Food Chain System Research Articles

Chaos to order — Effect of random predation in a Holling type IV tri-trophic food chain system with closure terms

Complex dynamics of modified Hastings–Powell (HP) model (phytoplankton-zoo-plankton-fish) with Holling type IV functional response and density-dependent mortality (closure terms) for top predator species is investigated in this paper. Closure terms describe the mortality of top predator in plankton food chain models. Modified HP model with Holling type IV functional response gives rise to similar type of chaotic dynamics (inverted “teacup attractor”) as observed in original HP model with Holling type II functional response. It is observed that introduction of nonlinear closure terms eliminate chaos and system dynamics becomes stable. Observation of this paper support the “Steele–Henderson conjecture” that, nonlinear closure terms eliminate or reduces limit cycles and chaos in plankton food chain models. Chaotic or stable dynamics are numerically verified by Lyapunov exponents (LE) method and Sil’nikov eigenvalue analysis and also illustrated graphically by plotting bifurcation diagrams. It is assumed that mortality of fish population, caused by higher-order predators (which are not explicitly included in the model) is not constant, rather it exhibits random variation throughout the year. To incorporate the effect of random mortality of fish population, white noise term is introduced into the original deterministic model. It is observed that the corresponding stochastic model is stable in mean square when the intensity of noise is small.

Extinction and persistence of a nonautonomous stochastic food-chain system with impulsive perturbations

In this paper, a nonautonomous stochastic food-chain system with functional response and impulsive perturbations is studied. By using Itô’s formula, exponential martingale inequality, differential inequality and other mathematical skills, some sufficient conditions for the extinction, nonpersistence in the mean, persistence in the mean, and stochastic permanence of the system are established. Furthermore, some asymptotic properties of the solutions are also investigated. Finally, a series of numerical examples are presented to support the theoretical results, and effects of different intensities of white noises perturbations and impulsive effects are discussed by the simulations.

Concept of food-chain control in the bioreactor fed with a mixture of substrates

Simulation of biological processes is a complex issue. This is especially true for such biological processes as the aquaculture food chain systems in the natural environment. The quality of the final product (e.g., fish and shellfish) largely depends on the conditions in the first level of the food chain. In the modern biotechnology, the biological processes are conducted in the automated continuous flow bioreactors. This is also possible for aquaculture systems, namely, the substrate-bacteria-protozoa process, which is taken under consideration in this paper. The main contribution of the authors is the idea of using a mixture of two substrates that have different properties and are fed into the continuous flow bioreactor in order to control process course and, especially, to control the oscillatory behaviour of the food chain system. The ratio of the inlet flow rates of the two substrates is proposed as a new control variable. The analysis of the system behaviour was performed based on the simulation experiments with the use of the modified mathematical model of the food chain system. The model is an extended version of the classical models defined for the single limiting substrates. The simulation results presets the possibility of using the new control variable for control of the oscillatory behaviour of the system.

Modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey

In this paper, a mathematical model is proposed and analyzed to study the effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey population. The model is formulated by using the system of non linear ordinary differential equations. In the model, it is assumed that the growth rate of prey population is affected by organismal toxicant. In this paper, we have introduced distributed delay in the environmental toxicant in the model. The distributed delay differential equations, though simple in structure, possess a rich array of solutions. The models are being analyzed by using variational matrix and Liapunov functions. The conditions for local and global stability of the equilibrium points are obtained. A region of attraction is being found for global asymptotic stability of the equilibrium points. Also, a Hopf bifurcation analysis has been performed with respect to key parameters for non-trivial equilibrium points. Furthermore, we support our analytical findings with numerical simulations.

Stability Analysis of a Fractional Order Food Chain Model

We provide an analytical proof of the local stability contained in the region of coexistence of the three species of a tritrophic fractional--order food chain model. An expansion formula for fractional derivatives given as in form of a series involving function and moments of its k-thderivative is used. The tritrophic fractional--order food chain system is converted to a system of ordinary differential equation of order 3M. Also stability analysis is studied by using the fractional Routh--Hurwitz stability conditions in origin. Numerical results show that the presented method is easy to implement and accurateto differential equations of fractional order.

Impulsive Control Strategy for a Nonautonomous Food-Chain System with Multiple Delays

A nonautonomous food-chain system with Holling II functional response is studied, in which multiple delays of digestion are also considered. By applying techniques in differential inequalities, comparison theorem in ordinary differential equations, impulsive differential equations, and functional differential equations, some effective control strategies are obtained for the permanence of the system. Furthermore, effects of some important coefficients and delays on the permanence of the system are intuitively and clearly shown by series of numerical examples.

