HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY

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1 Vol No. J. of Math. PRC HOPF BIFURCATION ANALYSIS OF A PREDATOR-PREY SYSTEM WITH NON-SELECTIVE HARVESTING AND TIME DELAY LI Zhen-wei, LI Bi-wen, LIU Wei, WANG Gan School of Mathematics and Statistics, Hubei Normal University, Huangshi 435, China Abstract: In this paper, we mainly study the Hopf bifurcation and the stability of modified predator-prey biological economic system with nonselective harvesting and time delay. By using the stability and bifurcation theory of differential-algebraic system, the conditions for stability of the positive equilibrium point are obtained, let time delay as bifurcation parameter, the existence of Hopf bifurcation and direction of Hopf bifurcation are obtained. We have improved the Leslie-Gower predator-prey system, make the system which we established more practical, so the conclusions are made more scientific. Keywords: stability; Hopf bifurcation; time delay; non-selective; predator-prey system; periodic solution 1 MR Subject Classification: 34D; 34K18; 34C3 Document code: A Article ID: Introduction In recent years, the increasingly serious problem of environmental degradation and resource shortage, made the analysis and modeling of biological systems more interested. The predator-prey system played a crucial role among the relationships between the biological population, and it naturally attracted much attention both for mathematicians and biologists, especially on predator-prey systems with or without time delay. As we know, delay differential equation models exhibit much more complicated dynamics than differential equation models without delay, see [1 1]. A lot of researchers studied the dynamics of predator-prey models with harvesting and obtained many dynamic behaviors, such as stability of equilibrium, Hopf bifurcation, periodic solution, Bogdanov-Takens bifurcation, Neimark-Sacker bifurcation, and so on, see [1 15]. In [16], Lucas studied the dynamic properties of the following Leslie-Gower predatorprey system { ẋ = xa y, ẏ = yd k y x, 1.1 Received date: Accepted date: Foundation item: Supported by the Funding Program of Higher School Outstanding Youth Scientific and Technological Innovation Team in Hubei of China T141. Biography: Li Zhenwei 1991, male, born at Qianjiang, Hubei, major in ordinary differential equations and control theory.

2 58 Journal of Mathematics Vol. 37 x and y denote prey and predator population densities at time t, respectively, a, d, and k are positive constants that represent the prey intrinsic growth rate, predator mortality rate, and the maximum value of the per capita reduction rate of x due to y, respectively. At present, economic profit is a very important factor for merchants, governments and even every citizen, so it is necessary to research biological economic systems, which are often described by differential-algebraic equations or differential difference-algebraic equations. In 1954, Gordon [13] studied the effect of the harvest effort on ecosystem from an economic perspective and proposed the following economic theory: Net Economic Revenue NER = Total Revenue TR - Total Cost TC. This provides theoretical evidence for the establishment of differential-algebraic equation. Based on the economic theory as mentioned above and system 1.1, Liu and Fu [1] considered the following Leslie-Gower predator-prey system ẋ = xa y E, ẏ = yd k y x, = Epx c v. 1. They investigated the Hopf bifurcation of the above system without considering the effect of time delay and the harvesting of predator. As is known to all, delay differential equation models exhibit much more complicated dynamics than ordinary differential equation models, see [1 1], as was pointed by Kuang [17] that any model of species dynamics without delays is an approximation at best. When we considered the model with non-selective harvesting, namely at the same time there are also the harvesting of predator and harvesting of the prey in the model, it will be more in line with the actual situation of the predator-prey systems. Motivated by the above discussion, in this paper, by choosing the time delay as a bifurcation parameter and consider the predator-prey systems with non-selective harvesting, we investigate a modified Leslie-Gower predator-prey systems with non-selective harvesting and time delay described by the following system dx = xtr dt 1 byt τ Et, dy = ytr dt k yt Et, xt = Etp 1 xt + p yt c 1 c m, 1.3 p 1 > and p > are harvesting reward per unit harvesting effort for unit prey and predator, respectively; c 1 and c are harvesting cost per unit harvesting effort for prey and predator, respectively; m is the economic profit per unit harvesting effort. In this paper, we mainly discuss the effects of economic profit on the dynamics of system 1.3 in the region R 3 + = {x, y, E x >, y >, E > }.

