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In this paper, we study the local dynamical behavior of a Brusselator-type model with state-dependent delay. First, we investigate the local stability of the positive equilibrium and the existence of the Hopf bifurcation. Secondly, the asymptotic of the solutions near the equilibria are constructed by using the perturbation method which further determines the properties of Hopf bifurcation. Finally, the numerical simulations are carried out to support the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global dynamics of a Lotka-Volterra competition-diffusion-advection system with nonlinear boundary conditions, including the existence, nonexistence and global stability of coexistence steady states. We start with the investigation of the principal eigenvalue of linearized system to get the local stability of steady states and then discuss the global dynamics in terms of competition coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a neural network model consisting of four neurons with delayed self and nearest-neighbor connections. We provide a full classification of all equilibria and periodic oscillations in the connection weight parameter space. Such a classification is essential for the description of spatio-temporal patterns of the model system and for the applications to dynamic memory storage and retrieval. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with a forager–exploiter model with homogeneous Neumann boundary condition: u t = d 1 Δ u - ∇ · (χ (w) u ∇ w) + a 1 u - b 1 u θ 1 x ∈ Ω , t > 0 , v t = d 2 Δ v - ∇ · (ρ (u) v ∇ u) + a 2 v - b 2 v θ 2 x ∈ Ω , t > 0 , w t = d 3 Δ w - α (u + v) w - β w + r (x , t) x ∈ Ω , t > 0 , ∂ ν u = ∂ ν v = ∂ ν w = 0 x ∈ ∂ Ω , t > 0 , u (x , 0) = u 0 (x) ≥ 0 , v (x , 0) = v 0 (x) ≥ 0 w (x , 0) = w 0 (x) ≥ 0 x ∈ Ω , where Ω ⊂ R n (n ≥ 2) is a bounded domain, the constants a 1 , a 2 , b 1 , b 2 , θ 1 , θ 2 , α , β are positive. The nonnegative functions χ (w) , ρ (u) , r (x , t) satisfy χ (w) ∈ C 1 ([ 0 , ∞)) , ρ (u) ∈ C 1 ([ 0 , ∞)) and r (x , t) ∈ C 1 (Ω ¯ × [ 0 , ∞)) ∩ L ∞ (Ω × (0 , ∞)) , respectively. The nonnegative initial functions satisfy u 0 , w 0 ∈ W 2 , ∞ (Ω) , v 0 ∈ W 1 , ∞ (Ω) and ∂ ν u 0 = ∂ ν v 0 = ∂ ν w 0 = 0 . When n > 2 , θ 1 > 2 and θ 2 > n 2 + 1 , we prove that this problem possesses a global classical solution, which is uniformly bounded. This result improves the work of Wang (Nonlinear Anal 222: 112985, 2022), in which the global boundedness of solution is established for θ 1 > n 2 + 1 , θ 2 > n 2 + 1 , χ (w) ≡ χ and ρ (u) ≡ ρ and χ , ρ > 0 are two constants. Moreover, when n = 2 , either θ 1 = 2 and θ 2 > 2 or 1 < θ 1 < 2 , θ 2 > 2 and ∫ t t + 1 ∫ Ω | ∇ r (x , t) | 2 < ∞ for all t > 0 , this problem also possesses a global bounded classical solution. This result partially generalizes previously known ones. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, multiplicity, stability, and Hopf bifurcation of steady-state solutions of a diffusive Lotka–Volterra type model for two species with nonlocal delay effect and nonlinear boundary condition. It is found that there is no Hopf bifurcation when the interior reaction term is weaker than the boundary reaction term, and that the interior reaction delay determines the existence of Hopf bifurcation only when the interior reaction term is stronger than the boundary reaction term. This observation helps us to understand the nonlinear balance between the interior reaction and boundary flux in nonlinear boundary problems. Moreover, the general results are illustrated by applications to a model with homogeneous kernels. [ABSTRACT FROM AUTHOR]

