Positive Steady-state Solutions Research Articles

Dirichlet problem for the Nicholson’s blowflies equation with density-dependent diffusion

This paper is concerned with the time delayed Nicholson’s blowflies equation with degenerate diffusion. We prove the existence and uniqueness of the positive steady state solution under the Dirichlet boundary condition and we show the stability of the nontrivial steady state.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

Dynamics analysis of a reaction–diffusion system with Beddington–DeAngelis functional response and strong Allee effect

In this paper, the dynamics of a diffusive predator–prey model with modified Leslie–Gower term and strong Allee effect on prey under homogeneous Neumann boundary condition is considered. Firstly, we obtain the qualitative properties of the system including the existence of the global positive solution and the local and global asymptotical stability of the constant equilibria. In addition, we investigate a priori estimate and the nonexistence of nonconstant positive steady state solutions. Finally, we establish the existence and local structure of steady state patterns and time-periodic patterns for the system.

Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

Dynamics of Modified Leslie-Gower Harvested Predator-Prey Model with Beddington-DeAngelis Functional Response

This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existence of non-constant positive steady state solutions.

Uniqueness and stability of positive solutions for a diffusive predator–prey model in heterogeneous environment

In this paper, we study a spatially heterogeneous predator–prey model where the interaction is governed by a Crowley-Martin type functional response. It is shown that the predator competition in this model plays an extremely important role. A good understanding of the existence, uniqueness and stability of positive solutions is obtained when the predator competition is strong. In particular, we show that when the predator competition is strong, the model has at most one positive steady-state solution for any $$\mu \in {{\mathbb {R}}}$$ , and it is globally asymptotically stable for any $$\mu >0$$ (if it exists). Additionally, we show that when the prey growth rate is over a critical value, the two populations stabilize at a unique coexistence state if the predator growth rate is strong.

Density-dependent effects on Turing patterns and steady state bifurcation in a Beddington\u2013DeAngelis-type predator\u2013prey model

In this paper, Turing patterns and steady state bifurcation of a diffusive Beddington–DeAngelis-type predator–prey model with density-dependent death rate for the predator are considered. We first investigate the stability and Turing instability of the unique positive equilibrium point for the model. Then the existence/nonexistence, the local/global structure of nonconstant positive steady state solutions, and the direction of the local bifurcation are established. Our results demonstrate that a Turing instability is induced by the density-dependent death rate under appropriate conditions, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion. Moreover, some specific examples are presented to illustrate our analytical results.

Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function

A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).

Population dynamics in river networks: analysis of steady states

We study the population dynamics of an aquatic species in a river network. The habitat is viewed as a binary tree-like metric graph with the population density satisfying a reaction-diffusion-advection equation on each edge, along with the appropriate junction and boundary conditions. In the case of a linear reaction term and hostile downstream boundary condition, the question of persistence in such models was studied by Sarhad, Carlson and Anderson. We focus on the case of a nonlinear (logistic) reaction term and use an outflow downstream boundary condition. We obtain necessary and sufficient conditions for the existence and uniqueness of a positive steady state solution for a simple Y-shaped river network (with a single junction). We show that the existence of a positive steady state is equivalent to the persistence condition for the linearized model. The method can be generalized to a binary tree-like river network with an arbitrary number of segments.

Dynamics and pattern formation in a diffusive predator-prey system

This paper is devoted to study a general diffusive predator-prey system with strong Allee effect in the prey. A priori estimate and nonexistence of nonconstant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. The Hopf bifurcation and steady-state bifurcation from the positive constant solution branch are investigated. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations.

Positive Solutions of Elliptic Boundary Value Problems and Applications to Population Dynamics

In this paper, we investigate the dynamics of population inhabiting a favorable environment surrounded by a region where survival is impossible. We use a parabolic partial differential equation with nonlinearity changing signs to describe the dynamics. By using a recent result on the existence of nonzero fixed point for r-nowhere normal-outward compact maps, we establish new results on existence of nonzero nonnegative or positive steady-state solutions of such boundary value problems. Additionally, we obtain the convergence of positive solutions of such parabolic boundary value problems in the Banach space $$C^1$$, which is different from the previous results that considered weak convergence or uniform convergence.

Steady-state solutions of a reaction–diffusion system arising from intraguild predation and internal storage

Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.

Qualitative analysis for a diffusive predator–prey model with hunting cooperative

This paper investigates a predator–prey model that adds a cooperation term to the attack rate of the predator population. For the non-spatial model, the existence and stability of non-negative equilibrium points, and bifurcations are studied by choosing cooperation coefficient as control parameter. Our analytical results show that hunting cooperation can be beneficial to the predator population. However, as the cooperation coefficient increases, hunting cooperation can also destabilize the model and promote a sudden collapse of the predator population. For the spatial model, the stability of positive constant steady state solution, Hopf bifurcation and Turing instability are discussed. It is obtained that when the predation diffusion is not smaller than the prey diffusion, the spatial model can reserve the stable stability of the positive constant steady state solution. It is noted that the model without hunting cooperative does not generate Turing instability, while the model with hunting cooperative may generate Turing instability.