3 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 59 For convenience, we let fxt, Et = f 1 Xt, Et f Xt, Et gxt, Et = Etp 1 xt + p yt c 1 c m, = xtr 1 byt τ Et ytr k yt xt Et, X t = x, y T. The rest of the paper is arranged as follows: in Section, the local stability of the positive equilibrium points are investigated by the corresponding characteristic equation of system 1.3. In Section 3, by using the normal form and Hopf bifurcation theorem, we study the Hopf bifurcation of the nonnegative equilibrium depending on the parameter we show that the positive equilibrium loses its stability and system 1.3 exhibits Hopf bifurcation in the second section. In Section 4, the theoretical result is supplied by a numerical example. Finally, this paper ends with a brief discussion. Local Stability Analysis of System In this section, we discuss the local stability of a positive equilibrium for system 1.3. Now, we try to find all possible positive equilibrium points of system 1.3. A point Y = x, y, E is an equilibrium point of system 1.3 if and only if Y satisfy the following equations xtr 1 byt τ Et =, ytr k yt Et =,.1 xt Etp 1 xt + p yt c 1 c m =. From.1, we can easy get E satisfy E 3 + γ 1 E + γ E + γ 3 =,. γ 1 = bc 1 + bc kp 1 p r 1 p r p, γ = kp 1r 1 + r 1 p r + bm bc 1 c bc r p, γ 3 = bmr p <. Based on the root and coefficient relationship of equation and γ 3 <, we can find at least one positive root E, so system 1.3 has at least one positive equilibrium point Y = x, y, E = kr1 E, r1 E br E, E b, r 1 > E, r > E. Now, we derive the formula for determining the properties of the positive equilibrium point of system 1.3. As in [13], first we consider the local parametric ψ of the third equation of system 1.3, which is defined as follows [x, y, Ē]T = ψzt = N T + U + V hzt, gψzt =,

4 6 Journal of Mathematics Vol. 37 U = 1 1, V = 1, Zt = y 1, y T, N = x, y, Ē, hzt = hh 1 y 1 t, y t, h y 1 t, y t, h 3 y 1 t, y t : R R is a smoothing mapping, that is xt = x + y 1 t, yt = y + y t, Ēt = Ē + h 3 y 1 t, y t. Then we can obtain the parametric system of system 1.3 as follows { dy1t dt dy t dt = y 1 t + x r 1 by t + y Ē + h 3 y 1 t, y t, = y t + y r k yt+y y Ē 1t+x + h 3 y 1 t, y t..3 Noticing that gψzt =, so we can get the linearized system of parametric system.3 at, as follows { dy1t = bx dt y t τ, dy t.4 = ky y dt 1 t ky x y t. x From.4, we can obtain the characteristic equation of the linearized system of parametric system. at, as follows λ + ky x λ + b ky x e λτ =..5 By eq..5, when τ =, it is obvious that ky x > and kby x >, then, two roots of eq..5 has always negative teal parts, i.e., the positive equilibrium point of system 1.3 is locally asymptotically stable. Now, based on the above discussion, we study the local stability around the positive equilibrium point for system 1.3 and the existence of Hopf bifurcation occurring at the positive equilibrium point when τ >. If iω is a root of eq..5, and substituting iω ω is a positive real number into eq..5, and separating the real and imaginary parts, two transcendental equations can be obtained as follows ω = kby cosωτ, x.6 ky ω = kby sinωτ. x x.7 Since sinωτ + cosωτ = 1 and adding.6 and.7, we obtain ω 4 + ky x ω kby x =..8

