© Hindawi Publishing Corp. DIFFUSIVE INSTABILITY IN A PREY-PREDATOR SYSTEM WITH TIME-DEPENDENT DIFFUSIVITY

An ecological model for prey-predator planktonic species has been 
considered, in which the growth of prey has been assumed to follow 
a Holling type II function. The model consists of two 
reaction-diffusion equations and we extend it to time-varying 
diffusivity for plankton population. A comparative study of local 
stability in case of constant diffusivity and time varying 
diffusivity has been performed. It has been found that the 
system would be more stable with time varying diffusivity 
depending upon the values of system parameter.


Introduction.
In 1952, Turing [10] proposed a diffusion-reaction theory of morphogenesis on the basis of well-known laws of physical chemistry.This concept has been extended to develop the theory of biological pattern formation.In an ecological context, Segel and Jackson [8] were the first to apply Turing's model to predator-prey system.Since then, diffusive instability has been playing an important role in the study of ecosystem [4,5,6].In the above-mentioned studies, the considered system parameters are all timeindependent and the ratio of diffusivities of predator and prey beyond a critical value determines the diffusive instability of the system.In case of oceanic diffusion, the diffusivities can vary with time.Horizontal currents in the sea depend upon the depth of the sea.Due to the mixing process of these horizontal currents with the vertical currents, horizontal dispersion occurs [2].As a result, the horizontal diffusivities of phytoplankton (prey) and zooplankton (predator) are not only to be different but also to vary with time [1].
In the present paper, we consider an ecological model for prey-predator, where the population size of prey is not very large.The growth of prey population is assumed to follow a Holling Type-II function.In this case, the diffusive instability of the system may occur and mortality resulting from intraspecies competition among prey is assumed to be negligible.In many real situations, due to severe intraspecies competition, the natural mortality of the predator may be ignored.We consider a basic prey-predator model, taking into account all the above situations with time-varying diffusivities and analyze its stability near equilibrium.We will also describe a general approach to the amplitude equations for perturbations and relate this to Hill's equation.Section 5 presents the stability analysis of the amplitude equation for small values of a parameter representing a level of variation in diffusivity.

Model system : diffusive instability with constant diffusion coefficients.
We consider a prey-predator system described by the system of equations (see [9]) where N 1 (x, t) and N 2 (x, t) are the concentrations of prey and predator at a position x and at time t and D 1 and D 2 are their diffusion coefficients, respectively.
In the absence of diffusion, the prey-predator system has spatially uniform steady states given by For the existence of the second stationary point, namely, E 2 , we must have εγ > 1 or γ > 1/ε.We now consider the effect of small inhomogeneous perturbations of the steady state E 2 .Let x 1 (x, t) and x 2 (x, t) be the perturbations such that Assuming x 1 and x 2 to be sufficiently small and linearizing (2.1) about E 2 , we get where In the absence of diffusion, the conditions for the stability of the system are (see [8]) Therefore, it follows that for ε < 1, the system will become stable.The system may be unstable for ε > 1.
We now consider the system with diffusion.We take the solution of the system in this case as where k is the wave number.Let where λ is the growth rate of perturbation in time t and α i is the amplitude at time t.
The characteristic equation of the system is (2.15) The system will not be stable if at least one of the roots of the above equation is positive, that is, the condition for diffusive instability is (see [3]) Now, H is a quadratic in k 2 and we suppose that D 1 and D 2 are constants.Then H(k 2 ) has a minimum for the value where (2.17) Then the inequality H(k m 2 ) < 0 gives Thus H(k m 2 ) will be negative when (2.18) is satisfied, and for the wave numbers close to k m 2 , the growth rate of perturbations λ can be positive.This criterion is equivalent to the dimensionless form where The second inequality in (2.19) is automatically satisfied as εγ 2 (εγ − 1) > 0 is true from the condition of existence of the second stationary point E 2 .
The first inequality in (2.19) gives the criterion of diffusive instability and from the first equation in (2.20), we get D 2 = βD 1 .Therefore, (2.18) gives (β + εγ 2 ) 2 > 4βε 2 γ 3 or (2.21) Therefore, the critical values of β are given by So the diffusive system given by (2.1) will be stable if The condition for diffusive instability can also be written as that is, (2.25)

