Positive Steady State Research Articles

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.

Asymptotic decay towards steady states of solutions to very fast and singular diffusion equations

We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having non-homogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic H 1 -convergence of the solutions to a unique non-homogeneous positive steady state generally to hold.

Spatiotemporal Dynamics of a Diffusive Immunosuppressive Infection Model with Nonlocal Competition and Crowley–Martin Functional Response

In this paper, we introduce the Crowley–Martin functional response and nonlocal competition into a reaction–diffusion immunosuppressive infection model. First, we analyze the existence and stability of the positive constant steady states of the systems with nonlocal competition and local competition, respectively. Second, we deduce the conditions for the occurrence of Turing, Hopf, and Turing–Hopf bifurcations of the system with nonlocal competition, as well as the conditions for the occurrence of Hopf bifurcations of the system with local competition. Furthermore, we employ the multiple time scales method to derive the normal forms of the Hopf bifurcations reduced on the center manifold for both systems. Finally, we conduct numerical simulations for both systems under the same parameter settings, compare the impact of nonlocal competition, and find that the nonlocal term can induce spatially inhomogeneous stable periodic solutions. We also provide corresponding biological explanations for the simulation results.

On the Schnakenberg model with crucial reversible reactions

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns.

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.

Sequential Geometric Programming Method for Parameter Estimation of a Nonlinear System in Microbial Continuous Fermentation

This paper addresses the problem of parameter estimation for the microbial continuous fermentation of glycerol to 1,3-propanediol. A nonlinear dynamical system is first presented to describe the microbial continuous fermentation. Some mathematical properties of the dynamical system in the microbial continuous fermentation are also presented. A parameter estimation model is proposed to estimate the parameters of the dynamical system. The proposed estimation model is a large-scale, nonlinear, and nonconvex optimization problem if the number of experimental groups is large. A sequential geometric programming (SGP) method is proposed to efficiently solve the parameter estimation problem. The results indicated that our proposed SGP method can yield smaller errors between the experimental and calculated steady-state concentrations than the existing seven methods. For the five error indices considered, that is, the concentration errors of biomass, glycerol, 1,3-propanediol, acetic acid, and ethanol, the results obtained using the proposed SGP method are better than those obtained using the methods in the literature (Xiu et al., Gao et al., Sun et al., Sun et al., Li and Qu, Wang et al., and Zhang and Xu), with improvements of approximately 71.86–95.03%, 52.08–94.87%, 99.70–99.98%, 5.39–90.29%, and 12.67–80.83%, respectively. This concludes that the established dynamical system can better describe the microbial continuous fermentation. We also present that our established dynamical system has multiple positive steady states in some fermentation conditions. We observe that there are two regions of multiple positive steady states at relatively high values of substrate glycerol concentration in feed medium.

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.

Spatiotemporal Dynamics and Bifurcation Analysis of a Generalized Two-Prey One-Predator System with Diffusion and Double Prey-Taxes

The presence of a predator can force the mediation of a coexistence state in three-species ordinary differential equation model, where two competing species are preyed on by a common predator. To understand how the addition of diffusion and prey-taxis affects predator-mediated coexistence in such an ecological system, we consider a general two-competing-prey and one-predator model with double prey-taxes under Neumann boundary conditions. We first show that there is a unique global classical solution to this model with ratio-dependent and nonratio-dependent predator functional responses. Then, we demonstrate the emergence of the so-called stationary patterns. Finally, in detail, we give some sufficient conditions for the existence, nonexistence, and stability of nonconstant positive steady states and time-periodic positive solutions. Surprisingly, we find that the combination of a repulsive prey-taxis and an attractive prey-taxis can also induce the emergence of pattern formations. The theoretical results imply that double prey-taxes play an extremely important part in biological control.

Absolute Concentration Robustness in Rank-One Kinetic Systems

A kinetic system has an absolute concentration robustness (ACR) for a molecular species if its concentration remains the same in every positive steady state of the system. Just recently, a condition that sufficiently guarantees the existence of an ACR in a rank-one mass-action kinetic system was found. In this paper, it will be shown that this ACR criterion does not extend in general to power-law kinetic systems. Moreover, we also discussed in this paper a necessary condition for ACR in multistationary rank-one kinetic system which can be used in ACR analysis. Finally, a concept of equilibria variation for kinetic systems which are based on the number of the system's ACR species will be introduced here.