5 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 61 From.8, ky x > and kby x <, we can easy find that eq..5 has a unique positive root ω, that is ω = ky x + Substituting ω into.6 and solving for τ, we get ky x + 4 kby x..9 τ n = 1 arccos x ω + nπ, n =, 1,,..1 ω kby ω Thus when τ = τ n, the characteristic equation.5 has a pair of purely imaginary roots iω. Lemma.1 Denote by λ n τ = η n τ + iω n τ the root of.5 such that η n τ n =, ω n τ n = ω, n =, 1,,. Then the following transversality condition η nτ n is satisfied. Proof Differentiating eq..5 with respect to τ, we obtain Noting that dλ dτ 1 = [λx + ky kby τe λτ ] kby τe λτ = 1 λ τ λ 1 λ + ky x λ. sign{re dλ dt } λ=iω = sign{re dλ dt 1 } λ=iω = sign{ ω + ky x ω ω + ky x } >. The proof is completed. From the above analysis and [17, 18], we have the following results. Theorem.1 i For system 1.3, its positive equilibrium point Y is locally asymptotically stable for τ [, τ and unstable for τ > τ. ii System 1.3 undergoes Hopf bifurcation at the positive equilibrium point Y for τ = τ n, n =, 1,,. 3 Direction and the Stability of Hopf Bifurcation In this section, we investigate the direction of bifurcation and the stability of bifurcation periodic orbits from the positive equilibrium point Y of system 1.3 at τ = τ by using the normal form approach theory and center manifold theory introduced by Hassard [15]. Now, we re-scare the time by t = t τ, ȳ 1 = y 1, ȳ = y, τ = τ + µ, Z = ȳ 1, ȳ for simplicity, we continue to use Z said Z, then the parametric system.3 of system 1.3 is equivalent to the following functional differential equation FDE system in C = C[ 1, ], R, ŻT = L µ ZT + fµ, Z t, 3.1

6 6 Journal of Mathematics Vol. 37 ZT = y 1 t, y t T, and L µ : C R, f : R C R are given, respectively, by [ ] [ L µ φ = τ + µ φ T + τ + µ a 1 a a 1 ] φ T 1, and fµ, φ = τ + µ a 1 = bx, a 1 = ky f 11 f, x, a = ky x, f 11 = bφ 1 φ 1, f = ky φ x 1 + ky φ 3 1 φ k φ x x +, and φ = φ 1, φ C. By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions θ [ 1, ] such that L µ φ = 1 dηθ, µφθ, φ C. In fact, choosing δθ = [ ] [ ηθ, µ = τ + µ δθ + τ + µ a 1 a {, θ, 1, θ =. For φ C 1 [ 1, ], R, define a 1 ] δθ + 1, Aµφθ = dφθ dθ 1, 1 θ <, dηθ, µφθ, θ =. 3. Then system 3.1 can be rewritten as Ż = AµZ t + RµZ t. 3.3 For ψ C 1 [, 1], R, the adjoint operator A of A as A ψs = dψs ds, < s 1, 1 dη T s, ψ s, s =, 3.4 η T is the transpose of the matrix η.

