Inhomogeneous Periodic Solutions Research Articles

Turing–Hopf Bifurcation Coinduced by Diffusion and Delay in Gierer–Meinhardt Systems

This paper investigates the impact of gene expression time delay and diffusion on the dynamic behavior of a class of Gierer–Meinhardt systems under Neumann boundary conditions. It provides necessary and sufficient conditions for the emergence of Hopf, Turing, and Turing–Hopf bifurcations. Utilizing the normal form of the Turing–Hopf bifurcation, the spatiotemporal dynamics close to the bifurcation point are categorized into six types, encompassing spatially inhomogeneous and homogeneous periodic solutions, as well as spatially homogeneous and inhomogeneous steady states, along with their transitions. Notably, it’s observed that the systems may lack stable spatially inhomogeneous periodic solutions under specific parameters, despite the emergence of Turing–Hopf bifurcation. These findings are supported by numerical simulations.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling

In this paper, we investigate the spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling. We first study the stability of positive equilibrium and Turing instability induced by diffusion. We then investigate the effect of delay, and it turns out that the value of the rate of recycling k plays an important role in the Hopf bifurcation induced by delay. The delay will and will not induce Hopf bifurcation with low and high level of k, respectively. To reveal the spatiotemporal dynamics, Turing–Hopf bifurcation is carried out, and normal form is derived. Many spatiotemporal dynamics are found, including the coexistence of two stable spatially inhomogeneous periodic solutions or two stable spatially inhomogeneous steadystate solutions.

Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.

Turing-Hopf Bifurcation Analysis in a Diffusive Ratio-Dependent Predator-Prey Model with Allee Effect and Predator Harvesting.

This paper investigates the complex dynamics of a ratio-dependent predator-prey model incorporating the Allee effect in prey and predator harvesting. To explore the joint effect of the harvesting effort and diffusion on the dynamics of the system, we perform the following analyses: (a) The stability of non-negative constant steady states; (b) The sufficient conditions for the occurrence of a Hopf bifurcation, Turing bifurcation, and Turing-Hopf bifurcation; (c) The derivation of the normal form near the Turing-Hopf singularity. Moreover, we provide numerical simulations to illustrate the theoretical results. The results demonstrate that the small change in harvesting effort and the ratio of the diffusion coefficients will destabilize the constant steady states and lead to the complex spatiotemporal behaviors, including homogeneous and inhomogeneous periodic solutions and nonconstant steady states. Moreover, the numerical simulations coincide with our theoretical results.

Global bifurcation in a general Leslie–Gower type predator[formula omitted]prey system with indirect prey-taxis

The influence of the indirect prey-taxis on the spatiotemporal dynamics of a class predator−prey system with predator functional response is studied. By analyzing the corresponding characteristic equation, the local stability of the positive equilibrium and the related bifurcations of the system are discussed. The critical values of the indirect prey-taxis coefficient leading to the Hopf bifurcation, steady state bifurcation, Turing-Turing bifurcation and double-Hopf bifurcation are derived. Taking the indirect prey-taxis coefficient as the bifurcation parameter, we obtain the global structure of nonconstant steady states bifurcating from the positive equilibrium by an abstract bifurcation theorem. Moreover, the stability condition of bifurcation solutions is carried out. Our results suggest that attractive indirect prey-taxis can destabilize the positive equilibrium and induced the emergence of spatially inhomogeneous periodic solution. And the higher the secretion level of chemoattractant by the prey, the more likely the system is to exhibit spatial patterns.

A migration-selection model in genetic engineering

We investigate a migration-selection system arising from CRISPR-Cas9 genetic engineering, which describes the evolution of the frequencies of a wild allele O, a drive allele D, and a brake allele B. The purpose is to see whether the drive allele D can persist in the population and whether its spread can be limited or stopped by the brake allele B when necessary. We give a complete classification of the dynamics of this system when there is no migration. We further show that migration may cause complex spatiotemporal patterns by demonstrating the existence of spatially inhomogeneous periodic solutions and steady state solutions.

Bifurcation analysis of a predator–prey model with memory-based diffusion

In this paper, we investigate the predator–prey model equipped with Fickian diffusion and memory-based diffusion of predators. The stability and bifurcation analysis explores the impacts of the memory-based diffusion and the averaged memory period on the dynamics near the positive steady state. Specifically, when the memory-based diffusion coefficient is less than a critical value, we show that the stability of the positive steady state can be destabilized as the average memory period increases, which leads to the occurrence of Hopf bifurcations. Moreover, we also analyze the bifurcation properties using the central manifold theorem and normal form theory. This allows us to prove the existence of stable spatially inhomogeneous periodic solutions arising from Hopf bifurcation. In addition, the sufficient and necessary conditions for the occurrence of stability switches are also provided.

