Stable Positive Equilibrium Research Articles

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.

Social activity and network formation

This paper develops a simple model in which a social hierarchy emerges endogenously when agents form a network for complementary interaction (“activity”). Specifically, we assume that agents are ex ante identical and their best response activity, as well as their value function, increases (strictly) concavely in the total activity of their neighbors in the network. There exists a unique and stable positive activity equilibrium on exogenous networks under mild conditions. When we endogenize network formation, equilibria become strongly structured: more active players have more neighbors, i.e., a higher degree, but tend to sponsor fewer links. Additionally, in strict equilibria, agents separate themselves into groups characterized by the symmetric activity of their members. The characteristic activity decreases in group size and the network is a complete multipartite graph.

The theorem of the carrying simplex for competitive system defined on the n-rectangle and its application to a three-dimensional system

First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle: {x ∈ Rn : 0 ≤ xi ≤ ki, i = 1, …, n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique (n - 1)-dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.

Modeling the role of altruism of antibiotic-resistant bacteria

Based on the new findings in a recent experimental study (Lee et al., Nature 467, 82-86, 2010) that antibiotic resistant mutants of bacteria produce indoles to protect the wild strain bacteria, we propose a mathematical model to describe the evolution of the wild strain, resistant strain and indoles with limited nutrient. We distinguish two cases: (i) mutation is negligible and a resistant strain preexists; (ii) mutation is not negligible. For (i), we establish conditions for co-persistence of both strains, which indicate that the wild strain can survive with the help from the altruistic resistant strain, whereas it dies out in the absence of such a benefit. This consolidates the experimental findings in Lee et al. (Nature 467:82-86 2010). Further analysis and simulations also reveal some new phenomena not reported in Lee et al. (Nature 467:82-86 2010), that is, periodic oscillations of the populations may occur within certain range of the parameters, and there exists bistability in the sense that a stable positive periodic solution coexists with a stable positive equilibrium.

A TWO-PARAMETER METHOD FOR CHAOS CONTROL AND TARGETING IN ONE-DIMENSIONAL MAPS

We investigate a method of chaos control in which intervention is proportional to the difference between the current state and a fixed value. We prove that this method allows to stabilize the most usual one-dimensional maps used in discrete-time models of population dynamics about a globally stable positive equilibrium. From the point of view of targeting, this technique is very flexible, and we show how to choose the control parameter values to lead the system towards the desired target. Another important feature of this control scheme in the ecological context is that it can be designed to prevent the risk of extinction in models with the so-called Allee effect. We provide a useful geometrical interpretation, and give some examples to illustrate our theoretical results.

Limit cycles in a Gause‐type predator–prey model with sigmoid functional response and weak Allee effect on prey

The goal of this work is to examine the global behavior of a Gause‐type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect.Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect.Then, for determined parameter constraints, the trajectories can have different ω − limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here.So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd.

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.

Impacts of predation on dynamics of age-structured prey: Allee effects and multi-stability

With a series of mathematical models, we explore impacts of predation on a prey population structured into two age classes, juveniles and adults, assuming generalist, age-specific predators. Predation on any age class is either absent, or represented by types II or III functional responses, in various combinations. We look for Allee effects or more generally for multiple stable steady states in the prey population. One of our key findings is the occurrence of a predator pit (low-density “refuge” state of prey induced by predation; the chance of escaping predation thus increases both below and above an intermediate prey density) when only one age class is consumed and predators use a type II functional response —this scenario is known to occur for an unstructured prey consumed via a type III functional response and can never occur for an unstructured prey consumed via a type II one. In the case where both age classes are consumed by type II generalist predators, an Allee effect occurs frequently, but some parameters give also rise to a predator pit and even three stable equilibria (one extinction equilibrium and two positive ones—Allee effect and predator pit combined). Multiple positive stable equilibria are common if one age class is consumed via a type II functional response and the other via a type III functional response—here, in addition to the behaviours mentioned above one may even observe three stable positive equilibria—“double” predator pit. Some of these results are discussed from the perspective of population management.