7 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 63 For φ C 1 [ 1, ], R and for ψ C 1 [, 1], R, in order to normalize the eigenvectors of operator A and adjoint operator A, we define a bilinear inner product θ ψs, φθ = ψφ ψξ θdηθφξdξ, 3.5 θ= 1 ξ= ηθ = ηθ,. It is easy to verify that A and A are a pair of adjoint operators. By the discussion in Section, we know that ±iωτ are eigenvalues of A. Thus they are also eigenvalues of A. Next we calculate the eigenvector qθ of A associated to the eigenvalue iωτ and the eigenvector q s of A associated to the eigenvalue iωτ. Then it is not difficult to show that iω β =, β = bx e iωτ qθ = 1, β T, q s = Gβ, 1e iωτs, Moreover, q s, qθ = 1 and q s, qθ =. iω ky x bx e iωτ, Ḡ = β + β bx τ β β e iωτ 1. In the reminder of this section, we use the same notations as those in [15]. We first compute the coordinates to describe the center manifold C at µ =. Define On the center manifold C, we have żt = q, Z t, W t, θ = Z t Reztqθ. 3.6 W t, θ = W zt, zt, θ = W θ z + W 11θz z + W θ z In fact, z and z are local coordinates for center manifold C in the direction of q and q. Note that W is real if z t is real. We consider only real solutions. For the solution z t C, since µ = and 3.3, we have ż = iωτ z + q θ, f, W z, z, θ + Re[ztqθ], = iωτ z + q f, W z, z, + Re[ztqθ], 3.8 rewrite it as ż = iωτ z + gz, ż, 3.9 gz, ż = g θ z + g 11θz z + g θ z From 3.3 and 3.8, we have { Ẇ = Żt żq zq = AW Re{ q fz, zqθ}, 1 θ <, AW Re{ q fz, zqθ} + f, θ =. 3.11

8 64 Journal of Mathematics Vol. 37 Rewrite 3.11 as Ẇ = AW + Hz, Z, θ, 3.1 Hz, z, θ = H θ z + H 11θz z + H θ z Substituting the corresponding series into 3.1 and comparing the coefficient, we obtain A iωτ W θ = H θ, AW 11 θ = H 11 θ Notice that qθ = 1, β T e iωτθ, q = Gβ, 1, and 3.6 we obtain y 1t = z + z + W 1 z + W 1 11z z + W 1 z +, y t = βz + β z + W z + W 11z z + W z +, y 1t 1 = βze iωτ + β ze iωτ + W 1 1 z + W z z + W 1 1 z +, y t 1 = βze iωτ + β ze iωτ + W 1 z + W 11 1z z + W 1 z +. According to 3.8 and 3.9, we know that By 3.7, it follows that gz, z = q f z, z = Ḡτ β, 1 f 11 = by 1t y t 1, f11 f f = ky y x 3 1t + ky y 1t y t k y t +. x x gz, z = Ḡτ { b β [z + z + W 1 z + W 1 11z z + W 1 z ], 3.15 [βze iωτ + β ze iωτ + W 1 z + W 11 1z z + W 1 z ] ky [z + z + W x 1 z 3 + W 11z z 1 + W 1 z ] + ky [z + z + W x 1 z + W 11z z 1 + W 1 z ] [βz + β z + W z + W 11z z + W z ] k x [βz + β z + W z + W 11z z + W z ] + }.

9 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 65 That is, gz, z = Ḡτ {z [ bβ β e iωτ ky x 3 + ky x β k x β ] +z z[ b β βe iωτ + βe iωτ ky x 3 + z [ b β βe iωτ ky x 3 + ky x β k x β ] + ky x β + β k β x β] +z z[ bβ β e iωτ ky + ky βw x ky k W x y 11 + bβ β e iωτ b β W x 3 ky + x 3 b β ky x β W 1] + }. Comparing the coefficients with 3.1, it follows that ky W 1 x + k βw x g = Ḡτ [ bβ β e iωτ ky x 3 + ky x β k x β ], g 11 = Ḡτ [ b β βe iωτ + βe iωτ ky x 3 k β β ], x g = [ b β βe iωτ ky x 3 g 1 = [ bβ β e iωτ ky x 3 + ky x + ky x β + β k β x β], + ky βw bβ β e iωτ x ky x 3 + β W 1 + ky k ky W x x y 11 + k βw x b β W b β W 1]. Now we compute W θ and W 11 θ. From 3.11 and 3.15, we have that for θ [ 1,, Hz, z, θ = Re{ q fz, zqθ} = gz, zqθ ḡz, z qθ Comparing the coefficients with 3.13, we can obtain that H θ = g qθ ḡ qθ, H 11 θ = g 11 qθ ḡ 11 qθ ky x Substituting the above equalities into 3.14, it follows that { Ẇ θ = iωw θ + g qθ + ḡ qθ, Ẇ 11 θ = g 11 qθ + ḡ 11 qθ Solving 3.18, we have { W θ = ig τ ω qeiωτθ + iḡ 3ωτ qe iωτθ + Ee iωτ, W 11 θ = ig11 τ ω qeiωτθ + iḡ11 ωτ qe iωτθ + F. 3.19