We focus upon the existence of periodic traveling waves in a nonlocal reaction–diffusion model with chemotaxis and nonlocal delay effect. Based on the perturbation method, the existence of periodic traveling waves with large wave speed are obtained by the Lyapunov–Schmidt reduction and the implicit function theorem. The trick of the proof is to treat the wave profile as a solution of an operator in a suitable Banach space. [ABSTRACT FROM AUTHOR]

A single-species model subject to nonlinear boundary conditions is investigated. Existence, nonexistence, and local/global asymptotical stability of positive steady-state solutions are shown to identify the ranges of parameters, the growth rate functions, and boundary reaction functions. In particular, we show that the model can have multiple equilibria and in some parameter ranges can support Allee effects. [ABSTRACT FROM AUTHOR]

We apply the 1 -equivariant degree method to a Hopf bifurcation problem for functional differential equations with a state-dependent delay. The formal linearization of the system at a stationary state is extracted and translated into a bifurcation invariant by using the homotopy invariance of 1 -equivariant degree. As a result, the local Hopf bifurcation is detected and the global continuation of periodic solutions is described. [ABSTRACT FROM AUTHOR]

This paper is concerned with a delayed reaction–diffusion equation on a unit disk. By means of the singularity theory and Lyapunov–Schmidt reduction, we not only derive universal conclusions about the existence of inhomogeneous steady-state solutions and the equivariant Hopf bifurcation theorems, but also obtain some more extraordinary properties of bifurcating solutions, which are produced by the radial symmetry through abstract methods based on the Lie group representation theory. Meanwhile, we illustrate our results by an application to a population model with a time delay. Furthermore, the methods established in this paper are applicable to specific delayed reaction–diffusion models with other symmetries. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a reaction–diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov–Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay τ changes. Finally, we illustrate our general theoretical results by an application to the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction. [ABSTRACT FROM AUTHOR]

In this paper, we study the local dynamics of a class of 3-dimensional Lotka-Volterra systems with a discrete delay. This system describes two predators competing for one prey. Firstly, linear stability and Hopf bifurcation are investigated. Then some regions of attraction for the positive steady state are obtained by means of Liapunov functional in a restricted region. Finally, sufficient and necessary conditions for the principle of competitive exclusion are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the existence and nonexistence of traveling waves for a diffusive epidemic model with a general nonlinear incidence rate and infection-age structure. It is observed that whether the disease can spread or not depends on the basic reproduction number R 0 and critical wave speed c ∗ . More precisely, the traveling wave solution exists when R 0 > 1 and c ≥ c ∗ , while the traveling wave solution vanishes when R 0 ≤ 1 or R 0 > 1 and c ∈ (0 , c ∗) . By constructing a new pair of upper and lower solutions, an open problem proposed by Ducrot and Magal (Proc R Soc Edinb-A 139:459–482, 2009; Nonlinearity 24:2891–2911, 2011) is solved. It is also shown that infection-age structure can reduce the speed of disappearance of infectious diseases. We also investigate the effects of nonlinear incidence rate and age structure on the basic reproduction number and critical wave speed. Our results generalize some known results. [ABSTRACT FROM AUTHOR]

