© Hindawi Publishing Corp. PERIODIC SOLUTION FOR A DELAYED THREE-SPECIES FOOD-CHAIN SYSTEM WITH HOLLING TYPE-II FUNCTIONAL RESPONSE

A delayed three-species periodic food-chain system with Holling 
type-II functional response is investigated. By using Gaines and 
Mawhin's continuation theorem of coincidence degree theory, a set 
of easily verifiable sufficient conditions is derived for the 
existence of positive periodic solutions to the system.


Introduction.
A rather characteristic behavior of population dynamics is the often-observed oscillatory phenomenon of the population densities.There are three typical approaches for modelling such a behavior: (i) introducing more species into the model and considering the higher-dimensional systems (like predator-prey interactions, see May [8]); (ii) assuming that the per capita growth function is time dependent and periodic in time; (iii) taking into account the time-delay effect on the population dynamics (Smith and Kuang [9], Zhao et al. [12]).In most of the models considered so far, it has been assumed that all biological and environmental parameters are constants in time.However, any biological or environmental parameters are naturally subject to fluctuation in time.The effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment.Thus, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (such as seasonal effects of weather, food supplies, mating habits, etc.); on the other hand, it is generally recognized that some kinds of time delays are inevitable in population interactions (see [1,2] and the references cited therein).Time delay due to gestation is a common example because, generally, the consumption of prey by the predator throughout its past history governs the present birth rate of the predator.The effect of time delays on the asymptotic behavior of populations has been studied by a number of authors (see, e.g., [3,11]).Therefore, more realistic models of population interactions should take into account the seasonality of the changing environment and the effect of time delays.
Recently, Wang and Fan [10] discussed a two-species periodic predator-prey system with infinite delay.Sufficient conditions are derived in [10] for the existence of a positive periodic solution to the system.Motivated by the work of Wang and Fan in [10], in the present paper, we are devoted to the study of the following three-species periodic food-chain predator-prey system with time delays: with initial conditions where It is well known that by the fundamental theory of functional differential equations [6], system (1.1) has a unique solution x(t) = (x 1 (t), x 2 (t), x 3 (t)) satisfying initial conditions (1.2).It is easy to verify that solutions of system (1.1) corresponding to initial conditions (1.2) are defined on [0, +∞) and remain positive for all t ≥ 0. In this paper, the solution of system (1.1) satisfying initial conditions (1.2) is said to be positive.

Existence of periodic solutions.
In this section, by using Gaines and Mawhin's continuation theorem of coincidence degree theory, we show the existence of positive ω-periodic solutions of (1.1) and (1.2).To this end, we first introduce the following notations.Let X, Y be real Banach spaces, let L : DomL ⊂ X → Y be a linear mapping, and let N : X → Y be a continuous mapping.The mapping L is called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Y .If L is a Fredholm mapping of index zero and there exist continuous projectors P : X → X and Q : Y → Y such that Im P = Ker L and Ker Q = Im L = Im(I −Q), then the restriction L P of L to Dom L ∩ Ker P : (I − P )X → Im L is invertible.Denote the inverse of L P by K P .
If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if QN( Ω) is bounded and For convenience of use, we introduce the continuation theorem of coincidence degree theory (see [5,  In what follows, we will use the notations where f is a continuous ω-periodic function. Lemma 2.2.Assume the following hold: (H1) a 32 − r 3 m M 2 > 0, (H2) r 1 a 21 − m M 1 r 2 − a 11 r 2 > 0. Then the system of algebraic equations ) A direct calculation shows that (2.4) Obviously, there exists a unique zero point u * 2 > 0 such that f (u * 2 ) = 0.The first equation of system (2.2) has a unique zero point u * 1 = r 1 /a 11 > 0. Furthermore, from the second equation of (2.2), we obtain which yields u * 3 > 0. The proof is complete.
We are now in a position to state our main result on the existence of a positive periodic solution to system (1.1).
Proof.Since solutions of (1.1) and (1.2) remain positive for all t ≥ 0, we let On substituting (2.6) into system (1.1), we derive is a positive ω-periodic solution of system (1.1).Therefore, to complete the proof, it suffices to show that system (2.7) has one ω-periodic solution.Take where and Define two projectors P and Q as It is clear that (2.13) Therefore, L is a Fredholm mapping of index zero.It is easy to show that P and Q are continuous projectors such that (2.14) Furthermore, the inverse K P of L P exists, that is, K P : ImL → Dom L ∩ Ker P , which is given by (2.16) Clearly, QN and K P (I − Q)N are continuous.In order to apply Lemma 2.1, we need to search for an appropriate open bounded subset Ω.

3.
Discussion.If all the biological and environmental parameters of system (1.1) are constants, then system (1.1) reduces to the following autonomous differential system: where a ij , r i , and m i are positive constants, τ 11 , τ 21 , and τ 32 are nonnegative constants.
Corresponding to Theorem 2.3, we have the following conclusion.Proof.Consider the following system of algebraic equations: The third equation of system (3.2) has a unique zero point 2), we obtain that is Hence it follows that x * 3 > 0. This completes the proof.
In this paper, we have combined the effects of periodicity of the environment and time delays on the dynamics of a food-chain model with Holling type-II functional response.By using Gaines and Mawhin's continuation theorem of coincidence degree theory, we have discussed the existence of positive periodic solutions of the model.
We note that assumptions (H1), (H2), and (H3) in Theorem 2.3 are equivalent to the following: ))e 2(a 21 /m 1 )ω .By Theorem 2.3, we see that system (1.1), with initial conditions (1.2), will have at least one periodic solution if the intrinsic growth rate of the prey species and the conversion rates of the predator and the top predator are high, and the density-dependent coefficient of the prey, the death rate of the predator and the top predator are low enough.By Theorem 2.3, we see that the time delays are harmless to the existence of positive periodic solutions.
An alternative method in proving the existence of positive periodic solutions of system (1.1) may be the application of Horn's asymptotic fixed-point theorem (see, e.g., [4,7]), while this method allows the investigator to address the stability issue of the periodic solutions.This may be our future work.
We would like to mention here that it is interesting but challenging to discuss the global attractiveness of positive periodic solutions of system (1.1) when all its coefficients are periodic functions with a common period.We leave this for our future work.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation