Non-constant Equilibrium Solutions Research Articles

A model of the burglar alarm hypothesis of prey alarm calls

When approached by a predator many prey species will emit an “alarm call” as a form of anti-predator behavior. One hypothesis for the function of alarm calls is the “burglar alarm” hypothesis whereby upon attack, a prey renders itself dangerous to a predator by generating an alarm call that attracts a predator at higher trophic levels in the food chain; that is, attracts a predator to the prey’s own predator. This paper concerns a model incorporating a mechanism to test the burglar alarm hypothesis. We prove in one space dimension global existence, of positive bounded classical solutions, and establish existence of non-constant equilibrium solutions and assess their stability. We provide some representative numerical simulations to emphasize the nature of pattern formation for this model and demonstrate the benefit achieved by a signal inducing prey species under the burglar alarm hypothesis.

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.

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.

Exploring Complex Dynamics of Spatial Predator–Prey System: Role of Predator Interference and Additional Food

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots[Formula: see text][Formula: see text][Formula: see text]stripes[Formula: see text][Formula: see text][Formula: see text]spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

Transport and concentration of wealth: Modeling an amenities-based-theory.

We derive a reaction-advection-diffusion equation based framework for analyzing the movement of wealth in urban environments. Gentrification is a core issue that affects many urban areas, and the dynamics of such are not fully understood. To understand the process using a few physically relevant variables, we develop an approach to model the interplay between local amenities and wealth from existing analyses of factors influencing gentrification. We conduct a linear stability analysis on model parameters that results in spatially homogeneous solutions and determine directions for parameter changes to induce instabilities. From these parameters, we determine quantities that lead to the formation of areas of wealth and amenity concentration in the long term on bounded and unbounded domains. We present a global bifurcation result in two dimensions, leading to the existence and stability of non-constant equilibrium solutions, which represents solutions with wealth and amenity hotspots. Finally, we present a theory for discerning the one-dimensional pattern formation in the transition between stability and instability and verify numerically through a weakly nonlinear analysis. The analysis provides a promising framework for further verification using publicly available geospatial data on the relevant variables.

Stability of Non-constant Equilibrium Solutions for Compressible Viscous and Diffusive MHD Equations with the Coulomb Force

We consider stability problems for the compressible viscous and diffusive magnetohydrodynamic (MHD) equations arising in the modeling of magnetic field confinement nuclear fusion. In the first part, we investigate the Cauchy problem to the barotropic MHD system. With the help of the techniques of anti-symmetric matrix and an induction argument on the order of the space derivatives of solutions in energy estimates, we prove that smooth solutions exist globally in time near the non-constant equilibrium solutions. We also obtain the asymptotic behavior of solutions when the time goes to infinity. The result shows that gradients of both the velocity and the magnetic field converge to the equilibrium solutions with the same norm $$ \Vert \cdot \Vert _{H^{s-3}}$$, while the density converge with stronger norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. In the second part, the initial value problem to the full MHD system is studied. By means of the techniques of choosing a non-diagonal symmetrizer and elaborate energy estimates, we prove the existence and uniqueness of global solutions to the system when the initial data are close to the non-constant equilibrium states. We find that both the density and temperature converge to the equilibrium states with the same norm $$ \Vert \cdot \Vert _{H^{s-1}}$$. These phenomena on the charge transport show the essential relationship of the equations between the barotropic and the full MHD systems.

Stability of non-constant equilibrium solutions for two-fluid non-isentropic Euler-Maxwell systems arising in plasmas

We consider the periodic problem for two-fluid non-isentropic Euler-Maxwell systems in plasmas. By means of suitable choices of symmetrizers and an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude is established near a non-constant equilibrium solution with asymptotic stability properties. This improves the results obtained in the work of Li et al. [Z. Angew. Math. Phys. 67(5), 133 (2016)] for models with temperature diffusion terms by using the pressure functions pν in place of the unknown variable densities nν.

Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus $$\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3$$ . This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.

Shadow system approach to a plankton model generating harmful algal bloom

Spatially localized blooms of toxic plankton species have negative impacts on other organisms via the production of toxins, mechanical damage, or by other means. Such blooms are nowadays a worldwide spread environmental issue. To understand the mechanism behind this phenomenon, a two-prey (toxic and nontoxic phytoplankton)-one-predator (zooplankton) Lotka-Volterra system with diffusion has been considered in a previous paper. Numerical results suggest the occurrence of stable non-constant equilibrium solutions, that is, spatially localized blooms of the toxic prey. Such blooms appear for intermediate values of the rate of toxicity $μ$ when the ratio $D$ of the diffusion rates of the predator and the two prey is rather large. In this paper, we consider a one-dimensional limiting system (we call it a shadow system) in $(0,L)$ as $D \to \infty $ and discuss the existence and stability of non-constant equilibrium solutions with large amplitude when $μ$ is globally varied. We also show that the structure of non-constant equilibrium solutions sensitively depends on $L$ as well as $μ$.

Bifurcation of stable equilibria under nonlinear flux boundary condition with null average weight

The bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described. In a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria. Herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case. For each value of the parameter we prove existence of an unique nonconstant equilibrium solution. It belongs to a globally parametrized bifurcation curve constructed using a Lyapunov–Schmidt type reduction combined with the Morse lemma. That solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi-orbits. The behavior of its trace, when the parameter is large, is established and complete diagrams containing all bifurcation and stability information are also provided.