We investigate the existence and branching patterns of wave trains (also called periodic traveling waves) in a 2D face-centered square lattice consisting of alternating light and heavy atoms, with linear coupling between nearest particles and a nonlinear substrate potential. In contrast to monatomic chains, we consider two different periodic waveform functions corresponding to light and heavy particles, respectively. As a result, we have to solve coupled advanced-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction. Then we obtain the small-amplitude solutions in the Hamiltonian system near equilibria in nonresonance and p : q resonance, respectively. In particular, the results can be applied to some one-dimensional diatomic lattices. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predatorâ€"prey system with nonlinear prey-taxis under Neumann boundary condition. For a class of chemotactic sensitive functions, we obtain the existence and boundedness of global classical solutions for initial boundary value problems in arbitrary dimensional space. In addition, we also study the local stability of the constant steady state solution, and obtain the global asymptotic stability of the steady state solution under different predation intensity by constructing appropriate Lyapunov functions. Furthermore, the steady state bifurcation, Hopf bifurcation and fold-Hopf Singularity are analysed in detail by using Lyapunovâ€"Schmidt reduction method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the long term behavior of solutions to stochastic delay differential equations with additive noise. We first establish the existence of an invariant measure by regarding the stochastic equation as a perturbation of a deterministic equation. If the deterministic part is stable it is plausible to expect the existence of an invariant measure under some conditions on the perturbation. Different from traditional methods, a retarded Ornstein-Uhlenbeck process is introduced to the study of the existence of a continuous random dynamical system and tempered pullback attractors. In particular, we prove pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors. We also establish the upper semicontinuity of the random attractors when noise intensity approaches zero, and investigate the exponential stability of stationary solutions in the mean square sense and almost surely. It is found that our theoretical results and methods can be generalized to other stochastic delay differential equations, even with multiplicative noise. Finally, our theoretical results are illustrated by an applications to a stochastic Nicholson’s blowflies model. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nutrient-microorganism system with nutrient-taxis in the sediment. The global existence and uniform-in-time bound of classical solutions in a domain with any dimension is established, and the linearized stability of the constant interior steady state is analyzed, which shows that different types of nutrient-taxis may (de-)stabilize the positive constant equilibrium. Moreover, over 1D domain, we obtain the local existence and stability of nonconstant steady states near the positive constant equilibrium via bifurcation theory. Finally, several numerical simulations on one- and two-dimensional spatial domains are presented, which demonstrate that this reaction-advection-diffusion system can admit numerous interesting spatio-temporal patterns. [ABSTRACT FROM AUTHOR]

This paper is devoted to an SEIR epidemic model with variable recruitment and both exposed and infected populations having infectious in a spatially heterogeneous environment. The basic reproduction number is defined and the existence of endemic equilibrium is obtained, and the relationship between the basic reproduction number and diffusion coefficients is established. Then the global stability of the endemic equilibrium in a homogeneous environment is investigated. Finally, the asymptotic profiles of endemic equilibrium are discussed, when the diffusion rates of susceptible, exposed and infected individuals tend to zero or infinity. The theoretical results show that limiting the movement of exposed, infected and recovered individuals can eliminate the disease in low-risk sites, while the disease is still persistent in high-risk sites. Therefore, the presence of exposed individuals with infectious greatly increases the difficulty of disease prevention and control. [ABSTRACT FROM AUTHOR]

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction. [ABSTRACT FROM AUTHOR]

A stage‐structure competition‐diffusion model with spatial heterogeneity is investigated in this paper. Under some suitable assumptions, the existence of spatially non‐homogeneous steady‐state solutions is established by investigating eigenvalue problems with indefinite weight and employing Lyapunov‐Schmidt reduction. The stability of spatially nonhomogeneous steady‐state solutions is obtained by analyzing the set of the spectrum of the associated infinitesimal generator. In particular, the global stability of the steady‐state solutions is described in weak competition case. [ABSTRACT FROM AUTHOR]

This paper is devoted to a stochastic regime-switching susceptible-infected-susceptible epidemic model with nonlinear incidence rate and Lévy jumps. A threshold λ in terms of the invariant measure, different from the usual basic reproduction number, is obtained to completely determine the extinction and prevalence of the disease: if λ > 0, the disease is persistent and there is a stationary distribution; if λ < 0, the disease goes to extinction and the susceptible population converges weakly to a boundary distribution. Moreover, some numerical simulations are performed to illustrate our theoretical results. It is very interesting to notice that random fluctuations (including the white noise and Lévy noise) acting the infected individuals can prevent the outbreak of disease, that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime, and that the prevalence of the disease can also be controlled by reducing the value of transmission rate of disease. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to the existence and uniqueness, and especially the global asymptotical stability of the coexistence steady state for a competitive Lotka-Volterra reaction-diffusion model with advection term arising in ecology. Our approaches utilized here include the monotone dynamical systems theory, the sub-super solutions method, the principal spectral theory, and some other nontrivial analytic skills. [ABSTRACT FROM AUTHOR]