Approaches of the German food industry for addressing the issue of food losses

In the food industry the subject of food losses is of great importance due to economic balance and an efficient application of resources as well as the development of an efficient food chain system. This paper presents the explorative results of a quantitative survey of leading companies of the German food industry to evaluate the relevance and handling of this issue. The investigation reveals that the topic food losses have a high significance in the food industry which will probably increase in future. A sample breakdown by branches indicates that the issue has the highest relevance for companies in the confectionery industry. These companies as well as those in the meat and fish industry want to consider the subject prospectively more powerful in their companies. Across the food industry, there is no communication to consumers of the efforts concerning food losses. And companies in the confectionery industry and in the fruit and vegetable industry rather want to engage more powerful in this topic if consumers’ interest increases. But in order to minimize food losses at all stages along the supply chain, communication and collaboration at all stages is essential, especially the communication to consumers. Thus, it has to be verified whether a suitable communication can lead to advantages in competition and become an important issue for companies to differentiate from competitors.

Chaos control via feeding switching in an omnivory system

Tanabe and Namba (Ecology, 86, 3411–3414) studied a three species Lotka–Volterra model with omnivory and explored that omnivory can create chaos. It is well documented that predator switching is a similar biological phenomenon to omnivory and likely to occur simultaneously. In the present paper, the tri-trophic Lotka–Volterra food web model with omnivory and predator switching is re-investigated. We observe that if we incorporate predator switching in the system and the intensity of predator switching increases above a threshold value, then the system will be stable from chaotic dynamics. To study the global dynamics of the system extensive numerical simulations are performed. Our analytical and numerical results suggest that predator switching mechanism enhances the stability and the persistence of a food chain system.

Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems

The Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) problem for port-controlled Hamiltonian systems is revisited. We propose a methodology that exploits the novel notion of algebraic solution of the so-called matching equation. This notion is instrumental for the construction of an energy function, defined on an extended state-space, which does not rely upon the solution of any partial differential equation. This yields, differently from the classical solution, a dynamic state feedback that stabilizes a desired equilibrium point. In addition, conditions that allow to preserve the port-controlled Hamiltonian structure in the extended closed-loop system are provided. The theory is validated on two physical systems: the magnetic levitated ball and a third order food-chain system. A dynamic control law is constructed for both these systems by assigning a damping factor that cannot be assigned by the classical IDA-PBC.

H∞filtering for a class of singular biological systems

This study deals with the H ∞ filtering problem for a class of singular biological systems with stochastic exogenous disturbance. Singular system theory is utilised to build a stochastic singular food chain system. The authors’ purpose is to estimate the unmeasurable state variables of the food chain system by the available measurements. The general condition is established for a class of non-linear singular systems to achieve input-to-state stable in mean and the prescribed H ∞ performance level. A sufficient condition of the solvability of filtering problem is proposed for the singular food chain system. Simulation example is provided to illustrate the effectiveness of the proposed filtering approach developed in this study.

Dynamical Analysis of a Food Chain System with Two Delays

This paper is concerned with a three species food chain system with the Beddington–DeAngelis functional response and two delays. The local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium are established by analyzing the distribution of the roots of the associated characteristic equation. Further, the properties of the bifurcating periodic solutions such as the direction and the stability are determined by using the normal form method and center manifold argument. Numerical simulations are presented for supporting the analytical results.

Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function

This paper is concerned with a 3-species and a 2-species food-chain reaction diffusion systems in a bounded domain where the diffusion coefficients may be density dependent and the reaction functions are ratio-dependent. These equations are quasilinear where the diffusion coefficients may be degenerate on the boundary of the domain. Three basic types of Dirichlet, Neumann and Robin boundary conditions are considered, and in each case some very simple conditions are obtained to ensure the dynamical behavior of the time-dependent solution in relation to some positive solutions or quasi-solutions of the steady-state problem, including the existence of these solutions. This dynamical behavior leads to the coexistence and global attractor of the food-chain systems. In the case of Neumann boundary condition sufficient conditions are given to ensure that the steady-state problem has a unique positive constant solution which is a global attractor of the time-dependent system.

Permanence and almost periodic solutions for an n-species Lotka–Volterra food chain system on time scales

By applying differential inequalities on time scales and the Lyapunov function method, we obtain some sufficient conditions which guarantee the permanence and the existence of an unique uniformly asymptotically stable almost periodic solution of an n-species Lotka–Volterra unidirectional food chain system on time scales.

How Complex, Probable, and Predictable is Genetically Driven Red Queen Chaos?

Coevolution between two antagonistic species has been widely studied theoretically for both ecologically- and genetically-driven Red Queen dynamics. A typical outcome of these systems is an oscillatory behavior causing an endless series of one species adaptation and others counter-adaptation. More recently, a mathematical model combining a three-species food chain system with an adaptive dynamics approach revealed genetically driven chaotic Red Queen coevolution. In the present article, we analyze this mathematical model mainly focusing on the impact of species rates of evolution (mutation rates) in the dynamics. Firstly, we analytically proof the boundedness of the trajectories of the chaotic attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. By using symbolic dynamics theory, we quantify the complexity of genetically driven Red Queen chaos computing the topological entropy of existing one-dimensional iterated maps using Markov partitions. Co-dimensional two bifurcation diagrams are also built from the period ordering of the orbits of the maps. Then, we study the predictability of the Red Queen chaos, found in narrow regions of mutation rates. To extend the previous analyses, we also computed the likeliness of finding chaos in a given region of the parameter space varying other model parameters simultaneously. Such analyses allowed us to compute a mean predictability measure for the system in the explored region of the parameter space. We found that genetically driven Red Queen chaos, although being restricted to small regions of the analyzed parameter space, might be highly unpredictable.

