Positive Constant Steady State Research Articles

Long-time behavior of the solutions of Murray–Thomas model for interacting chemicals

This paper is concerned with the Murray–Thomas model for interacting chemicals or species, under Robin boundary data. It is shown that the solutions are bounded and asymptotically converging toward an absorbing set of the phase-space. The stability of the positive constant steady states is discussed via a symmetrization procedure.

Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator‐Prey Model with Diffusion Effects

This paper is concerned with a delayed predator‐prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady‐state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).

ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS

We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration. Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.

Stability of positive constant steady states and their bifurcation in a biological depletion model

This paper is concerned with a biological depletion model in a bounded domain. The stability of the positive constant steady states is discussed. In one dimensional case, we make a detailed description for the global bifurcation structure from two positive constant solutions. The result indicates that if $d$ is properly small, the system has at least one non-constant positive steady-state. The main tools used here include the stability theory, bifurcation theory and simulations. From extensive numerical simulations, the predictions from linear theory are confirmed and the influence of parameters $d,D,\sigma$ on these patterns is depicted.

NON-CONSTANT POSITIVE STEADY STATE OF A DIFFUSIVE LESLIE-GOWER TYPE FOOD WEB SYSTEM

A three species food web comprising of two preys and one predator in an isolated homogeneous habitat is considered. The preys are assumed to grow logistically. The predator follows modified Leslie-Gower dynamics and feeds upon the prey species according to Holling Type II functional response. The local stability of the constant positive steady state of the corresponding temporal system and the spatio-temporal system are discussed. The existence and non-existence of non-constant positive steady states are investigated.

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form ut = Δu + f(t, u) on \({\mathbb{R}^N}\) . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.

A note on a diffusive predator-prey model and its steady-state system

In this note, a diffusive predator-prey model subject to the homogeneous Neumann boundary condition is investigated and some qualitative analysis of solutions to this reaction-diffusion system and its corresponding steady-state problem is presented. In particular, by use of a Lyapunov function, the global stability of the constant positive steady state is discussed. For the associated steady state problem, a priori estimates for positive steady states are derived and some non-existence results for non-constant positive steady states are also established when one of the diffusion rates is large enough. Consequently, our results extend and complement the existing ones on this model.

Non-constant positive steady states for the HP food chain system with cross-diffusions

We consider a strongly coupled reaction–diffusion system describing three interacting species in the HP food chain structure. Based on the Leray–Schauder degree theory, the non-existence and existence of non-constant positive steady states are investigated. The stability of the constant positive steady state is also discussed. The results indicate that strong cross-diffusions are helpful for the appearance of the non-constant positive steady state (stationary patterns).

Multiple bifurcations in a delayed predator–prey diffusion system with a functional response

The present paper is concerned with a delayed predator–prey diffusion system with a Beddington–DeAngelis functional response and homogeneous Neumann boundary conditions. If the positive constant steady state of the corresponding system without delay is stable, by choosing the delay as the bifurcation parameter, we can show that the increase of the delay can not only cause spatially homogeneous Hopf bifurcation at the positive constant steady state but also give rise to spatially heterogeneous ones. In particular, under appropriate conditions, we find that the system has a Bogdanov–Takens singularity at the positive constant steady state, whereas this singularity does not occur for the corresponding system without diffusion. In addition, by applying the normal form theory and center manifold theorem for partial functional differential equations, we give normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation and the explicit formula for determining the properties of spatial Hopf bifurcations.

Qualitative analysis on a reaction-diffusion prey-predator model and the corresponding steady-states

The authors study a diffusive prey-predator model subject to the homogeneous Neumann boundary condition and give some qualitative descriptions of solutions to this reaction-diffusion system and its corresponding steady-state problem. The local and global stability of the positive constant steady-state are discussed, and then some results for non-existence of positive non-constant steady-states are derived.

Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics

In this paper, an n -species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multi-species Lotka–Volterra type competition model and prey–predator model.

Qualitative analysis of a diffusive variable-territory prey-predator model

In this paper, we study thevariable-territory prey-predator model. We first establish theglobal stability of the unique positive constant steady state forthe ODE system and the reaction diffusion system, and then provethe existence, uniqueness and stability of positive stationarysolutions for the heterogeneous environment case.

Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey–predator model

In this work, we study a ratio-dependent prey–predator model with diffusion and homogeneous Neumann boundary condition. We prove that the unique positive constant steady state is locally and uniformly stable, and is globally asymptotically stable under some assumptions. The proof uses the iteration method.

Non-constant positive steady state of one resource and two consumers model with diffusion

In this paper, a predator–prey reaction–diffusion system with one resource and two consumers is considered. Assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington–DeAngelis functional response, and they compete for the common resource. First, it is proved that the unique positive constant steady state is stable for the ODE system and the reaction–diffusion system. Second, a prior estimates of positive steady state is given. Finally, the non-existence of non-constant positive steady state, the existence and bifurcation of non-constant positive steady state are studied.

Stationary patterns of the Holling–Tanner prey–predator model with diffusion and cross-diffusion

This paper is concerned with the Holling–Tanner prey–predator model subject to the homogeneous Neumann boundary condition. We will show that under certain hypotheses, even though the unique positive constant steady state is uniformly asymptotically stable for the dynamics with diffusion, non-constant positive steady states can exist due to the emergence of cross-diffusion. The biological implication of cross-diffusion means that the prey species exercise a self-defense mechanism to protect themselves from the attack of the predator. Our result also demonstrates that cross-diffusion can create Stationary Patterns.

Qualitative analysis for a diffusive predator–prey model

In this paper we study the qualitative properties of a diffusive predator–prey model subject to the homogeneous Neumann boundary condition. By a comparison argument and iteration technique, under some hypotheses we prove that the positive constant steady state is globally asymptotically stable. We also establish the existence and non-existence of non-constant positive steady states (stationary patterns) by use of the degree theory and the a priori estimates.

Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion

In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation. (An erratum to this article has been appended at the end of the pdf file.)

Qualitative analysis of a ratio-dependent predator–prey system with diffusion

Ratio-dependent predator–prey models are favoured by many animal ecologists recently as they better describe predator–prey interactions where predation involves a searching process. When densities of prey and predator are spatially homogeneous, the so-called Michaelis–Menten ratio-dependent predator–prey system, which is an ordinary differential system, has been studied by many authors. The present paper deals with the case where densities of prey and predator are spatially inhomogeneous in a bounded domain subject to the homogeneous Neumann boundary condition. Its main purpose is to study qualitative properties of solutions to this reaction-diffusion (partial differential) system. In particular, we will show that even though the unique positive constant steady state is globally asymptotically stable for the ordinary-differential-equation dynamics, non-constant positive steady states exist for the partial-differential-equation model. This demonstrates that stationary patterns arise as a result of diffusion.