. Nonlocal advection and time delay play important roles on the movement of macroscopic and microscopic substances. In this paper, we study a nonlocal advection-diffusion model with time delay being involved in the advection term. For the proposed model, the compactness of solution operators and global boundedness of solutions are proved. Based on these results, we further investigate the local dynamics near the positive steady state, when the kernel is taken as a triangular function. The results show that the sign and magnitude of the mean of kernel will cause Hopf bifurcation and bifurcation switches, while it has been shown in the literature that the positive steady state is always stable for symmetric kernel, such as top-hat function. Finally, we construct a Lyapunov functional to show the global asymptotic stability of the steady state, when the advection coefficient is less than a critical value.

In this paper, we have taken into account a model that incorporates two memory integrals with weak memory kernel and is delimited by a boundary consisting of Dirichlet boundary and no-flux boundary. We prove the existence of the global attractor and obtain conditions for the global stability of trivial steady-state via constructing the Lyapunov functional. Subsequently, the stability of the trivial steady-state solutions and associated bifurcation for this model are extensively investigated. It turns out that, for small parameter μ , the solutions of the equation will exhibit various dynamical behaviors when the parameters R and λ are altered. For the inhomogeneous steady-state solution bifurcated from the trivial solution, we also provided the conditions for its stability and instability.

A class of diffusive partial differential equations with strongly coupled time delays and diffusion is considered. The global boundedness of weak solutions of the equation is proved by an entropy method that was initially proposed for studying the global boundedness of reaction-diffusion equations with cross-diffusion. The presence of the time delays in the equation prevents the entropy method to be directly applied, and here we extend the entropy method to this class of diffusive partial differential equations with time delays by proving some key entropy inequalities, which further allows us to obtain the estimates of gradient of the solutions. The results can be used to show the global boundedness of solutions of population models with memory effect, which were recently proposed for describing the movement of highly-developed animal species. In addition, we show that the results are also applicable for the classic partial functional differential equations, where the time delays only appear in the reaction terms. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, a mathematical model for HIV-1 infection and immune response is considered, involving two discrete time delays in the intracellular as well as in activation of immune response. Using a recently developed geometric method for studying a class of transcendental equation with two time delays and delay dependent coefficients, we obtain the stability and bifurcation results at the non-trivial equilibrium. In particular, the crossing curves on the two-delays parameter plane can be completely characterized, on which Hopf and double-Hopf bifurcation will take place. In the case of Hopf bifurcation, there exist stability switches, and the direction and stability of delay induced Hopf-bifurcation can be determined using normal form theory and center manifold theorem. These results imply the model will exhibit complex temporal dynamics, such as period oscillations, quasi-periodic solutions, etc. Numerical examples are also carried out to verify these results.

In this paper, we propose and analyze a diffusive predator–prey model with memory-based diffusion, which is described by a delayed predator-taxis term. We first obtain the existence and uniform boundedness of solutions. Then we study the stability of positive constant equilibria. We find that the memory delay may induce Hopf bifurcation, generating spatially inhomogeneous periodic solutions. This result implies that the memory-based diffusion is beneficial to pattern formations and is contrary to the conclusion on predator–prey model only with predator-taxis. The normal form of Hopf bifurcation is also derived. The theoretical results are demonstrated through the case with the Holling II functional response. Numerical simulations also suggest the phenomenon of stability switches and the occurrence of double Hopf bifurcation.

In this paper, a diffusive competition model is formulated by considering the memory-based diffusions of species in their movement and spatial heterogeneity of the resource. The global existence of solution for the proposed model is proved. Under weak competition, the existence and stability of the positive steady state are obtained, when the random diffusion rates are large enough, by using implicit function theorem and some priori estimations. However, when memory-based diffusions dominate the dynamics, spatial-temporal patterns can be observed. We also consider the case that both competition coefficients equal to one, showing that the stabilities of semi-trivial steady states, existence of positive steady state by the method of upper and lower solutions, and bistable dynamics of the model. The results reveal that the memory-based diffusion plays an important role on the competition outcomes. Specifically, the species with slower random diffusion may not help itself to wipe out its competitor because of disadvantageous memory-based diffusion, while the species with faster random diffusion but with advantageous memory-based diffusion may win the competition.

In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results.

In this paper, a single reaction-diffusion population model with memory effect and the heterogeneity of the environment, equipped with the Neumann boundary, is considered. The global existence of a spatial nonhomogeneous steady state is proved by the method of super and subsolutions, which is linearly stable for relatively small memory-induced diffusion. However, after the memory-induced diffusion rate exceeding a critical value, spatial inhomogeneous periodic solution can be generated through Hopf bifurcation, if the integral of intrinsic growth rate over the domain is negative. Such phenomenon will never happen, if only memory-induced diffusion or spatially heterogeneity is presented, and therefore must be induced by their joint effects. This indicates that the memory-induced diffusion will bring about spatial-temporal patterns in the overall hostile environment. When the integral of intrinsic growth rate over the domain is positive, it turns out that the steady state is still linearly stable. Finally, the possible dynamics of the model is also discussed, if the boundary condition is replaced by Dirichlet condition.

