Positive Steady-state Solutions Research Articles

A reaction–diffusion model of major emerging infectious diseases in a spatially heterogeneous environment and case study

During the outbreak of major emerging infectious diseases, virus droplets can survive in the environment as aerosols for hours to days, and their impact on human infectious diseases is often overlooked. In addition, the speed of transmission of infectious diseases is often closely related to transportation. Therefore, studying the impact of environmental viruses and transportation on disease development is significant for effective infectious disease prevention and control. We proposed a degenerate reaction–diffusion infectious disease model ([Formula: see text]) considering environmental virus interference and established a well-posedness and threshold system for this model. We have obtained the system solution approaches the disease-free equilibrium ([Formula: see text]) when the basic reproduction number [Formula: see text]. The system has at least one positive steady state solution (PSS) when [Formula: see text]. This paper used a non-standard finite difference method discretization model while data fitting and parameter estimation were performed based on data provided by the Health Commission. Further sensitivity analysis was conducted on [Formula: see text]. At the same time, we also discussed the impact of various parameters in the early stages of the outbreak of major emerging infectious diseases on the development of the disease. Research found that even if the contact rate between people is controlled at a shallow level, the disease may persist. In the early stages of major emerging outbreak of infectious diseases, immediately reducing the use of transportation can effectively reduce the speed of disease spread.

Coexistence and bistability of a modified Leslie–Gower predator–prey model with mixed movement for the predator and Holling-type II schemes

This paper centers on the analysis of the dynamics of a modified Leslie–Gower predator–prey model employing Holling-type II schemes, with the prey exhibiting pure random diffusion and the predator undergoing a mixed form of movement. The extinction of species and uniform persistence of this system are explored, and several conditions for the stability, uniqueness and multiplicity of positive steady-state solutions are derived. In contrast to the specialist and generalist predator–prey systems in open advective environments, the dynamics of this system are more intricate. It emerges that multiple positive steady-state solutions and the bistable phenomenon exist for this system when a small advection rate and a moderate predation rate are imposed. Numerical simulations reveal that the increase of diffusion rate for prey disadvantages the survival of itself and has no impact on predator invasion, while the increase of diffusion rate for predators favors the invasion of itself.

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Steady States of a Diffusive Population-Toxicant Model with Negative Toxicant-Taxis

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4).

Some qualitative analyses on a vegetation-water model with cross-diffusion and internal competition

This paper is concerned with a vegetation-water model with cross-diffusion and intra-plant competitive feedback under Neumann boundary conditions. First, we found that the equilibrium with small vegetation density is always unstable, and if the cross-diffusion coefficient is suitably large, the equilibrium with relatively large vegetation density loses its stability, and Turing instability occurs. A priori estimates of positive steady-state solutions are also established by the maximum principle of elliptic equations. Moreover, some qualitative analyses on the steady-state bifurcations for both simple and double eigenvalues are conducted in detail. Space decomposition and the implicit function theorem are used for double eigenvalues. In particular, the global continuation is obtained, and the result shows that there is at least one non-constant positive steady-state solution when cross-diffusion is large. Finally, numerical simulations are provided to prove and supplement theoretic research results, and some vegetation patterns with the increase of the soil water diffusion feedback intensity are formed, where the transition appears: gap [Formula: see text] stripe [Formula: see text] spot.

Steady-state bifurcations and patterns formation in a diffusive toxic-phytoplankton–zooplankton model

In this paper, we study a diffusive toxic-phytoplankton–zooplankton model with prey-taxis under Neumann boundary condition. By analyzing the characteristic equation, we discuss the local stability of the positive constant solutions and show the repulsive prey-taxis is the key factor that destabilizes the solutions. By means of the abstract bifurcation theorem, we investigate the existence of non-constant positive steady-state solutions bifurcating from the constant coexistence equilibrium. Furthermore, we obtain the criterion for the stability of the branching solutions near the bifurcation point. Numerical simulations support our theoretical results, together with some interesting phenomena, stable heterogeneous periodic solutions emerge when prey-tactic sensitivity coefficient is well below the critical value, and zooplankton populations present extinction and continued transitions as habitat size increases.

Global dynamics of an indirect prey-taxis system with an anti-predation mechanism

In this paper, we shall study the following model:{Nt=dNΔN+λN−N2−NP,x∈Ω,t>0,Pt=Δ(d(S)P)+μP−P2+γNP,x∈Ω,t>0,St=dSΔS+τN−ηS,x∈Ω,t>0,∂N∂ν=∂P∂ν=∂S∂ν=0,x∈∂Ω,(N,P,S)(x,0)=(N0,P0,S0)(x),x∈Ω, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary. Based on semigroup estimates and Moser iteration, we first establish the existence of the global classical solution with a uniform-in-time bound. Moreover, we build the global stabilization of constant steady states by constructing suitable Lyapunov functionals and give the decay rates of solutions. Furthermore, we find that the anti-predation mechanism will generate steady state bifurcation. We further prove the global existence of non-constant positive steady state solutions under certain conditions. Finally, we numerically demonstrate spatially inhomogeneous coexistence patterns (i.e., non-constant positive steady state solutions) for the predator, the prey and the signal.