Model for the Dynamical Study of a Three-Species Food-Chain System under Toxicant Stress

The modeling investigation in this paper discusses the system level effects of a toxicant on a three species food chain system and the state variables of the models are prey and predator densities, concentration of toxicant in the environment and the concentration of toxicant in the prey population. In the models, we have assumed that the presence of top predator reduces the predatory ability of the intermediate predator. The stability analysis of the models is carried out and the sufficient conditions for the existence and extinction of the populations under the stress of toxicant are obtained. Further, it is also found that the predation rate of the intermediate predator is a bifurcating parameter and Hopf-bifurcation occurs at some critical value of this parameter. Finally, numerical simulation is carried out to support the analytical results.

From the Editor's Desk

How far does the system safety concept extend? Does system safety apply only to autos, trains, aircraft and weapon systems, or does it also apply to other types of systems, such as the ecosystem and the food chain system? Many years ago, Rachel Carson showed how DDT was an unsafe product, and how it contaminated our food chain and caused human diseases. It seems that there is now a similar controversy emerging. Genetically modified organics (GMOs) are common foods that have been genetically modified for various reasons — mostly for business gains rather than for health improvements. Monsanto, for example, has modified common food sources, such as soybeans, corn and wheat, to be able to survive the use of the commercial pesticide weed killer, Roundup. Farmers growing GMO foods can then regularly apply Roundup to kill bugs without damaging the crop. This sounds good and makes a lot of money for Monsanto, but the downside is that Roundup is now in these foods — and research is showing that the modified food has bad effects on test animals and, therefore, possibly on humans.

Optimal Harvesting Policies for a Stochastic Food-Chain System with Markovian Switching

An optimal harvesting problem for a stochastic food-chain system with Markovian switching is investigated in this paper. Firstly, the existence, uniqueness, and positivity of the food-chain system’s solution are proved. Secondly, persistent in mean of the system is obtained. Then the optimal harvesting policy is discussed. Finally, the main results are illustrated by several examples.

Chaos and Bifurcations of a Leslie-Gower Food Chain with Strong Allee Effect

Allee effect, as an important biological phenomenon, has been considered in many ecosystems, whereas, its influence on interactions of three or more species is little investigated. In this paper we modify a three-species Leslie-Gower type food chain system by incorporating the strong Allee effect into the source. Our results show that the existence of Allee effect contributes to the occurrence of more complex dynamics of the system, including Hopf, saddle-node, transcritical, saddle-node-Hopf, period-doubling, and period-halving bifurcations and chaos.

Direct and indirect effect of predators on Anopheles gambiae sensu stricto

The increased insecticides resistance by vectors and the ecological harm imposed by insecticides to beneficial organisms drawback mosquitoes chemical control efforts. Biological control would reduce insecticides tolerance and yet biodiversity friend. The predatory and non-predatory effects of Gambusia affinis and Carassius auratus on gravid Anopheles gambiae sensu strict and larvae survivorship were assessed. In determining predation rate, a single starved predator was exposed to third instar larvae of An. gambiae s.s. in different densities 20, 60 and 100. Six replicates in each of the densities for both predators, G. affinis and C. auratus, were set up. The larvae densities were monitored in every12 and 24h. In assessing indirect effects: An. gambiae s.s. first instar larvae of three densities 20, 60 and 100 were reared in water from a predator habitat and water from non-predator habitat. Larvae were monitored until they emerged to adults where larval survivorship and sex ratio (Female to total emerged mosquitoes) of the emerged adult from both water habitats were determined. Oviposition preference: twenty gravid females of An. gambiae s.s. were provided with three oviposition choices, one containing water from predator habitat without a predator, the second with water from a predator with a predator and the third with water from non-predatory habitat. The number of eggs laid on each container was counted daily. There were 20 replicates for each predator, G. affinis and C. auratus. Survivorship of An. gambiae s.s. larvae reared in water from non-predator habitat was higher than those reared in water from the predator habitats. Many males emerged in water from non-predatory water habitats while more females emerged from predator's habitats water. More eggs laid in tap water than in water from predator habitat and water from predator habitat with live predator. In 24h, a starved C. auratus and G. affinis were able to consume 100% of the 3rd instar larvae. The findings from this study suggest that G. affinis and C. auratus may be useful in regulating mosquito populations in favour of beneficial insects. However, a small scale trial shall be needed in complex food chain system to ascertain the observed predation and kairomones effects.

Singularly Perturbed Systems with Diffusion and Time Delay

The singularly perturbed systems for the nonlinear three-species food-chain reaction systems with time delay are considered. By using the method of the stretched variable, the formal asymptotic solution is obtained under suitable conditions.