Modeling economic growth with spatial migration: A stability analysis of the long-run equilibrium based on semigroup theory

We develop a new model of economic growth that fully accounts for the spatial dependence of the flows of workers and capital on salaries and returns. Considering a rather general setting in which we do not require the knowledge of the exact expressions of the production function and the population growth rate, we allow for a strictly positive steady state in which labor, capital, wages, and returns on capital are constant in space. Then, we establish the stability of such an equilibrium using the theory of abstract non-linear parabolic problems and analyzing the spectral properties of the derivative of the operator that describes the coupled dynamics of labor and capital. We present numerical simulations which agree with our theoretical investigation and show that the proposed model allows us to detect interesting transitional dynamics that the standard Solow model does not capture. In particular, we see how the migration of labor can slow down the process of wage convergence at some spatial locations and how the flow of workers, interacting with population dynamics, can reduce economic growth in countries where both developed and underdeveloped regions are present.

Cross diffusion induced spatially inhomogeneous Hopf bifurcation for a three species Lotka–Volterra food web model with cycle

In this article, we consider a three species Lotka–Volterra food web reaction–diffusion model with cycle, in which the intensity of the rate of preying on species 1 by species 2, the rate of preying on species 2 by species 3 and the rate of preying on species 3 by species 1 are in general asymmetrical. By suitably choosing cross diffusion coefficients as the bifurcation parameter, a spatially inhomogeneous Hopf bifurcation at a positive constant steady state is proved to occur for a sequence of critical values of the bifurcation parameter. In addition, we demonstrate that these spatially inhomogeneous bifurcating periodic solutions are stable when diffusion coefficients are located in suitable ranges. Our results show that cross diffusion plays a key role in the formation of spatially inhomogeneous periodic oscillatory patterns.

Bifurcation analysis of a delayed reaction–diffusion–advection Nicholson’s blowflies equation

In this paper, we investigate the dynamics of a reaction-diffusion Nicholson’s blowflies equation with advection. The stability of positive steady state and existence of Hopf bifurcation are obtained by analyzing the distribution of the eigenvalues. Moreover, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction and stability of the Hopf bifurcation is derived. Meanwhile, we find out that the bifurcation value is increasing with respect to the advection rate. Finally, numerical results demonstrate that the advection term causes the population to move from upstream to downstream, which also indicates that advection term plays a key role in the description and interpretation of some common natural phenomena.

Traveling wave solutions for three-species nonlocal competitive-cooperative systems

By using a two-point boundary-value problem and a Schauder's fixed point theorem, we obtain traveling wave solutions connecting \((0,0,0)\) to an unknown positive steady state for speed \(c\geq c^{\ast}=\max\{2,2\sqrt{d_2r_2},2\sqrt{d_3r_3}\}\). Then we present some asymptotic behaviors of traveling wave solutions. In particular we show that the nonlocal effects have a great influence on the final state of traveling wave solutions at \(-\infty\).
 For more information see https://ejde.math.txstate.edu/Volumes/2023/55/abstr.html

Dynamical behavior of solutions of a reaction–diffusion model in river network

In this paper, we study the long-time behavior of solutions of a reaction–diffusion model in a one-dimensional river network, where the river network has two branches, and the water flow speeds in each branch are the same constant β. We show the existence of two critical values c0 and 2 with 0<c0<2, and prove that when −c0≤β<2, the population density in every branch of the river goes to 1 as time goes to infinity; when −2<β<−c0, then, as time goes to infinity, the population density in every river branch converges to a positive steady state strictly below 1; when |β|≥2, the species will be washed down the stream, and so locally the population density converges to 0. Our result indicates that only if the water-flow speed is suitably small (i.e., |β|<2), the species will survive in the long run.