In this article, we study a predator-prey model with age structure, Holling-type IV response, and two time delays. By an algebraic method, we determine all the critical values for these two delays, such that the characteristic equation has purely imaginary roots. This provides a sharp stability region on the parameter plane of the positive equilibrium. Applying integrated semigroup theory and Hopf bifurcation theorem for abstract Cauchy problems with non-dense domain, we can show the occurrence of Hopf bifurcation as the time delays pass through these critical values. In particular, the phenomenon of stability switches can also be observed as the time delays vary. Numerical simulations are carried out to illustrate the theoretical results. For more information see https://ejde.math.txstate.edu/Volumes/2021/42/abstr.html

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented.

In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results.

Two n-species stochastic type K monotone Lotka-Volterra systems are proposed and investigated. For non-autonomous system, we show that there is a unique positive solution to the model for any positive initial value. Moreover, sufficient conditions for stochastic permanence and global attractivity are established. For autonomous system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant equals 1, then the corresponding species will be nonpersistent on average. To illustrate the theoretical results, the corresponding numerical simulations are also given.

In this paper, a predator-prey model with age structure, stocking rate and two delays is investigated. We show that Hopf bifurcation occurs when one of the time delay $τ$ crosses a sequence of critical values, by applyingHopf bifurcation theory for abstract Cauchy problems with non-dense domain. Numerical simulations are included to verify our results and a summary is also given.

In this paper, a multi-group SVIR epidemic model with age of vaccination is considered. The model allows the vaccinated individuals to become susceptible after the vaccine loses its protective properties, and the vaccination classes satisfy first-order the partial differential equations structured by vaccination age. Combining the Lyapunov functional method with a graph-theoretic approach, we show that the global stability of endemic equilibrium for the strongly connected system is determined by the basic reproduction number. In addition, the dynamics for non-strongly connected model are also investigated, depending on the basic reproduction numbers corresponding to each strongly connected component. Numerical simulations are carried out to support the theoretical conclusions.

In this paper, strongly connected and non-strongly connected multi-group viral models with time delays and general incidence functions are considered. Employing the Lyapunov functional method and a graph-theoretic approach, we show that the global dynamics of the strongly connected system are determined by the basic reproduction number under some reasonable conditions for incidence functions. In addition, we find a more complex and more interesting result for multi-group viral models with non-strongly connected networks because of the basic reproduction numbers corresponding to each strongly connected component. Finally, we provide simulations for non-strongly connected multi-group viral models to support our conclusion.

研究具有双线性发生率的带时滞的多组病毒模型,分别针对强连通和非强连通情形,得到基本再生数R0.利用Lyapunov泛函方法和LaSalle不变集原理,分别证明当R0＜1时无感染平衡点P0的全局渐近稳定性以及R0 ＞1时慢性感染平衡点P*的全局渐近稳定性.

This paper is devoted to the study of an eco-epidemiological model with stage structure in the predator and disease in the prey. To begin with, the positivity and boundedness of the solutions are obtained. This shows that the system possesses a bounded absorbing set. Then, by using the LaSalle-Lyapunov invariance principle, limit equation theory, and a geometrical criterion for analyzing the distribution of the eigenvalues, the stability of the boundary equilibria and interior equilibrium are established, respectively. Meanwhile, the existence of Hopf bifurcations is obtained when the delay τ varies in a limitary region. Furthermore, by employing center manifold theory and the normal form method, an algorithm for determining the direction and stability of the Hopf bifurcation is derived. At last, some numerical simulations are carried out for illustrating the analytic results.

This paper presents a one-layer neural network to solve nonsmooth convex optimization problems based on the Tikhonov regularization method. Firstly, it is shown that the optimal solution of the original problem can be approximated by the optimal solution of a strongly convex optimization problems. Then, it is proved that for any initial point, the state of the proposed neural network enters the equality feasible region in finite time, and is globally convergent to the unique optimal solution of the related strongly convex optimization problems. Compared with the existing neural networks, the proposed neural network has lower model complexity and does not need penalty parameters. In the end, some numerical examples and application are given to illustrate the effectiveness and improvement of the proposed neural network.

A stochastic Poincaré–Bendixson theorem and its generalized theorem are given in this paper. The aim of these theorems is to show the existence of a crater-like stationary distribution for a stochastic planar autonomous system. The main theorems are applied on a stochastic predator-prey system as an instruction. For example, stochastic Hopf bifurcation phenomenon is observed when the qualitative change of the shape of the stationary distribution is taken as an indicator of bifurcation.

In this paper, we mainly study the global robust exponential stability of the neural networks with possibly unbounded activation functions. Based on the topological degree theory and Lyapunov functional method, we provide some new sufficient conditions for the global robust exponential stability. Under these conditions, we prove existence, uniqueness and global robust exponential stability of equilibrium point. In the end, some examples are provided to demonstrate the validity of the theoretical results.