Analysis of a Lotka–Volterra food chain chemostat with converting time delays

A model of the food chain chemostat involving predator, prey and growth-limiting nutrients is considered. The model incorporates two discrete time delays in order to describe the time involved in converting processes. The Lotka–Volterra type increasing functions are used to describe the species uptakes. In addition to showing that solutions with positive initial conditions are positive and bounded, we establish sufficient conditions for the (i) local stability and instability of the positive equilibrium and (ii) global stability of the non-negative equilibria. Numerical simulation suggests that the delays have both destabilizing and stabilizing effects, and the system can produce stable periodic solutions, quasi-periodic solutions and strange attractors.     

Bifurcation and chaos in a Monod type food chain chemostat with pulsed input and washout

In this paper, we introduce and study a model of a Monod type food chain chemostat with pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halving.     

Dynamical behavior of a three species food chain model with Beddington–DeAngelis functional response

A three species food chain model with Beddington–DeAngelis functional response is investigated. The local stability analysis is carried out and global behavior is simulated numerically for a biologically feasible choice of parameters. The persistence conditions of a food chain model are established. The bifurcation diagrams are obtained for different parameters of the model after intensive numerical simulations. The results of simulations show that the model could exhibit chaotic dynamics for realistic and biologically feasible parametric values. Finally, the effect of immigration within prey species is investigated. It is observed that adding small amount of constant immigration to prey species stabilize the system.     

Analysis of a delayed Monod type chemostat model with impulsive input the polluted nutrient

In this paper, a Monod type chemostat model with delayed response in growth and impulsive input the polluted nutrient is considered. Using the discrete dynamical system determined by the stroboscopic map, we obtain a microorganism-extinction periodic solution. Further, it is globally attractive. The permanent condition of the investigated system is also obtained by the theory of impulsive delay differential equation. Our results reveal that the delayed response in growth plays an important role on the outcome of the chemostat.     

Creep of polymer materials under periodically varying loads

A method is proposed for constructing the creep curves of a material whose nonlinear memory properties are described by Rozovskii's nonlinear integral equation [2] (with allowance for the stress dependence of the relaxation time) under given periodic loading from known creep curves recorded at constant stress. In deriving the theoretical relation certain simplifying assumptions are made (the creep strain accumulated in 1–2 cycles is small, no vibration [4–6]). An experimental check shows that the proposed method can be used to predict the behavior of a material under periodic loading with an accuracy sufficient for practical purposes.Mekhanika Polimerov, Vol. 2, No. 3, pp. 330–336, 1966     

Stability analysis of nonlinear dynamic systems with slowly and periodically varying delay

On the basis of the geometric singular perturbation theory and the theory of delayed Hopf bifurcation in slow–fast systems with delay, the stability of nonlinear systems with slowly and periodically varying delay is investigated in this paper. Sufficient conditions ensuring asymptotic stability of those systems are obtained. Especially, though a time-varying delay usually increases complexity in the analysis of system dynamics and it usually deteriorates system stability as well, the study indicates that under certain conditions, the stability of the systems with a time-invariant delay only can be improved by incorporating a slowly and periodically varying part into the constant delay. Two illustrative examples are given to validate the analytical results.     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

A Lotka-Volterra type food chain model with stage structure and time delays

A three-species Lotka-Volterra type food chain model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators (immature top predators) do not have the ability to feed on prey (predator). By using some comparison arguments, we first discuss the permanence of the model. By means of an iterative technique, a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Persistence and global stability in a delayed Leslie-Gower type three species food chain

Our investigation concerns the three-dimensional delayed continuous time dynamical system which models a predator-prey food chain. This model is based on the Holling-type II and a Leslie-Gower modified functional response. This model can be considered as a first step towards a tritrophic model (of Leslie-Gower and Holling-Tanner type) with inverse trophic relation and time delay. That is when a certain species that is usually eaten can consume immature predators. It is proved that the system is uniformly persistent under some appropriate conditions. By constructing a proper Lyapunov function, we obtain a sufficient condition for global stability of the positive equilibrium.     

