Positive Equilibrium Research Articles

Turing instability analysis of a reaction–diffusion system for rumor propagation in continuous space and complex networks

Research on the propagation of online rumors necessitates careful consideration of rumor-spreading mechanisms. Media correction, self-correction, and the time delays associated with contact propagation are expressly introduced into the established reactive–diffusion rumor-spreading model in this paper. The conditions for a unique positive equilibrium point in the model are analyzed, laying the necessary groundwork for the occurrence of Turing instability in reaction–diffusion rumor propagation systems. A linearization approach is employed to derive the conditions for Turing instability in scenarios involving small delay to address the challenges posed by models with time delay. Furthermore, in order to anticipate the Turing pattern formation in advance, we employed a multiscale analysis approach to derive the amplitude equation within this model. The validity of the theoretical results is verified through numerical simulations, while the propagation dynamics of the model were analyzed within two complex network models, namely small-world networks and scale-free networks. Simulation results indicate that increasing time delay decelerates rumor propagation, while higher degrees of cross-diffusion, contact spreading, media correction, and self-correction expedite the spread of rumors. In small-world networks, the trust level in rumors has a narrower range, while in scale-free networks, the stability of rumors is faster. This indicates how to effectively adjust network structures for controlling or managing the spread of rumors.

Dynamics caused by the mean-reverting Ornstein–Uhlenbeck process in a stochastic predator–prey model with stage structure

There are many species that go through different life stages in nature, they have different ecological characteristics in different stages, besides, stochastic variation exists in the natural environment. Based on this biological phenomenon, we develop and investigate dynamics in a predator–prey system with stage structure and the mean-reverting Ornstein–Uhlenbeck processes. We prove that this system satisfies the existence and uniqueness of globally positive equilibrium. The sufficient conditions for the existence of a stationary Markov process in stochastic system are attained. We deal with explicit expression and the existence of density function of the system which make a great contribution to biological intrinsic essence for the system. In addition, the sufficient criteria for extinction of the predator populations is derived. Finally, numerical simulations are furnished to illustrate the analysis results.

Positive steady-state solutions for a vegetation–water model with saturated water absorption

In this paper, a vegetation–water model with saturated water absorption is considered. The global stability of boundary equilibrium is first obtained. Then the stability and Turing instability of positive equilibria are discussed and we find that the equilibrium with vegetation small is always unstable and if the vegetation diffusion is small or the water diffusion is large, the other positive equilibrium loses its stability and Turing instability occurs. In addition, a priori estimates of nonnegative steady-state solutions is obtained by the maximum principle. Moreover, some detailed qualitative analyses are carried out on the steady-state bifurcations at both simple and double eigenvalues. In particular, we establish the criterion to determine the bifurcation direction. Finally, some dynamics near the bifurcation point are showed numerically and we depict the evolution processes of vegetation patterns under different parameters. Our results show that the parameter 1/p, which represents the conversion of water absorption, has a great impact on the vegetation patterns formation: with the increase of p, the vegetation biomass decreases, and meanwhile it can induce the transition of different pattern structures.

Hopf bifurcation and hybrid control of a delayed diffusive semi-ratio-dependent predator-prey model

<p>A delayed diffusive predator-prey system with nonmonotonic functional response subject to Neumann boundary conditions is introduced in this paper. First, we analyze the associated characteristic equation to research the conditions for local stability of the positive equilibrium point and the occurrence of Turing instability induced by diffusion in the absence of delay. Second, we provide conditions for the existence of Hopf bifurcation driven by time delay. By utilizing the normal theory and center manifold theorem, we derive explicit formulas for Hopf bifurcation properties such as direction and stability from the positive equilibrium. Third, a hybrid controller is added to the system. By judiciously adjusting the control parameters, we effectively enhance the stability domain of the system, resulting in a modification of the position of the Hopf bifurcation periodic solutions. Numerical simulations demonstrate the presence of rich dynamical phenomena within the system. Moreover, sensitivity analysis was conducted using Latin hypercube sampling (LHS)/partial rank correlation coefficient (PRCC) to explore the impact of parameter variations on the output of prey and predator populations.</p>

Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

<p>In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.</p>

Multi-source partial discharge pattern recognition algorithm based on DCGAN-Yolov5

This study aims to overcome the complication that deep learning-based pattern recognition in partial discharge (PD) diagnosis of gas-insulated switchgear (GIS) can only identify single-source partial discharge but not multi-source partial discharge. Specifically, a GIS multi-source partial discharge detection algorithm based on Deep Convolution Generative Adversarial Networks and You Only Look Once (DCGAN-YOLOv5) is proposed. First, the Phase Resolved Partial Discharge (PRPD) features of multi-source PD are analyzed, a GIS experiment platform is established, and four typical PD defects are simulated. Besides, sample data are collected, and the DCGAN network is used for sample expansion. Then, a YOLOv5 network model is designed, and a spatial and channel attention mechanism is added to the feature extraction network with a positive sample equilibrium strategy. Finally, the effectiveness of the proposed algorithm is verified using laboratory data and field data collected from a 220 kV substation. The experimental results demonstrate that the proposed algorithm can effectively detect the multi-source PD features in PRPD patterns under complex noise and thus successfully identify the types of multi-source PD. The mean Average Precision (mAP) can reach 98.4%. The precision of single-source PD and multi-source PD can reach 95.2% and 89.3%, when testing with field data.

Mathematical analysis of synthesis chemical reactions for virus building block polymers in vivo.

For numerous viruses, their capsid assembly is composed of two steps. The first step is that virus structural protein monomers are polymerized to building blocks. Then, these building blocks are cumulative and efficiently assembled to virus capsid shell. These building block polymerization reactions in the first step are fundamental for virus assembly, and some drug targets were found in this step. In this work, we focused on the first step. Often, virus building blocks consisted of less than six monomers. That is, dimer, trimer, tetramer, pentamer, and hexamer. We presented mathematical models for polymerization chemical reactions of these five building blocks, respectively. Then, we proved the existence and uniqueness of the positive equilibrium solution for these mathematical models one by one. Subsequently, we also analyzed the stability of the equilibrium states, respectively. These results may provide further insight into property of virus building block polymerization chemical reactions in vivo.

Dynamics analysis of a diffusive prey-taxis system with memory and maturation delays

In this paper, a diffusive predator-prey system considering prey-taxis term with memory and maturation delays under Neumann boundary conditions is investigated. Firstly, the existence and stability of equilibria, especially the existence, uniqueness and stability of the positive equilibrium, are studied. Secondly, we prove that: ( i ) there is no spatially homogeneous steady state bifurcation as the eigenvalue of the negative Laplace operator is zero; ( i i ) as this system is only with memory delay τ 1 , the the spatially nonhomogeneous Hopf bifurcation appears; ( i i i ) when the model is only with maturation delay τ 2 , the system has spatially homogeneous and nonhomogeneous periodic solutions; ( i v ) for the case of two delays, the system has rich dynamics, for example, stability switches, whose curves have four forms. Finally, some numerical simulations are produced to verify and support the theoretical results.

Analysis of a prey-predator system incorporating the additive Allee effect and intraspecific cooperation

<abstract><p>To understand the influence of the Allee effect and intraspecific cooperation on the dynamics of a predator-prey system, we constructed a model using ordinary differential equations. Our research shows that the system exhibits more complex dynamics, including possible bistability between alternative semi-trivial states and an Allee effect for prey. The Allee effect can destabilize the system. The equilibrium points of the system could change from stable to unstable. Otherwise, even if the system were stable, it would take much longer time to reach a stable state. We also find that the presence of the Allee effect of prey increases the positive equilibrium density of the predator but has no effect on the positive equilibrium density of the prey. It should be noted that the influence of nonlinear predator mortality also causes the system to take a longer time to reach a steady state.</p></abstract>

