Positive Equilibrium Solution Research Articles

Dynamics of a nonlinear infection viral propagation model with one fixed boundary and one free boundary

In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line [0,∞), and that the virus and infected cells always die out when the Basic Reproduction NumberR0≤1, while the virus and infected cells have persistence properties when R0>1. To obtain the persistence properties of virus and infected cells when R0>1, the most work of this paper focuses on the existence and uniqueness of bounded positive equilibrium solutions for subsystems and the existence of positive equilibrium solutions for the entire system.

Global asymptotical stability and Hopf bifurcation for a three‐species Lotka‐Volterra food web model

In this article, we consider a delayed three‐species Lotka‐Volterra food web model with diffusion and homogeneous Neumann boundary conditions. We proved that the positive constant equilibrium solution is globally asymptotically stable for the system without time delays. By virtue of the sum of time delays as the bifurcation parameter, spatially homogeneous and inhomogeneous Hopf bifurcation at the positive constant equilibrium solution are proved to occur when the delay varied through a sequence of critical values. In addition, we consider the effect of cross‐diffusion on the system in the case that without time delays. By taking cross diffusion coefficients as the bifurcation parameter, our model undergoes inhomogeneous Hopf bifurcation around a positive constant equilibrium solution when the bifurcation parameter is varied through a sequence of critical values. A common feature in the most existing research work is that the bifurcation factor that induces Hopf bifurcation appears in the reaction terms (such as time delay) rather than diffusion terms. Our results demonstrate that the inhomogeneous Hopf bifurcation can be triggered by the effect of cross diffusion factors.

Analysis of Prey-Predator Scheme in Conjunction with Help and Gestation Delay

This paper presents a three-dimensional continuous time dynamical system of three species, two of which are competing preys and one is a predator. We also assume that during predation, the members of both teams of preys help each other and the rate of predation of both teams is different. The interaction between prey and predator is assumed to be governed by a Holling type II functional response and discrete type gestation delay of the predator for consumption of the prey. In this work, we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system, and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functions when the gestation delay is zero, and there is no periodic orbit within the interior of the first quadrant of state space around the interior equilibrium. As we introduced time delay due to the gestation of the predator, we also discuss the stability of the delayed model. It is observed that the existence of stability switching occurs around the interior equilibrium point as the gestation delay increases through a certain critical threshold. Here, a phenomenon of Hopf bifurcation occurs, and a stable limit cycle corresponding to the periodic solution of the system is also observed. This study reveals that the delay is taken as a bifurcation parameter and also plays a significant role for the stability of the proposed model. Computer simulations of numerical examples are given to explain our proposed model. We have also addressed critically the biological implications of our analytical findings with proper numerical examples.

Bifurcations and chaos control in a discrete Rosenzweig-Macarthur prey-predator model.

In this paper, we explore the local dynamics, chaos, and bifurcations of a discrete Rosenzweig-Macarthur prey-predator model. More specifically, we explore local dynamical characteristics at equilibrium solutions of the discrete model. The existence of bifurcations at equilibrium solutions is also studied, and that at semitrivial and trivial equilibrium solutions, the model does not undergo flip bifurcation, but at positive equilibrium solutions, it undergoes flip and Neimark-Sacker bifurcations when parameters go through certain curves. Fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by the center manifold theorem and bifurcation theory. We also studied chaos by the feedback control method. The theoretical results are confirmed numerically.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

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.

Dynamical analysis of a two-dimensional discrete predator–prey model

In this paper, we explore the existence of periodic solutions, local dynamics at equilibrium solutions, chaos and bifurcations of a discrete prey–predator model with Michaelis–Menten type functional response. More specifically, we explore local dynamical characteristics at equilibrium solutions, and existence of periodic solutions for the under consideration model. It is also studied the existence of bifurcations at equilibrium solutions, and investigated that at semitrivial and trivial equilibrium solutions model does not undergo flip bifurcation, but at positive equilibrium solution it undergoes flip and Neimark–Sacker bifurcations when parameters goes through the certain curves. It is also investigated that fold bifurcation does not exist at positive equilibrium, and we have studied these bifurcations by center manifold theorem and bifurcation theory. We also studied chaos by feedback control method. The theoretical results are confirmed numerically. Furthermore, we use 0–1 chaos test in order to quantify whether chaos exists in the model or not.

Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Event-triggered impulse control on reaction–diffusion Gilpin–Ayala competition model with multiple stationary solutions

In previous literature, the parameter restrictions for the reaction–diffusion Gilpin–Ayala competition ecosystem were limited to the control of the first eigenvalue of the Laplace operator. This paper considers extending the parameter to the control of the second eigenvalue. However, exceeding the first eigenvalue results in a loss of compactness in the energy functional of the determined ecosystem. Consequently, this paper discusses the orthogonal decomposition of a given Sobolev space and variational techniques, obtaining the existence of multiple equilibrium points. The presence of multiple equilibrium points is a common phenomenon in real-world ecosystems and implies that the system cannot be globally stable under normal circumstances. However, this study develops an effective event-triggered impulse mechanism (ETIM) that achieves global stability for the positive equilibrium solution in the ecosystem. Notably, the newly designed ETIM not only eliminates Zeno behaviour but also reduces the cost of impulse control. This is achieved by removing some unnecessary impulses in the fixed impulsive instants mechanism associated with the ecosystem in previous literature. Finally, a numerical simulation validates the effectiveness of the proposed method.

Cross-Diffusion-Induced Turing Instability in a Two-Prey One-Predator System

This paper focuses on a strongly coupled specific ecological system consisting of two prey species and one predator. We explore a unique positive equilibrium solution of the system that is globally asymptotically stable. Additionally, we show that this equilibrium solution remains locally linearly stable, even in the presence of diffusion. This means that the system does not follow classical Turing instability. However, it becomes linearly unstable only when cross-diffusion also plays a role in the system, which is called a cross-diffusion-induced instability. The corresponding numerical simulations are also demonstrated and we obtain the spatial patterns.

A Reaction–Diffusion–Advection Chemostat Model in a Flowing Habitat: Mathematical Analysis and Numerical Simulations

This paper is concerned with a reaction–diffusion–advection chemostat model with two species growing and competing for a single-limited resource. By taking the growth rates of the two species as variable parameters, we study the effect of growth rates on the dynamics of this system. It is found that there exist several critical curves, which may classify the dynamics of this system into three scenarios: (1) extinction of both species; (2) competitive exclusion; (3) coexistence. Moreover, we take numerical approaches to further understand the potential behaviors of the above critical curves and observe that the bistable phenomenon can occur, besides competitive exclusion and coexistence. To further study the effect of advection and diffusion on the dynamics of this system, we present the bifurcation diagrams of positive equilibrium solutions of the single species model and the two-species model with the advection rates and the diffusion rates increasing, respectively. These numerical results indicate that advection and diffusion play a key role in determining the dynamics of two species competing in a flow reactor.

Diffusion-Induced Instability of the Periodic Solutions in a Reaction-Diffusion Predator-Prey Model with Dormancy of Predators

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results.

Stationary distribution and global stability of stochastic predator-prey model with disease in prey population

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented.

Global boundedness and dynamics of a diffusive predator–prey model with modified Leslie–Gower functional response and density-dependent motion

This paper is devoted to studying the dynamical behaviors and stationary patterns of a diffusive modified Leslie–Gower predator–prey model with density-dependent motion in the predator population. We establish the existence of classical solutions with the uniform-in time bound and then analyze the local and global stability of the spatially homogeneous co-existence steady state under certain parametric conditions. By choosing the prey diffusion rate d2 as the bifurcation parameter, the steady state bifurcations from the positive constant equilibrium solution are investigated. Numerical simulations are performed to corroborate our analytical findings.

