In this paper, we introduce a reaction-diffusion equation incorporating memory and carrying capacity delays. We analyze the influence of the memory-based diffusion coefficient on the stability of positive constant equilibrium of the model. When the memory-based diffusion is small, the model can undergo Turing bifurcation. When the memory-based diffusion is big, the model exhibits richer dynamical behavior: (1) For the model with spatial memory delay tau(1) only, the positive constant equilibrium is unstable for all tau(1) > 0 and model can undergo Hopf bifurcation at the critical value. (2) For the model with carrying capacity delay tau 2 only, we distinguish two cases based on the magnitude of beta. For big beta, the model can undergo Hopf bifurcation at the critical values. (3) For tau(1), tau(2) > 0, spatiotemporal patterns exhibit richer dynamical regimes. Using the geometric approach proposed by Gu et al., (2005), we find richer dynamical behavior. Finally, we validate the conclusions through numerical simulations.

In this paper, we study the impact of memory delay and distributed delay on the dynamics of a diffusive predator-prey model with predator-taxis. For the scenario without memory delay, our findings reveal that the distributed delay can lead to Turing bifurcation for the negative diffusion coefficient and lead to Hopf bifurcation for the positive diffusion coefficient. Furthermore, for fixed diffusion coefficient and distributed delay, the existence of memory delay can induce Hopf bifurcation thereby the spatially periodic solutions are generated. Also the combinations of memory delay and distributed delay can induce rich dynamic behaviors via Turing-Hopf bifurcation. Finally, we apply our theoretical findings to a predator-prey model with Holling type-I functional response to explain the emergence of various spatiotemporal patterns.

In this paper, we investigate a delayed diffusive predator-prey model. It is assumed that the predators can cooperate to improve their hunting efficiency, and the prey can employ anti-predator mechanisms when facing with the predation. Also, considering the episodic-like spatial memory of the prey, we incorporate a delayed diffusion term into the reaction-diffusion model. Through the stability and bifurcation analysis, the spatial memory-based diffusion coefficient and the memory delay have important effects on the dynamics of the model and their combined impact can cause the destabilization of the positive constant steady state and give rise to steady state and Hopf bifurcations. We present our theories and study the effect of the fear level of prey and hunting cooperation of predator on pattern formation by taking numerical simulations.

In this paper, we investigate a delayed diffusive predator-prey model incorporating hunting cooperation in predator and antipredator behaviors in prey. Firstly, we regard time delay as a bifurcation parameter and explore its impact on the stability of the positive constant equilibrium and the occurrence of Hopf bifurcation. We obtain a minimum critical value of time delay and infer that below the critical value, the positive constant equilibrium remains stable, while it becomes unstable beyond the critical value, leading to the emergence of Hopf bifurcation precisely at the critical value. Subsequently, employing the normal form theory and center manifold theorem, we compute the direction and stability of the bifurcating periodic solutions. Finally, through numerical simulations, we validate the existence of both homogeneous and nonhomogeneous periodic solutions, thereby reinforcing our theoretical findings.

This paper addresses a phytoplankton-zooplankton model on time scales. Based on a fixed point theorem of strict-set-contraction, some sufficient conditions are established for global existence of positive periodic solution to the considered system. Finally, two examples are presented to illustrate effectiveness of the results. The main purpose of this paper is to study a more general class of biological system and to investigate the existence of positive solutions for the aforementioned system using strict-set-contraction theory. It is interesting that the results of this paper are applicable to both discrete and continuous systems. Periodic behavior is widely present in biological systems and also reflects symmetry in nature. Therefore, studying the periodic solutions of biological systems can help us better understand the dynamic behavior of biological systems.

In this paper, a three-component reaction-diffusion system originating from the classical May-Nowak model for viral infection is considered in a smoothly bounded domain Omega subset of & Ropf;(n),n >= 1. It is shown that for any suitably regular non-negative initial data and arbitrary superlinear dampening, the associated no-flux type initial-boundary value problem possesses at least one globally defined solution in an appropriate generalized sense. In addition, based on an analysis of a certain eventual Lyapunov-type functional, we prove that the corresponding generalized solution asymptotically enjoys relaxation by approaching the nontrivial homogeneous steady states in the large time limit.

In this paper, we study a diffusive predator–prey model with hyperbolic mortality and prey-taxis under homogeneous Neumann boundary condition. We first analyze the influence of prey-taxis on the local stability of constant equilibria. It turns out that prey-taxis has influence on the stability of the unique positive constant equilibrium, but has no influence on the stability of the trivial equilibrium and the semi-trivial equilibrium. We then derive Hopf bifurcation and steady state bifurcation related to prey-taxis, which imply that the prey-taxis plays an important role in the dynamics.