Complex dynamics of a diffusive predator–prey model with strong Allee effect and threshold harvesting

In this paper, complex dynamics of a diffusive predator–prey model is investigated, where the prey is subject to strong Allee effect and threshold harvesting. The existence and stability of nonnegative constant steady state solutions are discussed. The existence and nonexistence of nonconstant positive steady state solutions are analyzed to identify the ranges of parameters of pattern formation. Spatially homogeneous and nonhomogeneous Hopf bifurcation and discontinuous Hopf bifurcation are proved. These results show that the introduction of strong Allee effect and threshold harvesting increases the system spatiotemporal complexity. Finally, numerical simulations are presented to validate the theoretical results.

Effect of cross-diffusion on the stationary problem of a predator–prey system with a protection zone

This paper is concerned with the steady state problem of a predator–prey cross-diffusion system with herd effect, Allee effect and a protection zone. Some sufficient conditions for the existence of positive steady state solutions are given. Our proof is based on the local and global bifurcation theory and some a priori estimates. Some limiting behavior of positive steady states with respect to the Allee effect constant (α) or the diffusion coefficient (d), are discussed.

Global attractors of some predator–prey reaction–diffusion systems with density-dependent diffusion and time-delays

This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution.

Effect of the protection zone on coexistence of the species for a ratio-dependent predator-prey model

This paper deals with the effect of the protection zone Ω0 on coexistence of the species for a ratio-dependent predator-prey model. We obtain a critical value λ1N(bδ(x),Ω) which is less than the well-known critical value λ1D(Ω0) obtained in the previous literatures. Furthermore, we show that if λ>λ1N(bδ(x),Ω), then the prey persists regardless of the growth rates of the predator; while if λ≤λ1N(bδ(x),Ω), then there exists a real number μ⁎, such that the prey is ultimately extinct when μ>μ⁎. As for λ<λ1N(bδ(x),Ω) and μ<μ⁎, the curve of the positive steady state solutions of the model emanating from (λ,0;−c) ends at a singular point (0,0;μ2). Meantime, by using the Lyapunov–Schmidt reduction method, we obtain a fine profile of its bifurcation diagrams, and the uniqueness or multiplicity of its positive steady state solutions. In addition, as generally expected, the chances of survival of the prey will increase with the size of the protection zone.

A Kinetic Modelling of Enzyme Inhibitions in the Central Metabolism of Yeast Cells

Metabolic regulation plays an important role in the metabolic engineering of a cellular process. It is conducted to improve the productivity of a microbial process by identifying the important regulatory nodes of a metabolic pathway such as fermentation pathway. Regulation of enzymes involved in a particular pathway can be held to improve the productivity of the system. In the central metabolism of yeast cell, some enzymes are known as regulating enzymes that can be inhibited to increase the production of ethanol. In this research we study the kinetic modelling of the enzymes in the central pathway of yeast metabolism by taking into consideration the enzyme inhibition effects to the ethanol production. The existence of positive steady state solution and the stability of the system are also analysed to study the property and dynamical behaviour of the system. One stable steady state of the system is produced if some conditions are fulfilled. The conditions concern to the restriction of the maximum reactions of the enzymes in the pyruvate and acetaldehyde branch points. There exists a certain time of fermentation reaction at which a maximum and a minimum ethanol productions are attained after regulating the system. Optimal ethanol concentration is also produced for a certain initial concentration of inhibitor.

Dynamical behavior and bifurcation analysis of a homogeneous reaction\u2013diffusion Atkinson system

In this paper, we are concerned with a homogeneous reaction–diffusion Atkinson oscillator system subject to homogeneous Neumann boundary conditions on a bounded spatial domain. Using the comparison principle and the techniques of invariant rectangle, we prove the existence of the attraction region of the solutions. We thus prove that under certain conditions, the solutions of the PDE system converge to the unique positive equilibrium solutions. We also derive precise conditions such that the system does not have nonconstant positive steady-state solutions. Finally, we use the bifurcation technique to show the existence of Turing patterns. The results provide a clearer understanding of the mechanism of formations of patterns.

Bifurcation and spatio-temporal patterns in a diffusive predator–prey system

A diffusive predator–prey system with predator functional response subject to Neumann boundary conditions is considered. Existence, nonexistence, and boundedness of non-constant positive steady state solutions are shown to identify the ranges of parameters of spatial pattern formation. Hopf and steady-state bifurcation analyses are carried out in detail. These results provide theoretical evidences to the complex spatio-temporal dynamics found by numerical simulation.