Commensalism and syntrophy in the chemostat: a unifying graphical approach

<abstract><p>The aim of this paper is to show that Tilman's graphical method for the study of competition between two species for two resources can be advantageously used for the study of commensalism or syntrophy models, where a first species produces the substrate necessary for the growth of the second species. The growth functions of the species considered are general and include both inhibition by the other substrate and inhibition by the species' limiting substrate, when it is at a high concentration. Because of their importance in microbial ecology, models of commensalism and syntrophy, with or without self-inhibition, have been the subject of numerous studies in the literature. We obtain a unified presentation of a large number of these results from the literature. The mathematical model considered is a differential system in four dimensions. We give a new result of local stability of the positive equilibrium, which has only been obtained in the literature in the case where the removal rates of the species are identical to the dilution rate and the study of stability can be reduced to that of a system in two dimensions. We describe the operating diagram of the system: this is the bifurcation diagram which gives the asymptotic behavior of the system when the operating parameters are varied, i.e., the dilution rate and the substrate inlet concentrations.</p></abstract>

A double time-delay Holling Ⅱ predation model with weak Allee effect and age-structure

<abstract><p>A double-time-delay Holling Ⅱ predator model with weak Allee effect and age structure was studied in this paper. First, the model was converted into an abstract Cauchy problem. We also discussed the well-posedness of the model and the existence of the equilibrium solution. We analyzed the global stability of boundary equilibrium points, the local stability of positive equilibrium points, and the conditions of the Hopf bifurcation for the system. The conclusion was verified by numerical simulation.</p></abstract>

Investigating Global Stability and Bifurcation in an Ecological Dynamical System

We consider a continuous‐time model describing the interaction between phytoplankton and zooplankton using a Holling type‐II response. We then transform this continuous‐time model into a discrete‐time counterpart using a fractional‐order discretization method. The paper explores the local stability of this obtained system concerning all equilibrium points and establishes the global asymptotic stability of its positive fixed point. The study also demonstrates that, under specific mathematical conditions, the system undergoes a Neimark–Sacker bifurcation around its positive equilibrium point. To effectively manage this bifurcation, two modified hybrid control techniques are introduced. The paper concludes by presenting illustrative numerical examples that validate the theoretical findings and assess the effectiveness and feasibility of the newly proposed control strategies. In addition, a comparative analysis is conducted between the modified hybrid techniques and an existing hybrid approach.

Hopf bifurcation and stability analysis of a delay differential equation model for biodegradation of a class of microcystins

<abstract><p>In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data.</p></abstract>

ANALYSIS OF SKELLAM MODELS WITH A RIGID HARVESTING STRATEGY

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete Skellam model are most often used in practice. In the given paper the Skellam model of the form $N_{t+1}=a N_t/ (1+b N_t)$, $N_{t+1}=a N_t^2/ (b^2+ N_t^2)$, $N_{t+1}=a N_t/ (1+b N_t^2)$ is considered, where the parameters $a,b>0$ with taking an effect of harvesting. Positive equilibrium points and conditions for their stability for these equations were found. It is shown in analytical form that these equations do not have periodic solutions with period $T=2$, which means, according by the Sharkovskii theorem, periodic solutions of any periods. In the model with harvesting, only regimes with monotonic stabilization of the population size are observed. Therefore, in all models of Skellam with harvesting, the existence of a critical conception is show, beyond which the population will be completely destroyed. For practice it is important to know the permissible limits of harvesting intensity, which are found in this paper.