In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation in predator increases or the fear decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions.

This paper is concerned with a predator–prey model with prey-taxis and linear prey harvesting under the homogeneous Neumann boundary condition. The stability of the unique positive constant solution of the predator–prey model without prey-taxis is derived. Also, the emergence of Hopf bifurcation is concluded by choosing the proper Hopf bifurcation parameters. By the center manifold theorem and normal form, we compute the direction of Hopf bifurcation and the stability of the bifurcating solution. Moreover, the stationary pattern with prey-taxis is investigated. The conclusions show that prey harvesting and prey-taxis can enrich the dynamics.

This paper is concerned with a diffusive predator-prey model with prey-taxis and prey-structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator-prey model is derived. Hopf bifurcation and steady state bifurcation are also concluded.

In this paper, a predator-prey model with the tendency mechanism of volume-filling effect under homogeneous Neumann boundary conditions is studied. Firstly, in the case of predation tendency and spatial diffusion, the stability of non-negative equilibria is discussed by analysing the characteristic equation of the corresponding linearisation system. Secondly, by Hopf bifurcation theorem, the existence of periodic patterns is investigated. Next, we mainly study the steady state bifurcation when the parameter χ is selected. It is shown that the chemotactic mechanism has no effect on the existence of Hopf bifurcation when α is selected as a bifurcation parameter. Whereas, the parameter χ can induce the steady state bifurcation. Finally, to illustrate the theoretical analysis, numerical simulations are carried out.

This paper is concerned with a diffusive predator–prey model with prey‐taxis and prey‐stage structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator–prey model is derived. Hopf bifurcation and steady‐state bifurcation are also concluded.

In this paper, we study a toxin-mediated size-structured population model with nonlinear reproduction, growth, and mortality rates. By using the characteristic method and the contraction mapping argument, we establish the existence-uniqueness of solutions to the model. We also prove the continuous dependence of solutions on initial conditions.

This work studies a two-dimensional beam equation with a quintic nonlinear term and quasi-periodic forcing $$\begin{aligned} u_{tt}+\Delta ^2 u+ \varepsilon \phi (t)h(u)=0,\quad x\in {\mathbb {T}}^2,\quad t\in {\mathbb {R}} \end{aligned}$$ with periodic boundary conditions, where $$\varepsilon $$ is a small positive parameter; $$\phi (t)$$ is a real analytic quasi-periodic function in t with frequency vector $$\eta =(\eta _1 ,\eta _2 \ldots ,\eta _{n^*})\subset [\varrho , 2\varrho ]^{n^*}$$ for a given positive integer $$n^*$$ and some constant $$\varrho >0$$ ; and h is a real analytic function of the form $$\begin{aligned} h(u)=c_1u+c_5u^5+\sum _{{\hat{i}}\ge 6}c_{{\hat{i}}}u^{{\hat{i}}},\quad c_1,c_5\ne 0. \end{aligned}$$ Firstly, the linear part of Hamiltonian system corresponding to the equation is transformed to constant coefficients by a linear quasi-periodic change of variables. Then, a symplectic transformation is used to convert the Hamiltonian system into an angle-dependent block-diagonal normal form, which can be achieved by selecting the appropriate tangential sites. Finally, it is obtained that a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation by developing an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional Hamiltonian systems.

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $c^*$ such that the system possesses a traveling wave solution for any given $c> c^*$. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $c=c^*$ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $c<c^*$ is also discussed.

This article is devoted to the study of a two-dimensional $(2D)$ quasi-periodically forced beam equation $$u_{tt}+\Delta^2 u+ \varepsilon\phi(t)(u+{u}^3)=0,\quad x\in\mathbb{T}^2,\quad t\in\mathbb{R}$$ under periodic boundary conditions, where $\varepsilon$ is a small positive parameter, $\phi(t)$ is a real analytic quasi-periodic function in $t$ with frequency vector $\omega=(\omega_1,\omega_2 \ldots,\omega_m)$. We prove that the equation possesses a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional Hamiltonian system. The proof is based on an infinite dimensional KAM theorem and Birkhoff normal form. By solving the measure estimation of infinitely many small divisors, we construct a symplectic coordinate transformation which can reduce the linear part of Hamiltonian system to constant coefficients. And we construct some conversion of coordinates which can change the Hamiltonian of the equation into some Birkhoff normal form depending sparse angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. Lastly, we prove that there are many quasi-periodic solutions for the above equation via an abstract KAM theorem.