Spatiotemporal patterns and multiple bifurcations of a reaction- diffusion model for hair follicle spacing

<abstract><p>In this paper, the dynamical behaviors of a 2-component coupled diffusive system modeling hair follicle spacing is considered. For the corresponding ODEs, we not only consider the stability and instability of the unique positive equilibrium solutions, but also show the existence of unstable Hopf bifurcating periodic solutions. For the reaction-diffusion equations, we are mainly interested in the Turing instability of the positive equilibrium solution, as well as Hopf bifurcations and steady-state bifurcations. Our results showed that, under certain conditions, the reaction-diffusion system not only has Hopf bifurcating periodic solutions (both spatially homogeneous and non-homogeneous, all unstable), but it also has non-constant positive bifurcating equilibrium solutions. This allows for a clearer understanding of the mechanism for the spatiotemporal patterns of this particular system.</p></abstract>

SELF-DIFFUSIVE SPATIOTEMPORAL PATTERNS IN A FOOD CHAIN MODEL THROUGH WEAKLY NONLINEAR ANALYSIS

Density distributions of populations with self-diffusion and interaction in a spatial domain are dynamically visualized with coupled nonlinear reaction–diffusion equations. Incorporating self-diffusion terms creates a more pragmatic modeling paradigm and provides meaningful descriptions of influences on spatiotemporal pattern formation phenomena. This paper examines the effect of self-diffusion in a food chain system with a Holling type-IV functional response and the type of spatial structures forms on a geographical scale due to the random movement of species. We discussed the existence and uniqueness of a positive equilibrium solution and obtained the Turing instability conditions for the self-diffusive food chain model. Moreover, weakly nonlinear analysis close to the Turing bifurcation boundary is used to derive the amplitude equations. The stability of the amplitude equations and sufficient conditions for the emanation of spatiotemporal patterns (such as spots, stripes, and blended patterns) are investigated. The analytical results are verified with numerical simulations. The results are applicable to all environments and can be used to understand the effects of self-diffusion in other food chain models both qualitatively and quantitatively.

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.

Pattern Dynamics of Cross Diffusion Predator–Prey System with Strong Allee Effect and Hunting Cooperation

In this paper, we consider a Leslie–Gower cross diffusion predator–prey model with a strong Allee effect and hunting cooperation. We mainly investigate the effects of self diffusion and cross diffusion on the stability of the homogeneous state point and processes of pattern formation. Using eigenvalue theory and Routh–Hurwitz criterion, we analyze the local stability of positive equilibrium solutions. We give the conditions of Turing instability caused by self diffusion and cross diffusion in detail. In order to discuss the influence of self diffusion and cross diffusion, we choose self diffusion coefficient and cross diffusion coefficient as the main control parameters. Through a series of numerical simulations, rich Turing structures in the parameter space were obtained, including hole pattern, strip pattern and dot pattern. Furthermore, We illustrate the spatial pattern through numerical simulation. The results show that the dynamics of the model exhibits that the self diffusion and cross diffusion control not only form the growth of dots, stripes, and holes, but also self replicating spiral pattern growth. These results indicate that self diffusion and cross diffusion have important effects on the formation of spatial patterns.

Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity

In this paper, a delayed reaction–diffusion equation with carrying capacity-driven diffusion is investigated. The stability of the positive equilibrium solutions and the existence of the Hopf bifurcation of the equation are considered by studying the principal eigenvalue of an associated elliptic operator. The properties of the bifurcating periodic solutions are also obtained by using the normal form theory and the center manifold reduction. Furthermore, some representative numerical simulations are provided to illustrate the main theoretical results.

Bifurcation Analysis of Periodic Oscillation in a Hematopoietic Stem Cells Model with Time Delay Control

Underlying the state feedback control, the complex dynamical disease of the hematopoietic stem cells model based on Mackey’s mathematical description is analyzed. The bifurcating periodical oscillation solutions of the system are continued by applying numerical simulation method. The limit point cycle bifurcation and period doubling bifurcation are observed frequently in the continuation process. The attraction basins of the positive equilibrium solution shrink as the differentiate rate is ascending and the observed Mobiüs strain is simulated with boundary as the period-2 solution. The period doubling bifurcation leads to period-2, period-4, and period-8 solutions which are simulated. Starting from period doubling bifurcation point, the continuation of the bifurcating solution routes to homoclinic solution is finished. The simulation results improve the comprehension related to the spontaneous dynamical character manifested in the hematopoietic stem cells model.