Global dynamics of a predator-prey system with Holling type II functional response

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1 4 Nonlinear Analysis: Modelling and Control, 011, Vol. 16, No., 4 53 Global dynamics of a predator-prey system with Holling type II functional response Xiaohong Tian, Rui Xu Institute of Applied Mathematics Shijiazhuang Mechanical Engineering College Shijiazhuang , P R China tianh-008@163.com; ru88@yahoo.com.cn Received: 5 November 010 / Revised: 15 March 011 / Published online: 30 May 011 Abstract. In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The eistence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-etinction equilibrium is globally asymptotically stable when the coeistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coeistence equilibrium. Keywords: Holling type II functional response, stage structure, periodic solution, LaSalle invariance principle, global stability. 1 Introduction Mathematical modeling and computer simulation provide an effective tool in the study of contemporary population ecology [1, ]. In population dynamics, the functional response of predator to prey density refers to the change in the density of prey attacked per unit time per predator as the prey density changes [3]. In [4], based on eperiment, Holling suggested three kinds of functional response for different species to model the phenomena of predation, it seems more reasonable than the standard Lotka Voltera type predator-prey system. In [5], Bazykin proposed the following predator-prey system: ut = ut a εut vt = vt c + bvt 1 + αut,, dut 1 + αut ηvt This work was supported by the National Natural Science Foundation of China No and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and the Science Research Foundation of JCB No. JCB c Vilnius University, 011

2 Global dynamics of a predator-prey system with Holling type II functional response 43 where ut, vt represent the densities of prey and predator population, respectively. System 1 is called Holling type II predator-prey model in the literature. This system is an etension of the familiar Lotka Volterra system, in which the divisor 1 + αu is missing, i.e., α = 0. α is interpreted as a constant handling time for each prey captured. For low prey biomass αu 1 this response approimates the classical Lotka Volterra one. In [5], the stability of equilibrium, Hopf bifurcation, global eistence of limit cycles, global attractivity of equilibria, and codimension two bifurcations are investigated. The global behavior of system 1 has been discussed by many authors see, e.g., [5 7]. In system 1, it is assumed that each individual predator has the same ability to feed on prey. However, in the natural world, many species go through two or more life stages as they proceed from birth to death, and in different stages, they have different reactions to the environment. For eample, the immature predators are raised by their parents, and the rate they attack the prey and the reproductive rate can not be ignored. Stage-structured population models have received great attention in recent years see, e.g., [8 13]. In [1], Yu et al. studied the following strengthen type predator-prey model with stage structure ẋt = t r at a 1 y t, ẏ 1 t = ey t r 1 + Dy 1 t, ẏ t = Dy 1 t r y t + a ty t, where t represents the density of the prey at time t, y 1 t and y t represent the densities of the immature and the mature predator at time t, respectively; the parameters a, e, r, a 1, a, r 1, r and D are positive constants in which a is the intra-specific competition rate of the prey, a /a 1 is the rate of conversing prey into new mature predator, e is the birth rate of new predators, r is the intrinsic growth rate of the prey, r 1 is the death rate of the immature predator and r is the death rate of the mature predator, and D denotes the rate of immature predator becoming mature predator. In system, it was assumed that feeding on prey can only make contribution to the increasing of the physique of the predator and does not make contribution to the reproductive ability. In [1], the global asymptotic stability of the coeistence equilibrium was established by constructing suitable Lyapunov functions. Motivated by the works of Bazykin [5] and Yu et al. [1], in this paper, we are concerned with the effect of functional response and stage structure on the dynamics of a predator-prey system. To this end, we study the following differential equations ẋt = t r at a 1y t, 1 + mt ẏ 1 t = ey t r 1 + Dy 1 t, ẏ t = Dy 1 t r y t + a ty t 1 + mt, where a 1 /1 + m describes the Holling type II functional response, here a 1 and m represent the effects of capturing rate and handling time, respectively, and r m > a is 3 Nonlinear Anal. Model. Control, 011, Vol. 16, No., 4 53