10 66 Journal of Mathematics Vol. 37 In what follows, we seek appropriate E and F in From 3.11 and 3.15, we have H = g q ḡ q + τ H 11 = g 11 q ḡ 11 q + τ d 1 d d 1 d, 3., 3.1 d 1 = bβ e iωτ, d = ky x 3 + ky x β k x β, d 3 = breβ e iωτ, d 4 = ky x 3 + ky Reβ k β x x β. Substituting into 3.14 and noting that iωτ I iωτ I 1 e iωτθ dηθq =, 1 e iωτθ dηθq =. We obtain iω bx e iωτ E = ky iω + ky x x bx F = ky x ky x d 1 d d 3 d 4, It is easy to obtain E and F from 3. and 3.3, that is E 1 = ky iω + x d 1 bx e iωτ ky iω x + bky x e iωτ 4ω, E = F 1 = ky d x 3 + iωd 4 ky iω x + bky x e iωτ 4ω, Therefore we can compute the following quantities C 1 = ky x d 3 bx d 4, F = d 3. bky bx x i τ ω g g 11 g g + g 1, µ = ReC 1 Reλ τ, t = ImC 1 + µ Imλ τ τ ω, β = ReC 1,

11 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 67 which determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions of system 1.3 at the critical value τ. Theorem 3.1 i The direction of Hopf bifurcation is determined by the sign of µ : the Hopf bifurcation is supercritical if µ > and the Hopf bifurcation is subcritical if µ <. ii The stability of bifurcated periodic solution is determined by β: the periodic solution are stable if β > and unstable if β <. iii The period of bifurcation periodic solution is determined by t : the period increase if t >, decrease if t <. x prey time t.55 y predator time t.5 E harvest effort time t E harvest effort y predator x prey Figure 1: When τ =.6 < τ and with initial conditions x =.53, y = 3.148, E =.56, that show the positive equilibrium point Y is locally asymptotically stable..5 4 Numerical Simulations As an example we consider the differential-algebraic predator-prey system 1.3 with the parameters r 1 = 1.6, r = 1.3, b = k = m =.5, p 1 = 7, p = 6, c 1 = 5, c = 3, that is, = xt1.6.5yt τ Et, yt = yt1.3.5 Et, xt = Et7xt + 6yt dx dt dy dt 4.1 And by the discussions in Section and Section 3, we determine the stability of the positive equilibrium point and Hopf bifurcation. Here, for convenience, we only discuss one of the positive equilibrium point Y of system 4.1, and others positive equilibrium points of system 4.1 can be similar studied. we can easily get Y =.53, 3.148,.56, and by computing, we get ω =.994, τ = So by Theorem.1, the equilibrium point Y is asymptotically stable when τ [, τ = [,.6473 and unstable when τ >.6473.