Spatiotemporal Patterns of a Host-Generalist Parasitoid Reaction–Diffusion Model

In this paper, we study a delayed host-generalist parasitoid diffusion model subject to homogeneous Dirichlet boundary conditions, where generalist parasitoids are introduced to control the invasion of the hosts. We construct an explicit expression of positive steady-state solution using the implicit function theorem and prove its linear stability. Moreover, by applying feedback time delay [Formula: see text] as the bifurcation parameter, spatially inhomogeneous Hopf bifurcation near the positive steady-state solution is proved when [Formula: see text] is varied through a sequence of critical values. This finding implies that feedback time delay can induce spatially inhomogeneous periodic oscillatory patterns. The direction of spatially inhomogeneous Hopf bifurcation is forward when parameter [Formula: see text] is sufficiently large. We present numerical simulations and solutions to further illustrate our main theoretical results. Numerical simulations show that the period and amplitude of the inhomogeneous periodic solution increase with increasing feedback time delay. Our theoretical analysis results only hold for parameter [Formula: see text] when it is sufficiently close to 1, whereas numerical simulations suggest that spatially inhomogeneous Hopf bifurcation still occurs when [Formula: see text] is larger than 1 but not sufficiently close to 1.

Turing–Hopf bifurcation in a general Selkov–Schnakenberg reaction–diffusion system

The dynamics of a general Selkov–Schnakenberg reaction–diffusion model is investigated. We first study the globally stability of the positive equilibrium and the existence of Hopf bifurcation for the corresponding ordinary differential equation (ODE). Based on that, Turing and Turing–Hopf bifurcations due to diffusion of reaction–diffusion model are demonstrated. Using the center manifold theory and the normal form method, the bifurcation diagram of Turing–Hopf bifurcation is obtained and the spatial–temporal patterns in the different regions are considered. Numerical simulations show that the spatially homogeneous periodic, the spatially inhomogeneous and the spatially inhomogeneous periodic solutions appear.

Turing–Hopf bifurcation of a diffusive Holling–Tanner model with nonlocal effect and digestion time delay

In this paper, we study the codimension‐two Turing–Hopf bifurcation of a diffusive Holling–Tanner model with nonlocal effect and digestion time delay. The stability, Turing bifurcation, Hopf bifurcation, and Turing–Hopf bifurcation of this model are first investigated. Then, we derive the algorithm for calculating the normal form of Turing–Hopf bifurcation for this model. At last, we carry out some numerical simulations to verify our theoretical analysis results. The stable positive constant steady state and the stable spatially inhomogeneous periodic solutions are found. Furthermore, the evolution process from unstable spatially inhomogeneous steady states to stable positive constant steady state, the evolution process from unstable spatially inhomogeneous steady states to stable spatially inhomogeneous periodic solutions, and the evolution process from one unstable spatially inhomogeneous periodic solution to another stable spatially inhomogeneous periodic solution are also found.

Allee effect in a diffusive predator–prey system with nonlocal prey competition

Nonlocal competition and Allee effect have extensively been considered in modeling population dynamics of species independently. This paper introduces two aspects, which prey has nonlocal competition and predator is subject to Allee effect, in a predator–prey system to investigate the effects of predation on the spatial distribution of prey. The conditions for the coexistence equilibrium point to remain stable and to undergo spatially inhomogeneous Hopf bifurcation and Turing bifurcation have been studied. In the absence of Allee effect, we find that the coexistence equilibrium point of the system is locally asymptotically stable independent of the nonlocal competition. In the presence of Allee effect, nonlocal prey competition can destabilize the coexistence equilibrium point. Numerical simulations are carried out to illustrate the theoretical results. The amplitude of oscillation solution for nonlocal prey competition system is larger than local prey competition system until oscillation solution evolves to periodic solution. Also, nonlocal prey competition term can drive a spatially inhomogeneous Hopf bifurcation, and the spatially inhomogeneous periodic solution emerges. Moreover, it is showed that when the habitat domain is larger, comparing with local prey competition system, the prey diffusion coefficient of system with nonlocal prey competition needs to be larger for two species coexistence in the spatially homogeneous form.

Bifurcations in a diffusive predator–prey system with linear harvesting

Complex spatiotemporal dynamical behaviors of a diffusive predator–prey system with Michaelis–Menten type functional response and linear harvesting are investigated. Firstly, the critical conditions for the occurrence of Turing instability, which are necessary and sufficient, are derived in a novel way. Then, the existence conditions of codimension-1 Turing bifurcation, Hopf bifurcation, and codimension-2 Turing–Turing bifurcation, Turing–Hopf bifurcation are established. Furthermore, the detailed bifurcation set is given by calculating the amplitude equation with the method of the multiple time scale near the Turing–Hopf bifurcation. We find that the system may exhibit nonconstant steady-state solutions, spatially homogeneous periodic solutions, and spatially inhomogeneous periodic solutions, which can be verified by a series of numerical simulations. These investigations not only explain the effect of diffusion and harvesting on the dynamic behavior of the system, but also reveal the mechanism of spatiotemporal complexity in the diffusive predator–prey system.