Stability Analysis of Prey-Predator Model Migration with Holling Type-III Response Function in The Presence of Competition and Toxicity

In this paper, we propose and investigate a prey-predator model with two zones contaminated with heavy metal toxicity, especially copper (Cu), which enter the reservation zone and the unreserved zone in the aquatic environment. The dynamics of the prey population in the ecosystem can migrate from the reservation zone to the unreserved zone or vice versa, while predators are assumed to look for prey in the unreserved zone. The dynamic behavior of the population is expressed as a system of differential equations based on food intake capacity and other factors. We introduce a predator population with a Holling type III predation response function, coupled with inter-specific competition among prey due to overlapping diets and assuming the prey is contaminated with copper metal toxicity. The presence of a positive equilibrium point, namely the interior equilibrium point, is analyzed and investigated for its stability using the Routh-Hurwitz stability test. Numerical simulations are carried out to verify the results of the analysis and dynamics of the system solution. The results of the analysis of the interior equilibrium point T3 in each case is a stable point. This indicates a change in the balance of prey populations and predator populations.

Dynamical properties of a new SIR epidemic model

Taking into account the depletion of food supply by all individuals, and the fact that chronic infectious diseases will not cause the infected individuals to lose their fertility completely, we first propose a new SIR epidemic model of ODE. For this model, we derive its basic reproduction number $ \mathcal{R}_0 $, and show that the disease-free equilibrium point is globally asymptotically stable when $ \mathcal{R}_0\le1 $, while the unique positive equilibrium point is globally asymptotically stable when $ \mathcal{R}_0>1 $. Then we incorporate the spatial dispersion and free boundary condition into this ODE model. The well-posedness and longtime behaviors are obtained. Particularly, we find a spreading-vanishing dichotomy in which the basic reproduction number $ \mathcal{R}_0 $ plays a crucial role.

Patterns of a General Chemical Model Involving Degn-Harrison Reaction Scheme

This paper reports the pattern formation of a general Degn-Harrison system. We first determine the types and stability of the unique positive equilibrium for the spatially homogeneous system. The equilibrium may be node, focus, or center. Supercritical or subcritical Hopf bifurcation may occur if it is a center. In the sequel, we propose the conditions for the occurrence of Turing instability for the spatially inhomogeneous system. We can theoretically explain that Turing instability exists as the equilibrium transitions from homogeneous stable to inhomogeneous unstable states. Finally, we perform computational experiments to investigate the complex pattern formation of this chemical model. An interesting finding is that if one of the reactants has a high diffusion rate, the pattern formation will be inhibited. Conversely, a low diffusion rate may promote pattern formation.

The dynamics of a delayed predator-prey model with square root functional response and stage structure

<abstract> <p>In recent years, one of the most prevalent matters in population ecology has been the study of predator-prey relationships. In this context, this paper investigated the dynamic behavior of a delayed predator-prey model considering square root type functional response and stage structure for predators. First, we obtained positivity and boundedness of the solutions and existence of equilibrium points. Second, by applying the stability theory of delay differential equations and the Hopf bifurcation theorem, we discussed the system's local stability and the existence of a Hopf bifurcation at the positive equilibrium point. Moreover, the properties of the Hopf bifurcation were deduced by using the central manifold theorem and normal form method. Analytical results showed that when the time delay was less than the critical value, the two populations will coexist, otherwise the ecological balance will be disrupted. Finally, some numerical simulations were also included to verify the theoretical results.</p> </abstract>

Bifurcation analysis and chaos control of a discrete fractional-order Leslie-Gower model with fear factor

<p>This study focused on the dynamical behavior analysis of a discrete fractional Leslie-Gower model incorporating antipredator behavior and a Holling type Ⅱ functional response. Initially, we analyzed the existence and stability of the model's positive equilibrium points. For the interior positive equilibrium points, we investigated the parameter conditions leading to period-doubling bifurcation and Neimark-Sacker bifurcation using the center manifold theorem and bifurcation theory. To effectively control the chaos resulting from these bifurcations, we proposed two chaos control strategies. Numerical simulations were conducted to validate the theoretical results. These findings may contribute to the improved management and preservation of ecological systems.</p>

Qualitative analysis and traveling wave solutions of a predator-prey model with time delay and stage structure

<abstract><p>In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model.</p></abstract>