Bifurcation analysis of a predator–prey model with memory-based diffusion

In this paper, we investigate the predator–prey model equipped with Fickian diffusion and memory-based diffusion of predators. The stability and bifurcation analysis explores the impacts of the memory-based diffusion and the averaged memory period on the dynamics near the positive steady state. Specifically, when the memory-based diffusion coefficient is less than a critical value, we show that the stability of the positive steady state can be destabilized as the average memory period increases, which leads to the occurrence of Hopf bifurcations. Moreover, we also analyze the bifurcation properties using the central manifold theorem and normal form theory. This allows us to prove the existence of stable spatially inhomogeneous periodic solutions arising from Hopf bifurcation. In addition, the sufficient and necessary conditions for the occurrence of stability switches are also provided.

A diffusion-advection predator-prey model with a protection zone

In this paper, we investigate a diffusion-advection predator-prey model imposed the Danckwerts boundary conditions with three different types of protection zone. Some special cases have been extensively studied, such as: diffusion-advection predator-prey model without protection zone [22,29] and diffusive predator-prey model with protection zone without advection in high dimension [5,16]. Some ideas developed in the former works do not appear to work due to the difficulty caused by diffusion, advection and protection zone. By applying the comparison principle for parabolic equations [32] and persistence theory [26,37], we obtain almost complete long-time dynamics, which describes the optimal protection zone. If the predator functional response is Holling-type I, the uniqueness of the positive steady state has been derived, which improves the method proposed by López-Gómez and Pardo [20] and developed by Nie et al. [28]. Finally, new ways of coexistence have been obtained by investigating a special three-species model.

A network-based parametrization of positive steady states of power-law kinetic systems

A network-based parametrization of positive steady states of power-law kinetic systems

Spatial and temporal dynamics of COVID-19 with nonlocal dispersal in heterogeneous environment: Modeling, analysis and simulation

This paper proposes an infectious disease dynamics model with spatial heterogeneity to study the joint impact of nonlocal dispersal and vaccine on the transmission dynamics of COVID-19 (Corona Virus Disease 2019). Overcoming the difficulty of non-compactness caused by non-local operators, the functional expression of the next generation operator R is obtained by using the renewal equation. And then, the basic reproduction number R0 of the proposed model is obtained, i.e., the spectral radius of the next generation operator R. In terms of theoretical analysis, we prove that the global asymptotic stability of the disease-free steady state when R0<1; Using the uniform persistence theory of point dissipative systems, we prove the system is uniformly persistent and has at least one positive steady state when R0>1. We verify the theoretical results and study the influence of related factors on the transmission of COVID-19 pandemic in numerical simulation part. Finally, our numerical results show that: (1) Improving the coverage rate of COVID-19 complete vaccine and the vaccine efficiency is one of the optimal measures to effectively prevent and control the spread of COVID-19 under the current medical conditions; (2) The influence of the form of nonlocal dispersal kernel function and the virus in environment on the transmission of COVID-19 cannot be ignored Therefore, targeted prevention and control measures should be taken for the dispersal of the virus in urban areas and in the process of cold chain transportation to prevent COVID-19 from spreading to rural areas with relatively backward medical conditions.

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.

Spatial movement with memory-induced cross-diffusion effect and toxin effect in predator

It is well known that spatial memory exists in animal movement modelling. Since spatial memory is related to the past information, then delay arises. In view of the protection of ecological environment, more and more scholars are concerned about the effects of toxic substances on it. A prey–predator system with memory-induced cross-diffusion and toxin in predator is considered in this paper. First, we discuss the fundamental dynamics in detail. Then considering the memory-induced cross-diffusion coefficient and the averaged memory period delay in predator as the controlling parameters, we get that n-mode Hopf bifurcations exist at the positive steady state. Stability switches are generated, and there are spatially nonhomogeneous periodic solutions . Namely if the cross-diffusion coefficient is small, the populations always coexist. If the cross-diffusion coefficient is moderate, two kinds of critical values of the averaged memory period of Hopf bifurcations occur and stability switches may be induced by the delay. If the cross-diffusion coefficient becomes larger, one kind of critical value of the averaged memory period of Hopf bifurcation occur. From simulations, the memory-based cross-diffusion and toxin have vital influences on stability. The moderate toxin can be good for population coexistence. When the averaged memory period is small, the toxin does not change the stability. Once the averaged memory period is bigger, the toxin can change the stability. Moreover, it shows that the averaged memory delay could switch the stability of the system.