3 44 X. Tian, R. Xu reasonable for biological meaning, and implies that hunting has a reward in the sense of diminishing their mortality rate. It is easy to show that all solutions of system 3 are defined on [0, + and remain positive for all t 0. The organization of this paper is as follows. In the net section, by analyzing the corresponding characteristic equations, the local stability of each of nonnegative equilibria of system 3 is discussed. In Section 3, we present conditions for the permanence of system 3. Further, based on discussions above, by using the theory of monotone flows for three-dimensional competitive system, the eistence of orbitally asymptotically stable periodic solution is obtained. In Section 4, by using suitable Lyapunov functions and LaSalle invariance principle, sufficient conditions are derived for the global stability of the predator- etinction equilibrium and the coeistence equilibrium of system 3. A brief remark is given in Section 5 to conclude this work. Local stability In this section, we discuss the local stability of each equilibria of system 3 by analyzing the corresponding characteristic equations. It is easy to show that system 3 always has a trivial equilibrium E 0 0, 0, 0 and a predator-etinction equilibria E 1 r/a, 0, 0. Furthermore, if the following holds: H1 a r > a + rm[r ed D+r 1 ] > 0, then system 3 has a unique coeistence equilibrium E, y 1, y, where = y 1 = r ed a r m + emd, e y, y = a r a. a 1 [r ed] We now study the local stability of each of nonnegative equilibria of system 3. By analyzing the characteristic equation of system 3 at the equilibrium E 0 0, 0, 0, it is easy to show that E 0 is always unstable. The characteristic equation of system 3 at the equilibrium E 1 r/a, 0, 0 takes the form λ + r λ + g 1 λ + g 0 = 0, 4 where a r g 0 = r ed a + rm, g 1 = + r a + rm a r. a + rm Eq. 4 always has a negative real root λ = r. If a < a + rm[r D + r 1 ed], then g 0 > 0, g 1 > 0. It is easy to show that roots of λ + g 1 λ + g 0 = 0

4 Global dynamics of a predator-prey system with Holling type II functional response 45 have only negative real parts. Accordingly, the equilibrium E 1 of system 3 is locally asymptotically stable. If H1 holds, Eq. 4 has at least one positive real root. Therefore, E 1 is unstable. The characteristic equation of system 3 at the coeistence equilibrium E, y 1, y is of the form λ 3 + p λ + p 1 λ + p 0 = 0, 5 where p 0 = [ a r m + emd ] a 1 y 1 + m, a 1 y p 1 = + r 1 + m + r a a 1 + m + r, p = + a r + ed a 1 y m. It is readily seen that if p > 0, p 1 p p 0 > 0, by the Routh Hurwitz theorem, the coeistence equilibrium E of system 3 is locally asymptotically stable; and E is unstable if p 1 p p 0 < 0. Based on the discussions above, we have the following result. Theorem 1. For system 3, we have: i The equilibrium E 0 0, 0, 0 is always unstable. ii If a r < a + r m[r ed], then the equilibrium E 1 r/a, 0, 0 is locally asymptotically stable; if H1 holds, E 1 is unstable. iii Let H1 hold. If p > 0, p 1 p p 0 > 0, then the coeistence equilibrium E, y 1, y of system 3 is locally stable; if p 1 p p 0 < 0, E is unstable. 3 Permanence and stable periodic solution In this section, we establish the permanence of system 3 and the eistence of orbitally asymptotically stable periodic solution. Before giving our main results, we need the following lemmas. Lemma 1. Let r > ed. Then positive solutions of system 3 are ultimately bounded. Proof. Let t, y 1 t, y t be any positive solution of system 3. Define Nt = a t + ky 1 t + a 1 y t, where k = a 1 [r + ed]/e. Calculating the derivative of Nt along positive solutions of 3, it follows that Ṅt = a r a k a 1 Dy 1 a 1 r key a r a δky 1 + a 1 y, 6 Nonlinear Anal. Model. Control, 011, Vol. 16, No., 4 53