Synchronization of a periodically forced Duffing oscillator with a periodically excited pendulum

In this paper, the analytical conditions for a periodically forced Duffing oscillator synchronized with a chaotic pendulum are developed through the theory of discontinuous dynamical systems. From the analytical conditions, the synchronization invariant domains are developed. For a better understanding of synchronization of two different dynamical systems, the partial and full synchronizations of the Duffing oscillator with the chaotic pendulum are presented for illustrations. The control parameter map is developed from the analytical conditions. Under special parameters, the two systems can be fully and partially synchronized. Since the forced pendulum has librational and rotational chaotic motions, the periodically forced Duffing oscillator can be synchronized only with the librational chaotic motions of the pendulum. It is impossible for the forced Duffing oscillator to be synchronized with the rotational chaotic motions.     

Permanence for the Michaelis-Menten type discrete three-species ratio-dependent food chain model with delay

A discrete three trophic level food chain model with ratio-dependent Michaelis-Menten type functional response is investigated. It is shown that under some appropriate conditions the system is permanent. The results indicate that, to make the species coexist in the long run, it is a surefire strategy to keep the death rate of the predator and top predator rather small and the intrinsic growth rate of the prey relatively large.     

Fractional differential equations with a Krasnoselskii–Krein type condition

We consider an initial value problem for a fractional differential equation of Caputo type. The convergence of the Picard successive approximations is established by first showing that the Caputo derivatives of these approximations converge. The method utilized, originally introduced in [O. Kooi, The method of successive approximations and a uniqueness theorem of Krasnoselskii–Krein in the theory of differential equations, Nederi. Akad. Wetensch, Proc. Ser. A61; Indag. Math. 20 (1958) 322–327], is interesting in itself.     

Modeling the role of constant and time varying recycling delay on an ecological food chain

We consider a mathematical model of nutrient-autotroph-herbivore interaction with nutrient recycling from both autotroph and herbivore. Local and global stability criteria of the model are studied in terms of system parameters. Next we incorporate the time required for recycling of nutrient from herbivore as a constant discrete time delay. The resulting DDE model is analyzed regarding stability and bifurcation aspects. Finally, we assume the recycling delay in the oscillatory form to model the daily variation in nutrient recycling and deduce the stability criteria of the variable delay model. A comparison of the variable delay model with the constant delay one is performed to unearth the biological relevance of oscillating delay in some real world ecological situations. Numerical simulations are done in support of analytical results.     

Dynamics of a three-species food chain model with adaptive traits

A three-species food chain model is proposed with dynamically variable adaptive traits in the intermediate consumer. We prove that its solutions are non-negative and bounded, and we analyze the existence and stability of its equilibria. By applying Li and Muldowney’s [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equ 1993;106:27–39] high-dimensional Bendixson criterion, we show that the positive equilibrium is globally stable under specific conditions. We support our analytical findings with numerical simulations.     

Dynamics of a simple food chain model with a ratio-dependent functional response

This paper considers a diffusive ratio-dependent simple food chain model. The sufficient conditions for the existence and non-existence of coexistence states are provided. In addition, this paper investigates the uniqueness of coexistence states and examines the global attractor of the time-dependent system. Moreover, in view of extinction results, the domino effect and biological control are discussed.     

Analysis of a FitzHugh–Nagumo–Rall model of a neuronal network

Pursuing an investigation started in (Math. Meth. Appl. Sci. 2007; 30 :681–706), we consider a generalization of the FitzHugh–Nagumo model for the propagation of impulses in a network of nerve fibres. To this aim, we consider a whole neuronal network that includes models for axons, somata, dendrites, and synapses (of both inhibitory and excitatory type). We investigate separately the linear part by means of sesquilinear forms, in order to obtain well posedness and some qualitative properties. Once they are obtained, we perturb the linear problem by a nonlinear term and we prove existence of local solutions. Qualitative properties with biological meaning are also investigated. Copyright © 2007 John Wiley & Sons, Ltd.     

Persistence and global stability in a delayed predator–prey system with Michaelis–Menten type functional response

A delayed three-species predator–prey food-chain model with Michaelis–Menten type functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of constructing suitable Lyapunov functional, sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the system.     

Complex dynamics in a food chain with slow and fast processes

This paper is devoted to the analysis of the dynamic behavior of a three-species food chain model, in which two predators compete for the same prey while one of the predators feeds on the other. Under the assumption that the time responses of the three trophic levels are extremely diversified, the model is proved to have homoclinic orbit. We firstly use geometric singular perturbation method to detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Then, we show, numerically, that there exist also nonsingular homoclinic orbits that tend toward the singular ones for slightly different parameter values. This analysis is particularly helpful to understanding the chaotic behavior of the food chains.