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive Lotka-Volterra type competition system with advection terms under the homogeneous Dirichlet boundary condition. First, we obtain the existence, multiplicity and explicit structure of the spatially nonhomogeneous steady-state solutions by using implicit function theorem and Lyapunov-Schmidt reduction method. Secondly, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of spatially nonhomogeneous positive steady-state solutions and the non-existence of Hopf bifurcations at spatially nonhomogeneous positive steady-state solutions are given. Finally, two concrete examples are provided to support our previous theoretical results. It should be noticed that an elliptic operator with advection term is not self-adjoint, which causes some trouble in the spatial decomposition, explicit expressions of steady-state solutions and some deductive processes related to infinitesimal generators. Moreover, unlike other work, the advection rate here depends on the spatial position, which increases some difficulties in the investigation of the principal eigenvalue. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the dynamics of a stochastic susceptible-infected-susceptible epidemic model with nonlinear incidence rate and three independent Brownian motions. By defining a threshold λ, it is proved that if λ > 0, the disease is permanent and there is a stationary distribution. And when λ < 0, we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution. Moreover, the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results. As a result, appropriate intensities of white noises make the susceptible and infected individuals fluctuate around their deterministic steady–state values; the larger the intensities of the white noises are, the larger amplitude of their fluctuations; but too large intensities of white noises may make both of the susceptible and infected individuals go to extinction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic dynamics of Fisher-KPP equations with nonlocal dispersal operator and nonlocal reaction term in time periodic and space heterogeneous media. We first show the global existence and boundedness of nonnegative solutions. Next, we obtain some sufficient conditions ensuring the uniform persistence. Finally, we study the existence, uniqueness and global stability of positive time periodic solutions under several different conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to periodic travelling waves in a two-dimensional nonautonomous weakly damped lattice system with linear coupling between nearest particles and periodic nonlinear substrate potentials. Nonlinear functional analysis is employed to prove the existence and uniqueness of periodic travelling wave solutions. In the case of small forcing and damping, Lyapunov–Schmidt reduction is employed to study the bifurcation of periodic travelling wave solutions and the asymptotic expressions of the bifurcating solutions. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion model with delay effect and Dirichlet boundary condition is considered. Firstly, the existence, multiplicity, and patterns of spatially nonhomogeneous steady-state solution are obtained by using the Lyapunov–Schmidt reduction. Secondly, by means of space decomposition, we subtly discuss the distribution of eigenvalues of the infinitesimal generator associated with the linearized system at a spatially nonhomogeneous synchronous steady-state solution, and then we derive some sufficient conditions to ensure that the nontrivial synchronous steady-state solution is asymptotically stable. By using the symmetric bifurcation theory of differential equations together with the representation theory of standard dihedral groups, we not only investigate the effect of time delay on the pattern formation, but also obtain some important results on the spontaneous bifurcation of multiple branches of nonlinear wave solutions and their spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

In this paper, we consider the signal-dependent diffusion and sensitivity in a chemotaxis–competition population system with two different signals in a two-dimensional bounded domain. We consider more general signal production functions and assume that the signal-dependent diffusion is a decreasing function which may be degenerate with respect to the density of the corresponding signal. We first obtain the global existence and uniform-in-time bound of classical solutions and show that the blow-up effect can be precluded for signal-dependent diffusion and sensitivity with certain properties. Then, by constructing Lyapunov functionals, we study the global attractivity of nonzero (boundary/positive) homogeneous steady states under three different strengths of competition. In particular, we obtain that the nonzero boundary constant steady states are globally asymptotically stable when they are globally attractive, which means no pattern formation occurs, while for interior constant steady state, its global attractivity can imply the global stability for some special signal production functions. Finally, numerical simulations show that for large signal sensitivity, different signal production functions can lead to various complex spatial–temporal patterns around the positive homogeneous steady state. In particular, for a given signal production mechanism, various patterns are observed for different population growth rates. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment. The global existence and boundedness of solutions are shown. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global stability of semi-trivial solutions. The existence of positive steady state solution bifurcating from semi-trivial solutions is obtained by using local bifurcation theory. The stability analysis of the positive steady state solution is investigated in detail. In addition, we explore the asymptotic profiles of the steady state solution for small and large diffusion rates. [ABSTRACT FROM AUTHOR]

