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.     

Stability and bifurcation of a simple food chain in a chemostat with removal rates

In this paper we consider a model describing predator–prey interactions in a chemostat that incorporates genreal response functions and distinct removal rates. In this case, the conservation law fails. To overcome this problem, we use Liapunov functions in the study of the global stability of equlibria. Mathematical analysis of the model equations with regard to invariance of non-negativity, boundedness of solutions, dissipativity and persistence are studied. Hopf bifurcation theory is applied.     

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.     

Dynamical behavior of a stochastic food chain chemostat model with Monod response functions

This paper studies a food chain chemostat model with Monod response functions, which is perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then sufficient conditions for the existence of a unique ergodic stationary distribution are established by constructing suitable Lyapunov functions. Moreover, we consider the extinction of microbes in two cases. In the first case, both the predator and prey species are extinct. In the second case, only the predator species is extinct, and the prey species survives. Finally, numerical simulations are carried out to illustrate the theoretical results.     

Stationary distribution and extinction for a food chain chemostat model with random perturbation

In this paper, we study the dynamical behavior of a stochastic food chain chemostat model, in which the white noise is proportional to the variables. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we show the system has a unique ergodic stationary distribution. Furthermore, the extinction of microorganisms is discussed in two cases. In one case, both the prey and the predator species are extinct, and in the other case, the prey species is surviving and the predator species is extinct. Finally, numerical experiments are performed for supporting the theoretical results.     

Chaos in a three-species food chain model with a Beddington–DeAngelis functional response

This paper investigates a three-species food chain model with a Beddington–DeAngelis functional response, both analytically and through numerical simulations. First the equilibrium states of the system are identified and their stability analyzed analytically. The results of simulations demonstrate chaotic long-term behavior over a broad range of parameters. The existence of a strange attractor and computation of the largest Lyapunov exponent also demonstrate the presence of chaotic dynamics in the model.     

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake

In this paper, we consider a simple chemostat model involving a single species feeding on redundant substrate with a constant yield term. Many experiments indicate that very high substrate concentrations actually inhibit growth. Instead of assuming the prevalent Monod kinetics for growth rate of cells, we use a non-monotonic functional response function to describe the inhibitory effect. A detailed qualitative analysis about the local and global stability of its equilibria (including all critical cases) is carried out. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

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 tritrophic food chain model

We consider a tritrophic food chain model in which the classical assumption of the domino effect is supplemented with an adaptive parameter. Dynamical behaviours such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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     

Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease‐free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo‐rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Dynamics of modified Leslie–Gower-type prey–predator model with seasonally varying parameters

A modified Leslie–Gower-type prey–predator model composed of a logistic prey with Holling’s type II functional response is studied. The axial point (1, 0) is found to be globally asymptotically stable in a domain. Condition for stability of the non-trivial equilibrium point is obtained. The existence of stable limit cycle of the system is also established. The analysis for Hopf bifurcation is carried out. The numerical simulations are carried out to study the effects of seasonally varying parameters of the model. The system shows the rich dynamic behavior including bifurcation and chaos.     

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.     

On the explosive instability in a three‐species food chain model with modified Holling type IV functional response

In earlier literature, a version of a classical three‐species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.     

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.