The population dynamics of the predator and its prey brings to light diversity of patterns that have appeared in nature. Mathematical models have been designed to describe the predator-prey interaction. Analysis of the dynamical behavior of predator-prey systems is an area of interest for many researchers because of its complexity and challenging situation. The most noticeable element in the predator-prey relationship is functional response. Many of the predator-prey models have functional response that depends on prey density and their properties is well understood. A recent proposal by biologists infer that the functional response depends on the ratio of prey and predator. This kind of functional response is said to be ratio-dependent. In the past decades, researchers mathematically modeled the predator-prey behaviour having ratio-dependent functional response(see Arditi [1], Akchaya [3] and Abrams [4] and references citied there in).

It is pointed out that qualitative analysis of food chain and multispecies models based on ratio-dependent approach exists in Kesh [24], Gakkar [14], Baek [7]. It has been documented in the study of Kuang [25], Hsu [19] and Xiao [35] that ratio dependent models produce richer and more reasonable dynamics. Jost and Ellner [21] proposed and analysed a two species model with ratio dependent III functional response. Agarwal [2] generalized the three species model (one prey-two predators) with ratio-dependent III functional response. There are enormous numbers of food chain models consisting of two or more species with unique functional responses.

The system representing the interaction between three species shows complex dynamical behavior. For further reference see Gakkar [12,13,15], Kumar [27], Beak [6], Samantha [31], Tripathi [33], Fan [10], Patra[29], Freedman[9]. The interaction of species involving persistence and extinction have been the area of interest for the researchers Dubey [8], Kar [22,23], Naji [28].

The literature survey above infers that most models have same growth rates and functional response. But this is biologically unrealistic in nature. The reality is that predation happens on different preys in a number of consumption ways. To describe this happening, two different types of functional response are necessary. And it is also well-known that growth rate of different species is different. Sahoo [32] proposed that a real prey-predator model is constructed with different growth rates and different functional responses. So, in this paper, two prey species, one with Verhulst [34] logistic growth equation and other with Richards [30] growth equation is taken into account along with two types of functional responses namely Holling type I and Ratio-dependent III functional response.

This paper is organized as follows. We start in section 2 by defining the mathematical model of three species population which consists of two preys and one predator. The non linear system of differential equations that govern this system is introduced. Section 3 deals with the determination of equilibrium points and their existence conditions. In section 4, we analyzed dynamical behavior of these equilibrium points. Global stability and Persistence of the system is studied in section 5. In section 6, analysis of the model in presence of discrete delay is discussed. In section 7 is equipped with numerical simulation and discuss the problem.

Mathematical model considered is based on the predator-prey system with Holling type I and Ratio dependent type III functional response. The predator exhibits a Holling type I response to one prey and a Ratio dependent type III response to the other prey.

where X, Y denote population densities of prey and Z denote population density of the predator. In model (2.1) R and S are the intrinsic growth rate of two prey species, K and L are their carrying capacities, c is mortality rate of the predator, β is intraspecific competition factor, λ₁ and λ₂ denote prey species searching efficiency of the predator, a is the half-saturation co-efficient, b₁ and b₂ are the conversion factors denoting the number of newly born predators for each captured of first and second prey respectively.

In order to minimize the number of parameters involved with the model system, it is extremely useful to write the system in non-dimensionalized form. For this purpose we introduce the variables X, Y, Z and T as follows x→XK,y→YL,z→aZL and t → TRS.

In terms of the non-dimensionalized variables the model system (2.1) becomes

where r=1S,s=1R,c1=λ1LaRS,c2=λ2aRS,e=cRS,w1=b1KaL,w2=b2a.

Definition 2.1

The solution of ẋ = f(t, x) is said to be uniformly bounded if ∃c > 0 and for every 0 < a < c, ∃M = M(a) > 0 such that ||x(t₀)|| ≤ a ⇒ ||x(t)|| ≤ M, ∀t ≥ t₀ ≥ 0.

Theorem 2.2

All the solutions of the system (2.2) with positive initial condition (x₀, y₀, z₀) are uniformly bounded within a region Γ where

It can be checked that the system (2.2) has six non-negative equilibrium and two of them namely E₀(0, 0, 0), E₁(1, 0, 0) is always exists. We show that the existence of other equilibrium as follows

Existence ofE₂(x̃, ỹ, 0).

Here x̃, ỹ are the positive solutions of the following algebraic equations

Solving (3.1) and (3.2) we get x̃ = 1, ỹ = 1.

