Nonconstant Solutions Research Articles

Qualitative analysis of a diffusive predator–prey model with Allee and fear effects

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

Complex dynamics of a tumor-immune system with antigenicity

With the effect of antigenicity in consideration, we propose and analyze a conceptual model for the tumor-immune interaction. The model is described by a system of two ordinary differential equations. Though simple, the model can have complicated dynamical behaviors. Besides the tumor-free equilibrium, there can be up to three tumor-present equilibria, which can be a saddle or stable node/focus. Sufficient conditions on the nonexistence of nonconstant periodic solutions are provided. Bifurcation analysis including Hopf bifurcation and Bogdanov–Takens bifurcation is carried out. The theoretical results are supported by numerical simulations. Numerical simulations reveal the complexity of the dynamical behaviors of the model, which includes the subcritical/supercritical Hopf bifurcation, homoclinic bifurcation, saddle-node bifurcation at a nonhyperbolic periodic orbit, the appearance of two limit cycles with a singular closed orbit, and so on. Some biological implications of the theoretical results and numerical simulations are also provided.

Moving planes for domain walls in a coupled system

The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized to hyperfine spin states with a Rabi term coupling. This leads to domain wall solutions having a monotone structure for a non-cooperative system. We use the moving plane method to prove monotonicity and one-dimensionality of the phase transition solutions. This relies on totally new estimates for a type of system for which no Maximum Principle a priori holds. We also derive that one dimensional solutions are unique up to translations. When the Rabi coefficient is large, we prove that no non-constant solutions can exist.

Non-existence of patterns and gradient estimates in semilinear elliptic equations with Neumann boundary conditions

We call pattern any non-constant solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [20] and Matano [50] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the stability of patterns, and the C1 norm of the nonlinearity.In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincaré inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates.Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions.Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [54] and to extend some results on De Giorgi's conjecture to a larger class of domains.

Uniqueness and stability of activated dislocation shapes in crystals

Simplified models of thermally activated dislocation glide constitute an important link between atomic-level studies of isolated dislocations and macroscopic thermodynamic properties of materials. These models rest upon the activation enthalpy, which is the energy to transform an initially straight dislocation into its activated state at finite applied stresses. Minimizing this activation enthalpy leads to a boundary value problem for the shape of the dislocation line. Besides two constant solutions corresponding to a straight dislocation in its stable and unstable states at the applied stress, there exist an infinite number of non-constant solutions. We investigate the characters of these solutions for dislocations anchored at their ends. Using the second variation of the activation enthalpy, we derive a set of conditions that define a unique activated state of the dislocation. The corresponding analysis demonstrates that the shape of the dislocation in this activated state must change with the applied stress to maintain the state of minimum activation enthalpy.

Stability of Non-constant Equilibrium Solutions for the Full Compressible Navier–Stokes–Maxwell System

In this article we consider a Cauchy problem for the full compressible Navier–Stokes–Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic–parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier–Stokes–Poisson system in a three-dimensional torus.

Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establisha prioriestimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

Bifurcation and pattern formation in diffusive Klausmeier-Gray-Scott model of water-plant interaction

A reaction-diffusion model describing water and plant interaction proposed by Klausmeier is studied. The existence of non-constant steady state solutions is shown through bifurcation methods, and the existence of large amplitude spatial patterned solutions is proved using associated shadow system. It is rigorously shown that non-homogeneous patterned vegetation states exist when the rainfall is at a lower level in which homogeneous vegetation state cannot survive. Even when the rainfall is very low, slow plant diffusion and fast water diffusion can support a vegetation state with vegetation concentrating on a small area. This provides an example of diffusion-induced persistence that non-constant steady states may exist in a reaction-diffusion system when there are no positive constant steady states.

Further study on the Brück conjecture and some non-linear complex differential equations

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramaniket al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex numberain the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Letfbe a non-constant entire function such thatσ2(f)<∞,σ2(f)is not a positive integer andδ(0, f)>0. LetM[f]be a differential monomial offof degreeγMandα(z), β(z)∈S(f)be such thatmax{σ(α), σ(β)} <σ(f). IfM[f]+βandfγM−αshare the value 0 CM, thenM[f]+βfγM−α=c,wherec≠0is a constant.Originality/valueThis is an original work of the authors.