Global stability for a model of competition in the chemostat with microbial inputs

We propose a model of competition of n species in a chemostat, with constant input of some species. We mainly emphasize the case that can lead to coexistence in the chemostat in a non-trivial way, i.e., where the n − 1 less competitive species are in the input. We prove that if the inputs satisfy a constraint, the coexistence between the species is obtained in the form of a globally asymptotically stable (GAS) positive equilibrium, while a GAS equilibrium without the dominant species is achieved if the constraint is not satisfied. This work is round up with a thorough study of all the situations that can arise when having an arbitrary number of species in the chemostat inputs; this always results in a GAS equilibrium that either does or does not encompass one of the species that is not present in the input.

The replicability of oncolytic virus: Defining conditions in tumor virotherapy

The replicability of an oncolytic virus is measured by its burst size. The burst size is the number of new viruses coming out from a lysis of an infected tumor cell. Some clinical evidences show that the burst size of an oncolytic virus is a defining parameter for the success of virotherapy. This article analyzes a basic mathematical model that includes burst size for oncolytic virotherapy. The analysis of the model shows that there are two threshold values of the burst size: below the first threshold, the tumor always grows to its maximum (carrying capacity) size; while passing this threshold, there is a locally stable positive equilibrium solution appearing through transcritical bifurcation; while at or above the second threshold, there exits one or three families of periodic solutions arising from Hopf bifurcations. The study suggests that the tumor load can drop to a undetectable level either during the oscillation or when the burst size is large enough.

Influences of migrations from local competitive pressures on populations between patches

A mathematical model of two competitive populations with migrations between two patches is proposed, which incorporates the dispersals produced by local competitive pressures. It is shown that the density-dependent migrations enhance the persistence of the two competitive populations. Compared with constant migrations, the dispersals from local competitive pressures induce dramatic changes of dynamical behavior of the model. First, the model admits a transition from one stable positive equilibrium to bistable positive equilibria. Second, the model exhibits bifurcations with the existence of seven positive steady states, in which two stable positive equilibria coexist with two stable boundary equilibrium points. These changes alter the distribution of the two populations in the two patches and increase their survival possibilities.

Ecological consequences of global bifurcations in some food chain models

Food chain models of ordinary differential equations ( ode’s) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil’nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections ‘cut’ the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models.

Dynamical behaviors of a trimolecular response model with impulsive input

In the paper, we consider a trimolecular response model with impulsive input. Sufficient conditions for globally stable positive equilibrium point of the trimolecular response model with constant input are obtained. By impulsive differential theory, we obtain conditions for permanence of the trimolecular response model with impulsive input. By numerical analysis, we demonstrate complex phenomena such as limit cycles, periodic solutions, and chaos.

Global Analysis of Predator–Prey System with Hawk and Dove Tactics

Functional response of the Holling type II is incorporated into a predator–prey model with predators using hawk‐dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle‐node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.

How to control chaotic behaviour and population size with proportional feedback

We study the control of chaos in one-dimensional discrete maps as they often occur in modelling population dynamics. For managing the population, we seek to suppress any possible chaotic behavior, leading the system to a stable equilibrium. In this Letter, we make a rigorous analysis of the proportional feedback method under certain conditions fulfilled by a wide family of maps. We show that it is possible to stabilize the chaotic dynamics towards a globally stable positive equilibrium, that can be chosen among a broad range of possible values. In particular, the size of the population can be enhanced by control in form of population reduction. This paradoxical phenomenon is known as the hydra effect, and it has important implications in the design of strategies in such areas as fishing, pest management, and conservation biology.

A time-delay model for the effect of toxicant in a single species growth with stage-structure

In this paper, we investigate a single-species growth model with stage-structure consisting of immature and mature stages for the effects of toxicants with constant maturation time-delay. We study the dynamics of our model in three cases: an instantaneous emission of toxicant, a constant emission of toxicant, and a periodic emission of toxicant into the environment. We present results on positivity and boundedness of all solutions under appropriate conditions. The model equations are analyzed mathematically with regard to the nature of equilibria and their stabilities using the theory of nonlinear differential equations and computer simulations. It is shown that under suitable conditions, there exists a globally asymptotically stable positive equilibrium. It is concluded from the analysis that as the concentration of toxicant in the environment increases, equilibrium densities of both immature and mature populations decrease. It is also noted that the effects of toxicants are more on the equilibrium level of immature population in comparison to the mature population.