Variable diffusivities and amplitude equations.
To examine the stability of the uniform steady state to spatial and temporal perturbations in the presence of diffusion, we consider the system of (2.4) and define dimensionless time ωt = τ, where ω > 0 is the frequency of variation in D 2 .Here we consider D 1 as a constant and D 2 as a function of time.We now express the solutions of (2.4) in the form Then we obtain the system of equations for φ i as For simplicity, we consider the time-varying diffusivity D 2 in τ as where β 1 > 1 and β 1 > |α|.
The system of (3.2) can be rewritten as where As a reference state we take α = 0, that is, the case with constant diffusivities.
We have already seen that the criterion for diffusive instability with constant diffusion coefficient is In this case, the critical values of β 1 are and the corresponding critical wave number k cr for the first perturbation to grow is found by evaluating k m from (2.17), considering D 2 = β 1 D 1 at β 1 = β cr .This critical value of β 1 identifies the stable and unstable regions of the diffusive system (2.1).
We are now interested in finding the diffusive instability regions in the system of variable diffusivities and comparing the result with the reference system of constant diffusivities.
Substituting the first equation in (3.4) into the second equation and considering the transformation we get where Equation (3.8) is the standard form of Hill's equation [7].
Substituting the values of â11 , â12 , â21 , and â22 into (3.8),we get where The solution of (3.4) can then be written as 4. Linear stability for small variation in diffusion coefficient.We now study the linear stability of the system when the amplitude α of the variability in D 2 is small.For this, we first set β 1 = β cr and k m = k cr for marginal stability in the reference state and analyze the linear stability of the system when a small variation in D 2 is introduced.Equation (3.10) is then reduced to where The inequality in (4.2) holds since α is very small.
We seek a straightforward expansion for the solution of (4.1) in power series of η in the form of (see [7]) Since η 1, we have When δ > 0, say δ = r 2 , the solution of (4.5) can be written as where A and B are constant.Substituting from (4.7) in (4.6) and disregarding the homogeneous solution, we may write Therefore, the solution of the first equation in (3.4) is Now, ( â11 + â * 22 ) < 0 for ε < 1 and consequently φ 1 tends to zero when y, as well as τ, tends to infinity.Therefore, for δ > 0 and ε < 1, the diffusive system will be asymptotically stable.
From the analysis of Section 2, we have the condition of stability for constant diffusivity as Thus, the length of the interval of β for the stability of the system is [2εγ 2 In this section, we see that the stability criterion for variable diffusivity is 0 < β 1 < εγ 2 , where ε < 1.Therefore, in this case, the length of the interval of β 1 for which stability occurs is εγ 2 .So, if 2 (2εγ − 1) 2 − 1 < 1 or εγ < 1.06 (approximately), then the system will become more stable under variable diffusivity than under constant diffusivity.

Conclusion.
In the present paper, we have considered a model of preypredator ecosystem where the growth of prey is not directly proportional to the existing prey population and it is described by a Holling type II function.The stability analysis of the system reveals that the system without diffusion is stable when ε < 1, that is, the growth rate of prey is small.We have then studied the diffusive instability of the system with constant diffusion coefficient.From these studies, it follows that diffusive instability will certainly develop when (5.1)This condition will be satisfied when D 2 /D 1 1.Hence, for diffusive instability of the system, the mobility of predator should be much higher than that of the prey.
When the diffusivity of the predator is driven by time-varying diffusion coefficients, the stability criterion of the system is changed.In this case, we see that, for ε < 1, the system will become asymptotically stable if 0 < β 1 < εγ 2 .We also observe that, depending upon the values of ε and γ, that is, the growth rate of the prey and the competition rate of the predator, the system will become more stable with time-varying diffusivity than with constant diffusivity.If εγ < 1.06 (approximately), then the system with varying diffusivity will be more stable.

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

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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