Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.     

A semigroup approach to age-dependent population dynamics with time delay

We study the McKendrick type models of population dynamics with instantaneous time delay in the birth rate. The models involve first order partial differential equations with nonlocal and delayed boundary conditions. We show that a semigroup can be associated

to it and identify the infinistimal generator. Its spectral properties are analyzed yielding large time behaviour. An interesting result is that if the total population converges to an equilibrium it will converge to it in an oscillatory fashion. Further, we consider a logistic ara age-dependent model with delay. A nonlinear semigroup is constructed to describe the evolution of the population. Existence and uniqueness of the nonlinear equation are proved.     

Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice

In this paper, we study the existence of traveling waves of a delayed population model with age-structure on a 2-dimensional spatial lattice when the maturation time r is relatively small. Under the assumption that the birth function b satisfies the bistable condition without requiring monotonicity, we prove the persistence of traveling wavefronts by means of a perturbation argument based on the existing results on the asymptotic autonomous system and the Fredholm alternative theory.     

Propagation and its failure in a lattice delayed differential equation with global interaction

We study the existence, uniqueness, global asymptotic stability and propagation failure of traveling wave fronts in a lattice delayed differential equation with global interaction for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. In the bistable case, under realistic assumptions on the birth function, we prove that the equation admits a strictly monotone increasing traveling wave front. Moreover, if the wave speed does not vanish, then the wave front is unique (up to a translation) and globally asymptotic stable with phase shift. Of particular interest is the phenomenon of “propagation failure” or “pinning” (that is, wave speed c = 0), we also give some criteria for pinning in this paper.     

Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector

A delay ordinary deterministic differential equation model for the population dynamics of the malaria vector is rigorously analysed subject to two forms of the vector birth rate function: the Verhulst-Pearl logistic growth function and the Maynard-Smith-Slatkin function. It is shown that, for any birth rate function satisfying some assumptions, the trivial equilibrium of the model is globally-asymptotically stable if the associated vectorial reproduction number is less than unity. Further, the model has a non-trivial equilibrium which is locally-asymptotically stable under a certain condition. The non-trivial equilibrium bifurcates into a limit cycle via a Hopf bifurcation. It is shown, by numerical simulations, that the amplitude of oscillating solutions increases with increasing maturation delay. Numerical simulations suggest that the Maynard-Smith-Slatkin function is more suitable for modelling the vector population dynamics than the Verhulst-Pearl logistic growth model, since the former is associated with increased sustained oscillations, which (in our view) is a desirable ecological feature, since it guarantees the persistence of the vector in the ecosystem.     

Permanence of dispersal population model with time delays

Permanence of a dispersal single-species population model is considered where environment is partitioned into several patches and the species requires some time to disperse between the patches. The model is described by delay differential equations. The existence of food-rich patches and small dispersions among the patches are proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence ensures permanence if each food-poor patch is connected to at least one food-rich patch and if each pair in food-rich patches is connected. Furthermore, it is proved that partial persistence is ensured even under large dispersion among food-rich patches if the dispersion time is relatively small.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

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.     

Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model

Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Bonhoeffer et al. (1997) [1], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar type of approximate model incorporating time delay in the process of infection on the healthy T cells which, in turn, implies inclusion of a similar delay in the process of viral replication. The model is studied both analytically and numerically. We also include a similar delay in the killing rate of infected CD4⁺ T cells by Cytotoxic T-Lymphocyte (CTL) and in the stimulation of CTL and analyse two resulting models numerically.The models with no time delay present have two equilibria: one where there is no infection and a non-trivial equilibrium where the infection can persist. If there is no time delay then the non-trivial equilibrium is locally asymptotically stable. Both our analytical results (for the first model) and our numerical results (for all three models) indicate that introduction of a time delay can destabilize the non-trivial equilibrium. The numerical results indicate that such destabilization occurs at realistic time delays and that there is a threshold time delay beneath which the equilibrium with infection present is locally asymptotically stable and above which this equilibrium is unstable and exhibits oscillatory solutions of increasing amplitude.     

A host-vector model for malaria with infective immigrants

This paper considers a host-vector mathematical model for the spread of malaria that incorporates recruitment of human population through a constant immigration, with a fraction of infective immigrants. The model analysis is carried out to find the steady states and their stability. It is found that in the presence of infective immigrant humans, there is no disease-free equilibrium point. However, the model exhibits a unique endemic equilibrium state if the fraction of the infective immigrants ? is positive. When the fraction of infective immigrants approaches a small value, there is sharp threshold for which the disease can be reduced in the community. The unique endemic equilibrium for which there is a fraction of infective immigrants is globally asymptotically stable.     

Spreading speeds and traveling waves for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity

This paper is concerned with the traveling wave solutions and the spreading speeds for a nonlocal dispersal equation with convolution-type crossing-monostable nonlinearity, which is motivated by an age-structured population model with time delay. We first prove the existence of traveling wave solution with critical wave speed c = c*. By introducing two auxiliary monotone birth functions and using a fluctuation method, we further show that the number c = c* is also the spreading speed of the corresponding initial value problem with compact support. Then, the nonexistence of traveling wave solutions for c < c* is established. Finally, by means of the (technical) weighted energy method, we prove that the traveling wave with large speed is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm.     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

Novel dynamics of a predator–prey system with harvesting of the predator guided by its population

A predator–prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. The proposed model is a nonsmooth dynamic system with switches between the original predator-prey model (free subsystem) and a model with nonlinear harvesting (harvesting subsystem). We initially examine the dynamics of both the free and the harvesting subsystems, and then we investigate the dynamics of the switching system using theories of nonsmooth systems. Theoretical results showed that the harvesting subsystem undergoes multiple bifurcations, including saddle-node, supercritical Hopf, Bogdanov–Takens and homoclinic bifurcations. The switching system not only retains all of the complex dynamics of the harvesting system but also exhibits much richer dynamics such as a sliding equilibrium, sliding cycle, boundary node (saddle point) bifurcation, boundary saddle-node bifurcation and buckling bifurcation. Both theoretical and numerical results showed that, by implementing predator population guided harvesting, the predator and prey population could coexist in more scenarios than those in which the predator may go extinct for the continuous harvesting regime. They could either stabilize at an equilibrium or oscillate periodically depending on the value of the economic threshold and the initial value of the system.     

Nonlinear stability of traveling wavefronts in an age-structured population model with nonlocal dispersal and delay

This paper is concerned with the nonlinear stability of traveling wavefronts for a single species population model with nonlocal dispersal and age structure. By using the weighted energy method together with the comparison principle, we prove that the traveling wavefront is exponentially stable, when the initial perturbation around the wavefronts decays exponentially at –∞, but it can be arbitrarily large in other locations. In particular, our result implies that the time delay is harmless for stability of traveling wavefronts of the model.     

On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves

The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by Phillips (1960) is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips’ four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple-variables are properly defined with clear physical meanings.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

Some Results on the Lotka–Volterra Model and its Small Random Perturbations

The Lotka–Volterra model is a nonlinear sytem of differential equations representing competing species. We show that when the system is far from its equilibrium, then most of the time one of the populations is exponentially small. We then consider random perturbations of the classical model by noise. In the case of perturbation of coefficients averaging principle applies. In the case of perturbations leading to extinction of one of the populations large deviation principle is used to find the likely path to extinction.     

Introduction to diffusive logistic equations in population dynamics

The purpose of this paper is to provide a careful and accessible exposition of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. Moreover we prove that a population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to Thomas Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by P. F. Verhulst.     

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.