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough. [ABSTRACT FROM AUTHOR]

In this paper, we extend the equivariant Hopf bifurcation theory for semilinear functional differential equations in general Banach spaces and then apply it to reaction–diffusion models with delay effect and homogeneous Dirichlet boundary condition on a general open domain with a smooth boundary. In the process we derive the criteria for the existence and directions of branches of bifurcating periodic solutions, avoiding the process of center manifold reduction. [ABSTRACT FROM AUTHOR]

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to a two-species Lotka-Volterra model with general functional response. The existence, local and global stability of boundary (including trivial and semi-trivial) steady-state solutions are analyzed by means of the signs of the associated principal eigenvalues. Moreover, the nonexistence and steady-state bifurcation of coexistence steady-state solutions at each of the boundary steady states are investigated. In particular, the coincidence of bifurcating coexistence steady-state solution branches is also described. It should be pointed out that the methods we applied here are mainly based on spectral analysis, perturbation theory, comparison principle, monotone theory, Lyapunov-Schmidt reduction, and bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized predator–prey system with prey-taxis under Neumann boundary condition, that is, the predators can survive even in the absence of the prey species. It is proved that for an arbitrary spatial dimension, the corresponding initial boundary value problem possesses a unique global bounded classical solution when the prey-taxis is restricted to a small range. Moreover, the local stabilities of constant steady states (including trivial, semi-trivial and positive constant steady states) are investigated. A further study on the coexistence steady state implies that the prey-taxis term suppresses the global asymptotical stability and influences the steady-state/Hopf bifurcations (if they exist). Analyses of steady-state bifurcation, Hopf bifurcation, and even Hopf/steady-state mode interaction are carried out in detail by means of the Lyapunov–Schmidt procedure. In particular, we obtain stable or unstable steady states, time-periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. These results provide theoretical evidences to the complex spatiotemporal dynamics found in numerical simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the following two-species chemotaxis system {ut = ∆u − χ1∇ · (u∇v) + μ1u(1 − u − a1w), x ∈ Ω,t > 0, vt = ∆v − v + h(w), x ∈ Ω,t > 0, wt = ∆w − χ2∇ · (w∇z) + μ2w(1 − w − a2u), x ∈ Ω, t > 0, zt = ∆z − z+h(u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ Rn with smooth boundary. The parameters in the system are positive and the signal production function h is a prescribed C1-regular function. The main objectives of this paper are two-fold: One is the existence and boundedness of global solutions, the other is the large time behavior of the global bounded solutions in three competition cases (i.e., a weak competition case, a partially strong competition case and a fully strong competition case). It is shown that the unique positive spatially homogeneous equilibrium (u∗,v∗,w∗,z∗) may be globally attractive in the weak competition case (i.e., 0 < a1,a2 < 1), while the constant stationary solution (0, h(1), 1, 0) may be globally attractive and globally stable in the partially strong competition case (i.e., a1 >1 > a2 > 0). In the fully strong competition case (i.e. a1,a2 > 1), however, we can only obtain the local stability of the two semi-trivial stationary solutions (0, h(1), 1, 0) and (1, 0, 0, h(1)) and the instability of the positive spatially homogeneous (u∗,v∗,w∗,z∗). The matter which species ultimately wins out depends crucially on the starting advantage each species has. [ABSTRACT FROM AUTHOR]

