes depthwise and using the Galerkin projection method. Both transient and steady-state flows are investigated. Substrate movement is found to have a significant effect on the flow behavior. Initial conditions, decreasing with distance downstream, give rise to the formation of a wave that propagates with time and results in a shocklike structure (formation of a gradient catastrophe) in the flow. In this study, the substrate movement is found to delay shock formation. It is also found that there exists a critical substrate velocity at which the shock is permanently obliterated. Two substrate geometries are considered. For a continuous sinusoidal substrate, the disturbances induced by its movement prohibit the steady-state conditions from being achieved. However, for the case of a flat substrate with a bump, a steady state exists.

Properties of steady states for thin film equations

2000 ◽  
Vol 11 (3) ◽  
pp. 293-351 ◽  
Author(s):  
R. S. LAUGESEN ◽  
M. C. PUGH
Keyword(s):  
Contact Angle ◽  
Steady State ◽  
Blow Up ◽  
Contact Angles ◽  
Steady States ◽  
Period Function ◽  
Scaling Properties ◽  
Thin Film Equations ◽  
Monotonicity Result ◽  
Steady State Solutions

We consider nonnegative steady-state solutions of the evolution equationformula hereOur class of coefficients f, g allows degeneracies at h = 0, such as f(0) = 0, as well as divergences like g(0) = ±∞. We first construct steady states and study their regularity. For f, g > 0 we construct positive periodic steady states, and non-negative steady states with either zero or nonzero contact angles. For f > 0 and g < 0, we prove there are no non-constant positive periodic steady states or steady states with zero contact angle, but we do construct non-negative steady states with nonzero contact angle. In considering the volume, length (or period) and contact angle of the steady states, we find a rescaling identity that enables us to answer questions such as whether a steady state is uniquely determined by its volume and contact angle. Our tools include an improved monotonicity result for the period function of the nonlinear oscillator. We also relate the steady states and their scaling properties to a recent blow-up conjecture of Bertozzi and Pugh.

STABILITY OF STEADY-STATE SOLUTIONS IN MULTIDIMENSIONAL COUPLED MAP LATTICES

2004 ◽  
Vol 14 (08) ◽  
pp. 2689-2699
Author(s):  
MIROSLAVA DUBCOVÁ ◽  
DANIEL TURZÍK ◽  
ALOIS KLÍČ
Keyword(s):  
Steady State ◽  
Coupled Map Lattices ◽  
Multidimensional Lattice ◽  
Spatially Periodic ◽  
Spatially Homogeneous ◽  
Periodic Steady State ◽  
The Stability ◽  
Linearized Operator ◽  
Steady State Solutions

Coupled map lattices with multidimensional lattice are considered. A method for the determination of the stability of spatially homogeneous and spatially periodic steady-state solutions is derived. This method is based on the determination of the spectrum of the linearized operator by means of Gelfand transformation of some appropriate Banach algebra. The results are applied to several examples.

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

2021 ◽  
Author(s):  
Haixia Li ◽  
Wenbin Yang ◽  
Meihua Wei ◽  
Aili Wang
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
A Priori ◽  
Hopf Bifurcations ◽  
Predator Prey ◽  
Positive Steady State ◽  
Prey System ◽  
A Priori Bound ◽  
Steady State Solutions

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

MULTIPLE BIFURCATION ANALYSIS AND SPATIOTEMPORAL PATTERNS IN A 1-D GIERER–MEINHARDT MODEL OF MORPHOGENESIS

2010 ◽  
Vol 20 (04) ◽  
pp. 1007-1025 ◽  
Author(s):  
JIANXIN LIU ◽  
FENGQI YI ◽  
JUNJIE WEI
Keyword(s):  
Steady State ◽  
Bifurcation Analysis ◽  
Reaction Diffusion ◽  
Spatiotemporal Patterns ◽  
Fixed Boundary ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The One ◽  
Steady State Solutions ◽  
Fixed Boundary Condition

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis.

Sign in / Sign up

Export Citation Format

Share Document