Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov’s functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system’s behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Analysis of a predator–prey model with modified Holling–Tanner functional response and time delay

In paper, a predator–prey model with modified Holling–Tanner functional response and time delay is discussed. It is proved that the system is permanent under some appropriate conditions. The local stability of the equilibria is investigated. By constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the positive equilibrium of the model.     

Delays of schemes in a model that considers the input values of functional elements

A delay model for schemes of functional elements in arbitrary finite complete basis B is studied; in the model, delays of the basic element are given by random positive real numbers for each input and each input set of variables entering other inputs. Asymptotic estimates in the form τ_B n ± O(logn), where τ_B is a constant that depends only on basis B, are obtained for the delay of the multiplex function of order n. Based on these estimates, asymptotic estimates of the form τ_B n ± O(logn) for the corresponding Shannon function, i.e., for the delay of the worst function of logic algebra that depends on given n variables, are established.     

Traveling wavefront in a Hematopoiesis model with time delay

This paper is concerned with a reaction–diffusion equation with time delay, which describes the dynamics of the blood cell production. The existence of the traveling wavefront is given by using the upper–lower solution technique and the monotone iteration.     

A model of hysteresis with two inputs

We formulate and analyze a new model of vector hysteresis for the case of two-input signals. We prove the essential mathematical properties of this model and we present the solutions to two identification problems connected with our model.     

Continuity in functional differential equations with infinite delay

In this paper, we investigate the continuous dependence of solutions of the functional differential equation with infinite delayx(t)=f(t,x _t ) on initial functions. Endowing the phase space ag-norm as well as a supremum norm, we show that if the equation satisfies a mild fading memory dondition, then the continuity off in respect to the topology induced by the supremum norm can yield the continuity of solutions of the equation in respect to the topology induced by theg-norm which is stronger than the ahead one.This research was supported in part by an NSF grant with number NSF-DMS-8521408.On leave from South China Normal University, Guangzhou, PRC. This research was supported in part by the National Science Foundation of PRC.     

Analysis of a bacteria-immunity model with delay quorum sensing

A bacteria-immunity model with bacterial quorum sensing is formulated, which describes the competition between bacteria and immune cells. A distributed delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In this paper, we focus on a subsystem of the bacteria-immunity model, analyze the stability of the equilibrium points, discuss the existence and stability of periodic solutions bifurcated from the positive equilibrium point, and finally investigate the stability of the nonhyperbolic equilibrium point by the center manifold theorem.     

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.     

Travelling wave fronts in a vector disease model with delay

In this paper, we study the diffusive vector disease model with delay. This problem with strong biological background has attracted much research attention. We focus on the existence of traveling wave fronts, and find that there is a moving zone for the transition from the disease-free state to the infective state. To complete the theoretical analysis, we employ the mathematical tools including the monotone iteration technique as well as the upper and lower solution method.     

Stability and bifurcation analysis in a viral model with delay

In this paper, we consider a three‐dimensional viral model with delay. We first investigate the linear stability and the existence of a Hopf bifurcation. It is shown that Hopf bifurcations occur as the delay τ passes through a sequence of critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit formulaes that determine the stability, the direction, and the period of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the validity of the main results. Finally, some brief conclusions are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Hopf bifurcations in a reaction-diffusion population model with delay effect

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. The growth rate per capita can be logistic or weak Allee effect type. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of forward Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcation parameter. The general results are applied to a “food-limited” population model with diffusion and delay effects as well as a weak Allee effect population model.     

A multiple-criteria decision model with ranked inputs

Address for correspondence: G. Gregory, Laburnum Cottage, 10 Main Street, Milnthorpe, Cumbria, LA7 7PN. A model for multiple-criteria decision analysis is proposedwhere the input data consist of rankings of criteria and ofalternatives within each criterion. The analysis uses the approachof data-envelopment analysis, summing over the scores obtainedby this method. Examples of further assessment of the best alternativesare presented.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Measure functional differential equations with infinite delay

We introduce measure functional differential equations with infinite delay and an axiomatically described phase space. We show how to transform these equations into generalized ordinary differential equations whose solutions take values in a suitable infinite-dimensional Banach space. Even in the special case of functional equations with finite delay, our result improves the existing one by imposing weaker conditions on the right-hand side.