This paper is devoted to a spatial heterogeneous SIS model with the infected group equipped with a free boundary. Our main aim is to determine whether the disease is spreading forever or extinct eventually, and to illustrate, under the nonhomogeneous spatial environment, free boundaries can have a large influence on the infected behavior at the large time. For this purpose, we first introduce a basic reproduction number and then establish a spreading-vanishing dichotomy. Then by investigating the effect of the diffusion rate, initial domain and spreading speed on the asymptotic behavior of the infected group, we establish some sufficient conditions and even necessary and sufficient conditions for disease spreading or vanishing. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space-time white noise. Firstly, we establish an existence-uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence-uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition. [ABSTRACT FROM AUTHOR]

In this paper, a diffusive predator-prey model with nonmonotonic functional response is investigated. The stability of the positive spatially homogeneous steady states and bifurcations of spatially homogeneous/nonhomogeneous time-periodic solutions as well as steady-state solutions are studied. In particular, the formulas determining the direction and the stability of the bifurcating periodic solutions are obtained. These results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized predator‐prey system with prey‐taxis under the Neumann boundary condition. We investigate the local and global asymptotical stability of constant steady states (including trivial, semitrivial, and interior constant steady states). On the basis of a priori estimate and the fixed‐point index theory, several sufficient conditions for the nonexistence/existence of nonconstant positive solutions are given. [ABSTRACT FROM AUTHOR]

This paper is concerned with a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global attractivity of the semi-trivial solution. In addition, an attracting region was obtained by means of the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the existence, monotonicity, uniqueness, and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations. [ABSTRACT FROM AUTHOR]

Chemotaxis is an observed phenomenon in which a biological individual moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a reaction-diffusion model with chemotaxis and nonlocal delay effect under Dirichlet boundary condition by using Lyapunov-Schmidt reduction and the implicit function theorem. The existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady state solutions are investigated. Moreover, our results are illustrated by an application to the model with a logistic source, homogeneous kernel and one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlocal dispersal problem with a nonlocal term arising in population dynamics: where is a bounded domain, , represents the nonlocal dispersal operator with continuous and non-negative dispersal kernel. The kernel is assumed to be non-negative and is allowed to have a degeneracy in a smooth subdomain of . When is either positive or vanishes in a subdomain, we respectively investigate the existence, multiplicity and asymptotical stability of positive steady states under the local/global variation of parameter by means of sub-supersolution method, Lyapunov-Schmidt reduction, and bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous) steady-state solution and periodic solutions for a reaction-diffusion model with nonlocal delay effect and Dirichlet/Neumann boundary condition are investigated by using Lyapunov-Schmidt reduction. Moreover, we illustrate our general results by applications to population models with one-dimensional spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a Nicholson's blowflies equation with state-dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state-dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper focuses on a Lotka-Volterra model with delay and cross-diffusion. By using Lyapunov-Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov-Schmidt reduction. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a Mimura-Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov-Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady-state solutions bifurcating from the nonzero steady-state solution. Moreover, we illustrate our general results by applications to models with a one-dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the stability and bifurcation of a class of two-dimensional stochastic differential equations with multiplicative excitations are investigated. Firstly, we employ Taylor expansions, polar coordinate transformation and stochastic averaging method to transform the original system into an Itô averaging diffusion system. Secondly, we apply the maximum Lyapunov exponent and the singular boundary theory to analyze the local and global stability of the fixed point. Thirdly, we explore the stochastic dynamical bifurcation of the Itô averaging amplitude equation by studying the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. Finally, an example is given to illustrate the effectiveness of our analyzing procedure. [ABSTRACT FROM AUTHOR]

This paper is devoted to a reaction cross-diffusion system under Neumann boundary conditions. Firstly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are investigated by means of Lyapunov-Schmidt reduction. Next, the linear stability and Hopf bifurcations of homogeneous steady-state solutions are described in detail. In particular, the Hopf bifurcation direction and the stability of bifurcating time-periodic solutions are determined by using center manifold reduction and normal form theory. Finally, some of the main results are illustrated by an application to a predator-prey model with Allee effect and one-dimensional spatial domain . [ABSTRACT FROM AUTHOR]