Constant Equilibrium Solution Research Articles

Global asymptotical stability and Hopf bifurcation for a three‐species Lotka‐Volterra food web model

In this article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Bifurcation Solutions to the Templator Model in Chemical Self-Replication

In this paper, we are concerned with a diffusive templator model in chemical self-replication, which describes the process of an individual molecule duplicating itself. Firstly, the stability of non-negative constant equilibrium solution is introduced. Then the existence of Hopf bifurcation is proved. Particularly, the stability and the direction of Hopf bifurcation for the spatially homogeneous model are discussed. Furthermore, by space decomposition and implicit function theorem, it is shown that the system may undergo a steady-state bifurcation with a two-dimensional kernel. Finally, several numerical simulations are completed to demonstrate the theoretical results.

The FitzHugh–Nagumo Model Described by Fractional Difference Equations: Stability and Numerical Simulation

The aim of this work is to describe the dynamics of a discrete fractional-order reaction–diffusion FitzHugh–Nagumo model. We established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employed a Lyapunov functional to show that the constant equilibrium solution is globally asymptotically stable. Furthermore, numerical simulations are shown to clarify and exemplify the theoretical results.

Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Stability and Pattern Formation in a General Class of Reaction-Diffusion-Advection System

In this paper, we systematically study two-species reaction-diffusion-advection system with linear cross-diffusion and cross-advection. Firstly, we provide sufficient conditions for cross-diffusion, self-advection and cross-advection driven instability, which implies that cross-diffusion, self-advection and cross-advection can give rise to pattern formation for the same diffusion coefficients. Secondly, we focuses on a class of general reaction-diffusion-advection system. By investigating the linearized stability of the constant equilibrium solution, we prove that the self-diffusion and self-advection terms have no effect on the stabilization of the constant steady state, the linear cross terms favor the destabilization of the constant steady state and mechanism of pattern formation. Furthermore, the theoretical results are applied to predator-prey and water-vegetation systems with cross-diffusion and cross-advection.

On Fractional-Order Discrete-Time Reaction Diffusion Systems

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type of system. However, analyzing these systems using discrete fractional calculus is novel and requires significant research in a diversity of disciplines. Thus, in this paper, we investigate the discrete-time fractional-order Lengyel–Epstein system as a model of the chlorite iodide malonic acid (CIMA) chemical reaction. With the help of the second order difference operator, we describe the fractional discrete model. Furthermore, using the linearization approach, we established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employ a Lyapunov functional to show that when the iodide feeding rate is moderate, the constant equilibrium solution is globally asymptotically stable. Finally, numerical models are presented to validate the theoretical conclusions and demonstrate the impact of discretization and fractional-order on system dynamics. The continuous version of the fractional-order Lengyel–Epstein reaction–diffusion system is compared to the discrete-time system under consideration.

Global boundedness and dynamics of a diffusive predator–prey model with modified Leslie–Gower functional response and density-dependent motion

This paper is devoted to studying the dynamical behaviors and stationary patterns of a diffusive modified Leslie–Gower predator–prey model with density-dependent motion in the predator population. We establish the existence of classical solutions with the uniform-in time bound and then analyze the local and global stability of the spatially homogeneous co-existence steady state under certain parametric conditions. By choosing the prey diffusion rate d2 as the bifurcation parameter, the steady state bifurcations from the positive constant equilibrium solution are investigated. Numerical simulations are performed to corroborate our analytical findings.

A meshless numerical method for a system with intraspecific and interspecific competition

In this paper we study a novel mathematical model with intraspecific and interspecific competition between two species consisting of a non-linear parabolic–ODE–parabolic system. It describes the evolution of two populations in competition for a resource, one of which is subject to chemotaxis. We analyze the local stability of the constant equilibrium solutions and we obtain the periodic behavior of the solution for certain data of the problem. For this purpose, we apply the meshless numerical method of Generalized Finite Differences (GFDM) and we prove the conditional convergence of the discrete solution to the analytical one. The conditional convergence of the numerical method is demonstrated and, thought its implementation, we obtain numerical solutions whose asymptotic behavior agrees with the analytically one expected. We give several numerical examples on the applications of this meshless method over regularly and irregularly distributed nodes to illustrate its potential.

A single bounded input‐feedback control to the generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters and , where is a positive parameter and is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when , Lyapunov stable when , and globally exponentially stable when . Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear ‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear ‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory.

On the dynamics of the diffusive Field-Noyes model for the Belousov-Zhabotinskii reaction

In this paper, a diffusive Field-Noyes model for the Belousov-Zhabotinskii reaction is considered. We are mainly concerned with the global existence, boundedness and the asymptotic behaviors of the solutions. Of our particular interests, we focus on the existence of attraction region which attracts all the solutions of the system (regardless of the initial values), the global asymptotic stability of the constant positive equilibrium solution, the lumped parameter phenomenon, as well as Turing instability of the spatially homogeneous periodic solutions. In particular, a general formula in terms of the diffusion rates for the general 3×3 reaction-diffusion system is derived to determine Turing instability of the Hopf bifurcating periodic solutions, which extends our earlier results on general 2×2 reaction-diffusion system.