A reaction-diffusion system and its shadow system describing harmful algal blooms

The occurrence of harmful algal blooms (HAB) in ecosystems is a worldwide environmental issue that currently needs to be addressed. An attempt to theoretically understand the mechanism behind the formation of HAB has led to the proposal of a reaction-diffusion model of the Lotka--Volterra type. In particular, a shadow system, as a limiting system of the model in which the diffusion rate tends to infinity, has been proposed to study whether or not stable nonconstant equilibrium solutions of the system exist, because these solutions are mathematically associated with HAB. In this paper, we discuss the convergence property between solutions of the full system and its shadow system from the point of view of an evolutional problem.

Stability of non-constant equilibrium solutions for two-fluid Euler–Maxwell systems

In this work we consider periodic problems for two-fluid compressible Euler–Maxwell systems for plasmas. The initial data are supposed to be in a neighborhood of non-constant equilibrium states. Mainly by an induction argument used in Peng (2015), we prove the global stability in the sense that smooth solutions exist globally in time and converge to the equilibrium states as the time goes to infinity. Moreover, we obtain the global stability of solutions with exponential decay in time near the equilibrium states for two-fluid compressible Euler–Poisson systems.

Stability of non-constant equilibrium solutions for Euler–Maxwell equations

We consider a periodic problem for compressible Euler–Maxwell equations arising in the modeling of magnetized plasmas. The equations are quasilinear hyperbolic and partially dissipative. It is proved that smooth solutions exist globally in time and converge toward non-constant equilibrium states as the time goes to infinity. This result is obtained for initial data close to the equilibrium states with zero velocity. The proof is based on an induction argument on the order of the derivatives of solutions in energy and time dissipation estimates. We also show the global stability with exponential decay in time of solutions near the equilibrium states for compressible Euler–Poisson equations.

Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation

We are dealing with a two-component system of reaction–diffusion equations with mass conservation in a bounded domain with the Neumann boundary conditions. We prove the global boundedness of the solution in L∞-norm for t⩾0 under a condition, and then the existence of a Lyapunov function. Moreover, by studying the linearized eigenvalue problem of a nonconstant equilibrium solution, we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation. As an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one. It is also shown that a modified system with a new parameter, which covers the present model, possesses a Lyapunov function.

Infinite dimensional relaxation oscillation in aggregation-growth systems

Two types of aggregation systems with Fisher-KPP growth are proposed.One is described by a normal reaction-diffusion system, and the other isdescribed by a cross-diffusion system. If the growth effect is dominant, aspatially constant equilibrium solution is stable. When the growth effectbecomes weaker and the aggregation effect become dominant, the solutionis destabilized so that spatially non-constant equilibrium solutions, whichexhibit Turing's patterns, appear. When the growth effect weakens further,the spatially non-constant equilibrium solutions are destabilized throughHopf bifurcation, so that oscillatory Turing's patterns appear. Finally, whenthe growth effect is extremely weak, there appear spatio-temporal periodicsolutions exhibiting infinite dimensional relaxation oscillation.

Stability of the Stochastic Reaction‐Diffusion Neural Network with Time‐Varying Delays and p‐Laplacian

The main aim of this paper is to discuss moment exponential stability for a stochastic reaction‐diffusion neural network with time‐varying delays and p‐Laplacian. Using the Itô formula, a delay differential inequality and the characteristics of the neural network, the algebraic conditions for the moment exponential stability of the nonconstant equilibrium solution are derived. An example is also given for illustration.

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henryʼs approach to the problem. Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided. The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.

Qualitative analysis of a strongly coupled prey-predator model

This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non-existence of the non-constant positive solutions are obtained. Moreover, we prove that the nonlinear diffusion terms can create non-constant positive equilibrium solutions when the corresponding model without nonlinear diffusion term fails.

Stability and bifurcation of nonconstant solutions to a reaction–diffusion system with conservation of mass

We deal with a two-component system of reaction–diffusion equations with conservation of mass in a bounded domain with the Neumann or periodic boundary conditions. This system is proposed as a conceptual model for cell polarity. Since the system has conservation of mass, the steady state problem is reduced to that of a scalar reaction–diffusion equation with a nonlocal term. That is, there is a one-to-one correspondence between an equilibrium solution of the system with a fixed mass and a solution of the scalar equation. In particular, we consider the case when the reaction term is linear in one variable. Then the equations are transformed into the same equations as the phase-field model for solidification. We thereby show that the equations allow a Lyapunov function. Moreover, by investigating the linearized stability of a nonconstant equilibrium solution, we prove that given a nondegenerate stable equilibrium solution of the nonlocal scalar equation, the corresponding equilibrium solution of the system is stable. We also exhibit global bifurcation diagrams for equilibrium solutions to specific model equations by numerics together with a normal form near a bifurcation point.

Dynamics of solution for a class of delayed diffusive neural networks with mixed boundary conditions

In this paper, we investigate a class of cellular neural networks model with delays and diffusive terms. By using the method of upper and lower solutions, we obtain that if the neuronal output signal functions in system possess mixed quasimonotone property and the corresponding elliptic system has upper and lower solutions the model has a unique nonconstant equilibrium solution. Under some additional conditions we further obtain that the solution of the neural networks converges to this nonconstant equilibrium solution.