Existence ofE₃(χ̄, 0, z̄).

Here χ̄, z̄ are the positive solutions of the following algebraic equations

Solving (3.3) and (3.4) we get

We see that E₃(χ̄, 0, z̄) exists if w₁c₁ > e.

Existence ofE₄(0, ŷ, ŷ)

Here ỹ, ẑ are the positive solution of the following algebraic equations

Solving (3.5) and (3.6) we get

We see that the equilibrium E₄(0, ŷ, ẑ) exists if w2s>(c2e(w2c2-e)).

Existence ofE₅(x^*, y^*, z^*)

Here (x^*, y^*, z^*) is the positive solution of the system of algebraic equation given below:

Solving (3.7), (3.8) and (3.9) we get

We shall examine the stability of the system (2.2), the variational matrix relating to every equilibrium steady state is measured.

Theorem 4.1

The trivial equilibrium point E₀is stable in z direction and unstable in x − y direction.

Theorem 4.2

The equilibrium point E₁is stable in x − z direction and unstable in y direction, if w₁c₁< e. Otherwise unstable in y − z direction and stable in x direction.

Theorem 4.3

The equilibrium point E₂is locally asymptotically stable if w₁c₁ + w₂c₂ < e. Otherwise unstable in z direction and stable in x − y direction.

Theorem 4.4

The equilibrium point E₃is locally asymptotically stable if satisfy the condition p₁ > 0, p₃ > 0 and p₁p₂ − p₃ > 0 otherwise unstable.

Theorem 4.5

The equilibrium point E₄is locally asymptotically stable if and only if A^* + B^* + C^* < 0 and Δ > 0 otherwise unstable.

Theorem 4.6

The equilibrium point is E₅(x^*, y^*, z^*) locally asymptotically stable if and only if the inequalities of (4.2) are satisfied.

Theorem 5.1

The interior equilibrium E₂is globally asymptotically stable in the interior of the quadrant of the x − y plane.

Theorem 5.2

The interior equilibrium E₃is globally asymptotically stable in the interior of the quadrant of the x − z plane.

Theorem 5.3

The interior equilibrium E₄is globally asymptotically stable in the interior of the quadrant of the y − z plane.

Theorem 5.5

Let the hypotheses of theorems 5.1, 5.2 and 5.3 hold, and then the system (2.2) is uniformly persistent if the following inequalities hold

We apply differential equations for any system involving time delay. Time delay may arise naturally or artificially. Delay differential equations involves complex dynamics compared to ordinary differential equations as time delay may cause stability fluctuations. without time delay a real system may not be well established. The application of time-delay in realistic models is detailed in the books of Gopalsamy [16], Kuang [26].

In this section, we analyze the model system (2.2) with delay τ (discrete time delay in the predator response function). Then the model system (2.2) takes the following form

with the initial densities

The main purpose of this section is to study the stability behavior of E₅(x^*, y^*, z^*) in the presence of discrete delay (τ ≠ 0). Now to prove the stability behavior of E₅(x^*, y^*, z^*) for the system (6.1), first we linearize the system (6.1) by using following transformation

The linear system is given by

where

We look for solution of the model (6.1) of the form A(τ) = ρe^−λτ, ρ ≠ 0. This leads to the characteristic equation

where

The eigen values are the roots of the characteristic equation (6.3) of the system (6.1) that has infinitely many solutions. We wish to find periodic solution of the system (6.1), for the periodic solution eigenvalues will be purely imaginary. Substituting λ = iω in equation (6.3) we get

Comparing real and imaginary parts, we get

Squaring and adding we get

where

Putting ω² = δ equation becomes

Now equation (6.5) will be positive if s₁ > 0, s₃ < 0.

By Descartes rule of sign, the cubic equation (6.5), has at least one positive root. Consequently the stability criteria of the system for τ ≠ 0, will not necessarily ensure the stability of system for τ ≠ 0. The critical value of delay that is given as

So corresponding to λ = iω₀ there exists τ0n* such that

Hopf Bifurcation

We observe that the condition’s for Hopf bifurcation (Hale [18]) are satisfied yielding the required periodic solution, that is

This signifies that there exists at least one Eigen value with positive real part for τ > τ^*. Now, we show the existence of Hopf bifurcation near E₅(x^*, y^*, z^*) by taking τ as bifurcating parameter.

Differentiating equation (6.3) with respect to τ,