Stability and patterns of the nutrient-microorganism model with chemotaxis

Abstract In this paper, the stability and the bifurcations of the nutrient-microorganism model with chemotaxis are analyzed, subject to no-flux boundary conditions. By choosing the chemotaxis coefficient as the control parameter, it is found that the steady state bifurcation, the Hopf–Turing bifurcation, can happen in the model. The induced spatially homogeneous periodic solution, the non-constant steady state, and the spatially inhomogeneous periodic solution are exhibited. The results suggest that chemotaxis assimilated into the model could give rise to rich spatiotemporal dynamical behaviors.

Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect

<abstract><p>Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable.</p></abstract>

Spatiotemporal dynamics of a diffusive predator-prey model with delay and Allee effect in predator.

It has been shown that Allee effect can change predator-prey dynamics and impact species persistence. Allee effect in the prey population has been widely investigated. However, the study on the Allee effect in the predator population is rare. In this paper, we investigate the spatiotemporal dynamics of a diffusive predator-prey model with digestion delay and Allee effect in the predator population. The conditions of stability and instability induced by diffusion for the positive equilibrium are obtained. The effect of delay on the dynamics of system has three different cases: (a) the delay doesn't change the stability of the positive equilibrium, (b) destabilizes and stabilizes the positive equilibrium and induces stability switches, or (c) destabilizes the positive equilibrium and induces Hopf bifurcation, which is revealed (numerically) to be corresponding to high, intermediate or low level of Allee effect, respectively. To figure out the joint effect of delay and diffusion, we carry out Turing-Hopf bifurcation analysis and derive its normal form, from which we can obtain the classification of dynamics near Turing-Hopf bifurcation point. Complex spatiotemporal dynamical behaviors are found, including the coexistence of two stable spatially homogeneous or inhomogeneous periodic solutions and two stable spatially inhomogeneous quasi-periodic solutions. It deepens our understanding of the effects of Allee effect in the predator population and presents new phenomena induced be delay with spatial diffusion.

Turing–Hopf Bifurcation Analysis of the Sel’kov–Schnakenberg System

In this paper, we investigate the spatiotemporal dynamics of the Sel’kov–Schnakenberg system. The stability of the positive constant steady state is studied by the linear stability theory. Hopf bifurcation and Turing–Hopf bifurcation are generated by varying two parameters in the model. The normal form near the Turing–Hopf singularity is calculated to explore the complex dynamics of the system. Finally, numerical simulations are carried out to verify the theoretical results. Our results show that the Sel’kov–Schnakenberg system exhibits complex dynamics near the Turing–Hopf singularity, including the existence of inhomogeneous steady states, homogeneous periodic solutions and inhomogeneous periodic solutions.

Complex dynamics induced by harvesting rate and delay in a diffusive Leslie-Gower predator-prey model

<abstract><p>In this paper, under homogeneous Neumann boundary conditions, the complex dynamical behaviors of a diffusive Leslie-Gower predator-prey model with a ratio-dependent Holling type III functional response and nonlinear prey harvesting is carefully studied. By scrupulously analyzing and comprehending the distribution of the eigenvalues, the existence and stability (balance) of the extinction and coexistence equilibrium states are determined, and the bifurcations exhibited by the system are investigated by a mathematical analysis. Additionally, based on the theoretical analysis and numerical simulation, (Harvesting rate-induced, Delay-induced), Turing-Hopf bifurcations points are derived. Our results show that delay and nonlinear prey harvesting rates can create spatially inhomogeneous periodic solutions.</p></abstract>

Bifurcation analysis and spatiotemporal patterns in delayed Schnakenberg reaction-diffusion model

The diffusive Schnakenberg model with gene expression time delay is considered. In this paper, the stability, diffusion-driven instability, and time delay-induced Hopf bifurcation have been investigated. By linear stability analysis, we find the parameter areas where the unique positive equilibrium is stable and Turing instability can occur for a certain relationship of diffusion rates. Then we obtain a series of critical values for the time delay at which the spatially homogeneous and inhomogeneous periodic solutions may emerge. Based on the explicit formula determining the properties of the Hopf bifurcation, we employ numerical simulations for parameters both in the stable region and Turing instability region. The numerical simulations show that delay can destabilize the stability of the positive equilibrium solution and eventually induce spatially homogeneous and inhomogeneous periodic solutions. Furthermore, the spatiotemporal patterns in the two spaces dimension from the Turing instability regime provide an indication of the wealth of patterns that the delayed system can exhibit.