The global dynamics of a discrete juvenile–adult model with continuous and seasonal reproduction

A general discrete juvenile–adult population model with time-dependent birth rate and nonlinear survivorship rates is studied. When breeding is continuous, it is shown that the model has a unique globally asymptotically stable positive equilibrium provided the net reproductive number is larger than one. If it is smaller than one, then the extinction equilibrium is globally asymptotically stable. When breeding is seasonal, it is shown that there exists a unique globally asymptotically stable periodic solution provided the net reproductive number is larger than one. When this value is less than one, the population goes to extinction. Conditions on the birth rate where the population with seasonal breeding survives while the population with continuous breeding becomes extinct are provided.

Coexistence in competition models with density-dependent mortality

We consider a two-competitor/one-prey model in which both competitors exhibit a general functional response and one of the competitors exhibits a density-dependent mortality rate. It is shown that the two competitors can coexist upon the single prey. As an example, we consider a two-competitor/one-prey model with a Holling II functional response. Our results demonstrate that density-dependent mortality in one of the competitors can prevent competitive exclusion. Moreover, by constructing a Liapunov function, the system has a globally stable positive equilibrium. To cite this article: S. Ruan et al., C. R. Biologies 330 (2007).

Pest management through continuous and impulsive control strategies

In this paper, we propose two mathematical models concerning continuous and, respectively, impulsive pest control strategies. In the case in which a continuous control is used, it is shown that the model admits a globally asymptotically stable positive equilibrium under appropriate conditions which involve parameter estimations. As a result, the global asymptotic stability of the unique positive equilibrium is used to establish a procedure to maintain the pests at an acceptably low level in the long term. In the case in which an impulsive control is used, it is observed that there exists a globally asymptotically stable susceptible pest-eradication periodic solution on condition that the amount of infective pests released periodically is larger than some critical value. When the amount of infective pests released is less than this critical value, the system is shown to be permanent, which implies that the trivial susceptible pest-eradication solution loses its stability. Further, the existence of a nontrivial periodic solution is also studied by means of numerical simulation. Finally, the efficiency of continuous and impulsive control policies is compared.

Territorial Dynamics: Persistence in Territorial Species

The widely studied and very controversial northern spotted owl, along with many other threatened and endangered species, exhibits territorial behavior. That is, adult pairs claim and defend a home range encompassing sufficient resources and of sufficient size to allow the pair to survive and reproduce successfully. Readers may be familiar with population models such as the logistic growth model, the Gompertz model, the Ricker model, and the Beverton-Holt model. These all capture the basic concept of limited growth (carrying capacity); however, they fail to exhibit some fundamental characteristics of the dynamics of territorial species. In particular, they do not exhibit a threshold in the density of suitable habitat below which the species is destined for extinction even if some suitable habitat is still available. In this paper, we develop a model first proposed by Lamberson and Carroll [1] for the dynamics of a territorial animal or bird population. It consists of a continu ous model for dispersal which distinguishes between adults?individuals who hold territories?and juveniles?those (nonterritorial) individuals that have not yet secured a home range. Here we think of birth not as the time of physical birth, but the time at which juveniles leave their natal territory and begin the search for their own home range. The model explicitly considers the cost of dispersal by including an ongoing rate of mortality due to pr?dation and starvation while animals search for a territory. We establish that there is a threshold for density of suitable habitat, below which the population must decrease to extinction and above which the population tends to a stable positive equilibrium size. Within populations of territorial animals we frequently find individuals that have not had the good fortune to secure a home range. These individuals, sometimes called floaters, usually occupy habitat of marginal quality and not suitable for attracting a mate. Usually, they eke out a secretive existence on the fringes of territories already claimed by other individuals. The dynamics of this floater population is important in understanding the overall behavior of the species, especially if the species is threat ened or endangered. In our model, the floaters are considered part of the population of juveniles since they have not yet secured a suitable territory. In this paper, we use a simple system of differential equations to describe the dy namics of a territorial species. The behavior of the system will be studied under both equilibrium and nonequilibrium conditions. For equilibrium conditions, we will es tablish: the fixed points, their stability, and the critical threshold in habitat density for persistence of the population.