Positive steady-state solutions for a vegetation–water model with saturated water absorption

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Dynamics of a competition model with intra- and interspecific interference in the unstirred chemostat

This paper deals with an unstirred chemostat system with intra- and interspecific interference. We first investigate the uniqueness and asymptotic behaviors of the positive steady state solution of the single species model. The results show that the species cannot survive when the intraspecific interference is large enough. For two species system, we find that the two species can coexist when the influence of interspecific interference is relatively small, and that the system has no positive steady-state solution when the strength of interspecific interference is large enough. Furthermore, we study the influence of diffusion rate on species coexistence. The results show that for large diffusion rate, both species will be washed out, and the competition exclusion occurs at small diffusion rate. This indicates that species coexistence only occurs at intermediate diffusion rate.

Positive steady-state solutions for a water-vegetation model with the infiltration feedback effect

Positive steady-state solutions for a water-vegetation model with the infiltration feedback effect

A qualitative analysis of positive steady-state solutions for a mussel-algae model with diffusion

A qualitative analysis of positive steady-state solutions for a mussel-algae model with diffusion

Dynamics of a Reaction–Diffusion–Advection System with Nonlinear Boundary Conditions

In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.

Generic algebraic conditions for the occurrence of switch-like behavior of a chemical kinetic system of the hypoxia pathway.

Weakly reversible chemical reaction networks with zero deficiency associated with mass-action kinetics admit, within each positive stoichiometric compatibility class, one positive steady state which is locally asymptotically stable and this irrespective of the values of the kinetics constants. Networks which do not enjoy these structural properties potentially exhibit more diverse dynamical behaviors. In this article, we consider a chemical reaction network associated with mass-action kinetics which is not weakly reversible and has a deficiency larger than one. The chemical reactions are at most bimolecular, but inflow and outflow reactions are present. Our results are as follows. We establish the existence of positive steady-state solutions and obtain their analytic expressions in the concentration space and in convex coordinates. We show that the system fulfills necessary conditions for a saddle-node and for a bifurcation into a saddle and a node. We apply a constructive approach to obtain a set of numerical values for the state variables and kinetic parameters, not reported previously, such that the reduced Jacobian is characterized by a zero eigenvalue with all other eigenvalues having negative real parts. The bifurcation diagram confirms the presence of the switch-like behavior.

Positive Steady-State Solutions for a Class of Prey-Predator Systems with Indirect Prey-Taxis

Positive Steady-State Solutions for a Class of Prey-Predator Systems with Indirect Prey-Taxis

A diffusive Monod-Haldane predator-prey system with Smith growth and a protection zone

In predation, too large intrinsic growth rate of the predator will usually drive prey to extinction. In this paper, we consider a diffusive Monod-Haldane predator-prey system with Smith growth to study the effect of an internal protection zone. We mainly focus on the corresponding steady system to locate a critical patch size of the protection zone. We find that if the size of the protection zone is below the critical patch size, then the system has no positive steady state solutions for too large intrinsic growth rate of the predator, while if it is above the critical patch size, then there exist positive steady state solutions no matter how large the intrinsic growth of the predator is. We further get the asymptotic behavior of the system when the protection zone is large enough, which indicates that the predator and the prey are spatially segregated as the intrinsic growth rate of the predator tends to infinity.

Stationary Patterns of a Cross-Diffusion Prey-Predator Model with Holling Type II Functional Response

In this paper, we consider positive steady-state solutions of a cross-diffusions prey-predator model with Holling type II functional response. We investigate sufficient conditions for the existence and the nonexistence of nonconstant positive steady state solutions. It is observed that nonconstant positive steady states do not exist with small cross-diffusion coefficients, and the constant positive steady state is global asymptotically stable without cross-diffusion. Furthermore, we show that if natural diffusion coefficient or cross-diffusion coefficient of the predator is large enough and other diffusion coefficients are fixed, then under some conditions, at least one nonconstant positive steady state exists.

Pattern formation and qualitative analysis for a vegetation-water model with diffusion

In this paper, a diffusive vegetation-water model under Neumann boundary conditions is considered. Firstly, the stability and the diffusion-induced Turing instability are studied. Then, some a priori estimates of positive steady-state solutions are obtained by the maximum principle. Moreover, the bifurcations at both simple and double eigenvalues are investigated in detail. Finally, numerical simulations are shown to support and supplement theoretical analysis results. In particular, the evolution processes of vegetation patterns are depicted under different parameters.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Dynamics of a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment

In this paper, we study a diffusion-advection Lotka-Volterra competition model with stage structure in a spatially heterogeneous environment. The existence and local asymptotic stability of spatially non-homogeneous semi-trivial steady-state solutions and spatially non-homogeneous positive steady-state solutions are obtained by the implicit function theorem and spectral analysis. We show that three scenarios can occur: if random diffusion rates of two species are sufficiently large, both species go extinct; if random diffusion rates of two species are relatively small, two competing species coexist; if one species has a large random diffusion rate and the other has a small random diffusion rate, the species with a large random diffusion rate are driven to extinction. An interesting finding is that a large delay does not lead to Hopf bifurcation at the spatially non-homogeneous steady-state solution, but makes this steady-state solution approach zero. We numerically demonstrate the effects of spatial heterogeneity on spatiotemporal dynamics.