12 68 Journal of Mathematics Vol. 37 x prey time t.6.55 y predator time t E harvest effort time t y predator x prey Figure : When τ =.68 > τ, with the same initial conditions above that shows the bifurcating periodic solutions from the positive equilibrium point Y. E harvest effort When τ =, we can easily show that the positive equilibrium point is asymptotically stable. Y =.53, 3.148,.56 By the theory of Hassard [15], as it is discussed in former section, we also determine the direction of Hopf bifurcation and the other properties of bifurcating periodic solution. By computing, we can obtain the following values C 1 = i, λ τ = i, it follows that µ =.343 <, β = 1.67 >, t = >, from which and Theorem 3.1 we conclude that the Hopf bifurcation of system4.1 occurring at τ =.6473 is subcritical and the bifurcating periodic solution exists when τ cross τ to the left and the bifurcating periodic solution is unstable. By Theorem 3.1, the positive equilibrium point Y of system 4.1 is locally asymptotically stable when τ =.6 < τ as is illustrated by computer simulation in Fig.1. And periodic solutions occur from Y when τ =.68 > τ as is illustrated by computer simulation in Fig.. 5 Discussion Nowadays, economic profit is a very important factor for governments, merchants, and even citizen, and the harvested biological resources in the predator-prey systems are usually sold as commodities in the market in order to achieve the economic interest. So modelling and qualitative analysis for bio-economic system are necessary.

13 No. Hopf bifurcation analysis of a predator-prey system with non-selective harvesting and time delay 69 Compared with most other researches on dynamics of predator-prey population, see [1, 5, 1, 18], the main contribution of this paper lies in the following aspect. The predator-prey system we consider incorporate delay and non-selective harvesting, which could make our model more realistic and the analysis result in this paper is more scientific. So our paper provide a new ideal and a efficacious method for the qualitative analysis of the Hopf bifurcation of the differential-algebraic biological economic system. In addition, stage structure, diffusion effects, disease effects may be incorporated into our bio-economic system, which would make the bio-economic system exhibit much more complicated dynamics. References [1] Li K, Wei J. Stability and Hopf bifurcation analysis of a prey-predator system with two delays[j]. Chaos Solitions Fract., 9, 4: [] Ma Y F. Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays[j]. Nonlinear Anal. Real World Appl., 1, 13: [3] Sadhukhan D, Mondal B, Maiti M. Discrete age-structured population model with age dependent harvesting and its stability analysis[j]. Appl. Math. Comput., 8, 1: [4] Huo H F, Li W T. Existence and global stability of periodic solutions of a discrete predator-prey system with delays[j]. Appl. Math. Comput., 4, 153: [5] Zhang G D Shen Y Chen B S. Bifurcation analysis in a discrete differential-algebraic predator-prey system[j]. Appl. Math. Model., 14, 38: [6] Li X, Ruan S, Wei J. Stability and bifurcation in delay-differential equations with two delays[j]. J. Math. Anal. Appl., 1999, 36: [7] Pan K, Li B W. Existence of positive periodic solution for two-patches predator-prey impulsive diffusion delay systems with functional response[j]. J. Math.,. 1, 31: [8] Zhao H, Wang L, Ma C. Hopf bifurcation in a delayed Lotka-Volterra predator-prey system[j]. Nonl. Anal. RWA., 8, 91: [9] Wei J, Ruan S. Stability and bifurcation in a neural model with two delays[j]. Phys. D., 1999, 13: [1] Celik C. The stability and Hopf bifurcation for a predator-prey system with time delay[j]. Chaos Solitons Fract., 8, 37: [11] Pan S X. Asymptotic spreading in a Lotka-Volterra predator-prey system[j]. J. Math. Anal. Appl., 13, 47: [1] Liu W, Fu C J, Chen B S. Hopf bifurcation and center stability for a predator-prey biological economic model with prey harvesting[j]. Commun. Nonlinear. Sci., doi:1.116/j.cnsns [13] Gordon H S. Economic theory of a common property resource: the fishery[j]. J. Polit., 1954, 6: [14] Li P L, Yu C C, Zeng X W. The qualitative analysis of a class of predator-prey system with functional responses[j]. J. Math., 6, 6: 17. [15] Hassard N D, Kazarinoff Y H, Wan Y H. Theory and application of Hopf bifurcation[m]. Cambridge: Lodon Mathematical Society Lecture Notes, vol. 41, Cambridge Univ. Press, [16] Lucas W F. Modules in applied mathematics: differential equation models[m]. New York: Springer, 1983.

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