Analysis on a diffusive multiple Allee effects predator–prey model induced by fear factors

This paper is concerned with a diffusive predator–prey system with multiple Allee effects induced by fear factors subject to Neumann boundary conditions. Firstly, we introduce a priori estimates of non-negative and non-trivial solutions. Moreover, nonexistence of non-constant positive solutions are shown for certain parameters ranges. Secondly, we study the stability of non-negative constant solutions by the linearized theory. Then, by choosing the Allee threshold θ as the bifurcation parameter, we investigate the spatially homogeneous and non-homogeneous periodic solutions as well as non-constant steady-state solutions. Finally, some theoretical results are illustrated by numerical simulations.

Global Bifurcation of Stationary Solutions for a Volume-Filling Chemotaxis Model with Logistic Growth

In this paper, we show how the global bifurcation theory for nonlinear Fredholm operators (Theorem 4.3 of [Shi & Wang, 2009]) and for compact operators (Theorem 1.3 of [Rabinowitz, 1971]) can be used in the study of the nonconstant stationary solutions for a volume-filling chemotaxis model with logistic growth under Neumann boundary conditions. Our results show that infinitely many local branches of nonconstant solutions bifurcate from the positive constant solution [Formula: see text] at [Formula: see text]. Moreover, for each [Formula: see text], we prove that each [Formula: see text] can be extended into a global curve, and the projection of the bifurcation curve [Formula: see text] onto the [Formula: see text]-axis contains [Formula: see text].

Pattern Dynamics in a Diffusive Gierer–Meinhardt Model

The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

Connected sets of solutions of symmetric elliptic systems

The purpose of this paper is twofold. First we study bifurcations of connected sets of critical orbits of some invariant functional from a given family of critical orbits. We use techniques of equivariant bifurcation theory to obtain a Rabinowitz type alternative for symmetric gradient operators. The second aim is to apply the abstract results to study orbits of nonconstant solutions of a nonlinear Neumann problem.

Further study on the Brück conjecture and some non-linear complex differential equations

The current paper deals with the study of nonconstant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. Our results generalize the results due to Pramanik et al. (2017).

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

In this paper we study the Degn–Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behavior of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincaré–Bendixson theorem and the direct Lyapunov method. Numerical simulations are also shown to corroborate the asymptotic stability predictions.Moreover, we determine the constraints on the size of the reactor and the diffusion coefficient such that the system does not admit non-constant positive steady state solutions.

Global stability of nonhomogeneous equilibrium solution for the diffusive Lotka–Volterra competition model

A diffusive Lotka-Volterra competition model is considered for the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion. First it is shown that in a two-species system in which the diffusion coefficients, resource functions and competition rates are all spatially heterogeneous, the positive equilibrium solution is globally asymptotically stable when it exists. Secondly the existence and global asymptotic stability of the positive and semi-trivial equilibrium solutions are obtained for the model with arbitrary number of species under the assumption of spatially heterogeneous resource distribution. A new Lyapunov functional method is developed to prove the global stability of a non-constant equilibrium solution in heterogeneous environment.

A construction of patterns with many critical points on topological tori

We consider reaction–diffusion equations on closed surfaces in \({{\mathbb {R}}}^3\) having genus 1. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants.

Dynamics of Holling-type II prey–predator system with a protection zone for prey

ABSTRACT In this paper, a diffusive predator–prey model with Holling type II (Michaelis–Menten) functional response and a protection zone for prey is investigated. Dynamics and steady state solutions of the system are analyzed. We give the a priori estimates and obtain the nonexistence of nonconstant positive solution as the diffusion coefficients are large enough. Moreover, we demonstrate the existence and stability of nonconstant steady state solutions branching from constant semi-trivial solution by using bifurcation theory.

On the Functional Inequality $$f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y)$$

We prove that all solutions $$f: {\mathbb {R}} \rightarrow {\mathbb {R}}$$ of the functional inequality $$\begin{aligned} (*) \quad f(x)f(y)-f(xy)\le f(x)+f(y)-f(x+y), \end{aligned}$$ which are convex or concave on $${\mathbb {R}}$$ and differentiable at 0 are given by $$\begin{aligned} f(x)=x\quad \text{ and } \quad f(x)\equiv c, \quad \text{ where } \quad 0\le c\le 2. \end{aligned}$$ Moreover, we show that the only non-constant solution $$f: {\mathbb {R}} \rightarrow {\mathbb {R}}$$ of $$(*)$$ , which is continuous on $${\mathbb {R}}$$ and differentiable at 0 with $$f(0)=0$$ is $$f(x)=x$$ .

Stationary solutions of advective Lotka–Volterra models with a weak Allee effect and large diffusion

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint) which stems from the original model as diffusion and advection of one of the species tend to infinity. Finally, we classify the global bifurcation structure of nonconstant solutions of the simplified 1D case.