The mechanism of formation and transformation of white-eye square patterns in dielectric barrier discharge system is investigated numerically, using the two-layer Lengyel–Epstein model with asymmetric and symmetric coupling. When the scale of the simulation system [Formula: see text] is two to three times of pattern wavelength [Formula: see text], it is found that an obvious intermediate state with square distribution appears by adjusting the ratio of diffusion coefficients [Formula: see text]/[Formula: see text]. When it is coupled with a suitable short-wavelength Turing mode in the range of [Formula: see text] to [Formula: see text], a new spatial resonance structure can be formed in the short-wavelength mode subsystem, and the pattern evolves from a simple square pattern to a white-eye square pattern. Although the two coupling methods achieve the same results, the duration time of the white-eye square pattern in the symmetric coupling method is significantly longer than that in the asymmetric coupling method. Because the quadratic coefficient of the amplitude equation in the reaction–diffusion system is not zero, the simple square pattern of the long wavelength mode subsystem gradually transits into a stable hexagon pattern gradually. As a result, the white-eye pattern transits from a square to a hexagon.

随着我国许多油田进入高含水开发阶段,剩余油挖潜难度增大,复合分流河道带和单河道的划分已不能满足生产要求,迫切需要进行分流河道内部有利砂体的识别.现代沉积研究是认识分流河道内砂体发育特征的有效手段.通过对鄱阳湖赣江三角洲的现场考察,发现汊口滩是三角洲分流河道中发育的重要砂体类型,以发育在分流河道的分汊口处为典型特征.根据水动力条件、沉积物组合、沉积构造等特征,将汊口滩划分为滩头、滩中、滩尾3个沉积单元.从滩头到滩尾具有水动力条件减弱、沉积物层理规模减小、单砂层厚度减小、粒度变细、泥质夹层增多的特点.汊口滩主要是由于在分汊口处,水流受到汊口的顶托作用流速降低,沉积物按粒度分异堆积形成;堆积方式主要以向上游方向的逆流加积为主.与水道砂体相比,汊口滩发育的层理类型多,而且内部夹层发育,非均质性更强;由于夹层的遮挡作用,砂体不易发生水淹,有利于形成剩余油的富集区.

In this paper, we study the spatiotemporal dynamics of a diffusive Leslie-type predator–prey system with Beddington–DeAngelis functional response under homogeneous Neumann boundary conditions. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the diffusive model, firstly, it is shown that Turing (diffusion-driven) instability occurs which induces spatial inhomogeneous patterns. Next, it is proved that the diffusive model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the diffusive model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Numerical simulations are also presented to verify the theoretical results.

In this paper, the simple and superlattice square patterns in two-dimensional space are investigated numerically by the two-layer coupled Lengyel-Epstein model. When the wave number ratio of Turing modes is greater than one, our results show that the spatial resonance form of the fundamental mode is changed with the increase of coupling strength, and simple hexagon pattern evolves spontaneously into a new pattern with a complicated structure. In addition to the reported superlattice hexagonal pattern, simple square pattern and superlattice square pattern are obtained, such as the complicated big-small spot, spot-line, ring and white-eye square pattern. The characteristics of simple and complicated superlattice square pattern are investigated by the intermediate process of evolution. When the coupling parameters \begin{document}$\alpha $\end{document} and \begin{document}$\beta $\end{document} increase synchronously within a certain range, the type I square patterns of the same wavelength are obtained in the two subsystems. When the coupling parameters \begin{document}$\alpha $\end{document} and \begin{document}$\beta $\end{document} increase asynchronously, the type I square pattern can evolve into the type II square pattern on the same spatial scale through phase transition. Then, the new subharmonic modes are generated, and the complicated superlattice square patterns are obtained due to the resonance between the two Turing modes in a short wavelength mode subsystem. The influence of coupling between two subsystems on the square pattern is investigated. When the type I square pattern of wavelength \begin{document}$\lambda $\end{document} emerges, the square pattern will quickly lose its stability in the short wavelength mode subsystem, since the coupling coefficient is equal to zero. Finally a new square pattern of wavelength \begin{document}$\lambda $\end{document}/N is formed. The type I square patterns of two subsystems successively evolve into the type II square patterns through the phase transition. The spots move relatively with the extension of simulation time, and a new mode is generated and forms three-wave resonance in two subsystems, and then the hexagonal pattern dominates the system. Our results also show that the type II square pattern spontaneously transforms into a hexagonal pattern.