Diffusion-driven instability of both the equilibrium solution and the periodic solutions for the diffusive Sporns-Seelig model

<abstract><p>In this paper, a reaction-diffusion Sporn-Seelig model subject to homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. Of our particular interests, we are concerned with diffusion-driven instability of both the positive constant equilibrium solution and the Hopf bifurcating spatially homogeneous periodic solutions. To strengthen our analytical results, we also include some numerical simulations. These results allow for the clearer understanding the mechanisms of the spatiotemporal pattern formations of this chemical reaction model.</p></abstract>

A novel structure preserving semi‐implicit finite volume method for viscous and resistive magnetohydrodynamics

AbstractIn this work we introduce a novel semi‐implicit structure‐preserving finite‐volume/finite‐difference scheme for the viscous and resistive equations of magneto‐hydrodynamics (VRMHD) based on an appropriate 3‐split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field, and a third subsystem involving the pressure‐velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfvén waves and the magneto‐acoustic waves are treated implicitly. Thanks to this, the final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence‐free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual (staggered) meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix‐free conjugate gradient algorithm. The final scheme can be regarded as a novel shock‐capturing, conservative, and structure preserving semi‐implicit scheme for VRMHD. Several numerical tests are presented to show the main features of our novel divergence‐free semi‐implicit FV/FD solver: linear‐stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a second‐order of convergence is obtained for a smooth time‐dependent solution; shock‐capturing capabilities are proven against a standard set of stringent MHD shock‐problems; accuracy and robustness are verified against a non‐trivial set of two‐ and three‐dimensional MHD problems.

Steady-state solutions of the templator model in chemical self-replication

In this paper, we study the dynamics of the diffusive templator model in chemical self-replication. Firstly, the stability of nonnegative constant equilibrium solution and the stationary patterns induced by diffusion are investigated. Next, we study the properties and existence of nonconstant steady states. The global structures of bifurcation emitting from the positive constant solution are also established. Finally, numerical simulations are done to complemented theoretical results.

Analysis and simulation on dynamics of a partial differential system with nonlinear functional responses

We introduce a reaction–diffusion system with modified nonlinear functional responses. We first discuss the large-time behavior of positive solutions for the system. And then, for the corresponding steady-state system, we are concerned with the priori estimate, the existence of the nonconstant positive solutions as well as the bifurcations emitting from the positive constant equilibrium solution. Finally, we present some numerical examples to test the theoretical and computational analysis results. Meanwhile, we depict the trajectory graphs and spatiotemporal patterns to simulate the dynamics for the system. The numerical computations and simulated graphs imply that the available food resource for consumer is very likely not single.

Asymptotic dynamics of hermitian Riccati difference equations

In this paper, we consider the hermitian Riccati difference equations. Analogous to a Riccati differential equation, there is a connection between a Riccati difference equation and its associated linear difference equation. Based on the linear difference equation, we can obtain an explicit representation for the solution of the Riccati difference equation and define the extended solution. Further, we can characterize the asymptotic state of the extended solution and the rate of convergence. Constant equilibrium solutions, periodic solutions and closed limit cycles are exhibited in the investigation of asymptotic behavior of the hermitian Riccati difference equations.

Spatiotemporal Dynamics Induced by Michaelis–Menten Type Prey Harvesting in a Diffusive Leslie–Gower Predator–Prey Model

This paper is devoted to study the spatiotemporal dynamics of a diffusive Leslie–Gower predator–prey model with Michaelis-Menten type harvesting in the prey population. The existence and stability of possible non-negative constant equilibria are investigated. By regarding [Formula: see text] as a bifurcation parameter, the Hopf bifurcation from the positive constant equilibrium solution is investigated. The necessary and sufficient conditions of Turing instability are explicitly obtained. We show that at the critical value of the bifurcation parameter [Formula: see text] a Turing bifurcation occurs (i.e. a pattern arises). The conditions for the stability of the pattern are also derived in detail. Moreover, the global steady state bifurcation from the positive constant equilibrium solution is investigated. In particular, the local steady state bifurcation from double zero eigenvalues is also obtained by the techniques of space decomposition and the implicit function theorem. Our results show that Michaelis–Menten type harvesting in our model plays a crucial role in the formation of spatiotemporal dynamics, which is a strong contrast to the case without harvesting.

Solving a chemotaxis–haptotaxis system in 2D using Generalized Finite Difference Method

We study a mathematical model of cancer cell invasion of tissue (extracellular matrix) consisting of a system of reaction–diffusion-taxis partial differential equations which describes the interactions between cancer cells, the matrix degrading enzyme and the host tissue. We analyze the local stability of the constant equilibrium solutions to the chemotaxis–haptotaxis system, we derive a discretization of the system by means of the Generalized Finite Difference Method (GFDM) and we prove the convergence of the discrete solution to the analytical one. Also, we provide several numerical examples on the applications of this meshless method over regular and irregular domains.

Dynamic behaviors of a predator–prey model with weak additive Allee effect on prey

This paper deals with a reaction–diffusion system modeling predator–prey interaction with additive Allee effect. We first analyze nonnegative constant equilibrium solutions and study the stability of the unique positive solution. Then we investigate the steady state bifurcation and Hopf bifurcation from the unique positive constant solution, respectively. The main tools used here include the bifurcation theory, the space decomposition and the implicit function theorem. Moreover, we describe the global structure from a simple eigenvalue. Finally, for the case d1=0, we prove the existence of far-from-equilibrium steady states with jump discontinuities. Moreover, by applying the singular perturbation method, we give a proof of the existence of large amplitude solutions when d1 is sufficiently small.

Spatiotemporal complexity of a diffusive planktonic system with prey-taxis and toxic effects

In this paper, we propose a three-species reaction-diffusion planktonic system with prey-taxis and toxic effects, in which the zooplankton can recognize the nontoxic and toxin-producing phytoplankton and can make proper response. We first establish the existence and stability of the unique positive constant equilibrium solution by utilizing the linear stability theory for partial differential equations. Then we obtain the existence and properties of nonconstant positive solutions by detailed steady state bifurcation analysis. In addition, we obtain that change of taxis rate will result in the appearance of time-periodic solutions. Finally, we conduct some numerical simulations and give the conclusions.