5 46 X. Tian, R. Xu where δ = min{ a 1 D/k, r ke/a 1 }. Noting that r > ed, it follows from 6 that We derive from 7 that Ṅt = a r + δ a δnt a r + δ δnt. 7 4a lim sup Nt a r + δ := M. t + 4aδ Hence, for ε > 0 sufficiently small, there eists a T > 0 such that, if t > T, This completes the proof. t < M + ε, y 1 t < M + ε, y t < M + ε. Let X be a complete metric space. Suppose that X 0 X, X 0 X, X 0 X 0 = φ. Assume that T t is a C 0 semigroup on X satisfying T t : X 0 X 0, T t : X 0 X 0. 8 Let T b t = T t X0 and let A b be the global attractor for T b t. The following lemma was introduced by Wang and Chen [10]. Lemma. [10] Suppose that T t satisfies 8 and the following: i there is a t 0 0 such that T t is compact for t > t 0 ; ii T t is point dissipative in X; iii à b = A b ω is isolated and has an acyclic covering M, where iv W s M i X 0 = φ for i = 1,..., n. M = {M 1, M,..., M n }; Then X 0 is uniform repeller with respect to X 0, i.e., there is an ε > 0 such that for any X 0, lim inf t + dt t, X 0 ε, where d is the distance of T t from X 0. We now investigate the permanence of system 3. Theorem. If H1 holds, then system 3 is permanent. Proof. Define U 1 = {, y 1, y R 3 +: 0 }, U = {, y 1, y R 3 +: y 1 0, y 0 }. If X 0 = U 1 U and X 0 = intr 3 +, it is easy to show that X 0 and X 0 are coeistently invariant. Moreover, by Lemma 1, conditions i and ii of Lemma are clearly satisfied.

6 Global dynamics of a predator-prey system with Holling type II functional response 47 Hence, we only need to verify the conditions iii and iv. There are two constant solutions E 0 0, 0, 0 and E 1 r/a, 0, 0 in X 0, corresponding, respectively, to t = y 1 t = y t = 0 and t = r/a, y 1 t = y t = 0. If t, y 1 t, y t is a solution of system 3 initiating from U 1, then ẏ 1 t = ey t y 1 t, ẏ t = Dy 1 t r y t. Obviously, if H1 holds, y 1 t 0 and y t 0 as t +. If t, y 1 t, y t is a solution of system 3 initiating from U with 0 > 0, it is easy to see that t r/a as t +. This shows that if invariant set E 0 and invariant set E 1 are isolated, {E 0, E 1 } is isolated and is an acyclic covering. It is obvious that E 0 is isolated invariant. The isolated invariance of E 1 will be a consequence of the following proof. We now prove that W s E 0 X 0 = φ and W s E 1 X 0 = φ. We restrict our attention to the second equation, since the proof for the first is simple. Assume the contrary. Then there eists a positive solution t, ỹ 1 t, ỹ t of system 3 such that t, ỹ1 t, ỹ t r a, 0, 0 Choose ξ > 0 sufficiently small satisfying Let T > 0 be sufficiently large such that Then we have, for t T,, as t +. r ed < a r/a ξ. 9 mr/a ξ r a ξ < t < r + ξ for t T. a ỹ 1t = eỹ t ỹ 1 t, ỹ t Dỹ 1 t r ỹ t + a r/a ξ 1 + mr/a ξỹt. We consider the matri A ξ defined by [ ] D r1 e A ξ = a D r/a ξ 1+mr/a ξ r. Since A ξ admits positive off-diagonal elements, the Perron Frobenius theorem implies that there is a positive eigenvector υ for the maimum eigenvalue α of A ξ. Moreover, by computing, we see that the maimum eigenvalue α is positive since we have 9. We now consider the following system: y 1t = y 1 t + ey t, y t = Dy 1 t r a r/a ξ y t. 1 + mr/a ξ Nonlinear Anal. Model. Control, 011, Vol. 16, No., 4 53

7 48 X. Tian, R. Xu Let υ = υ 1, υ and let l > 0 be small enough such that ỹ 1 t 0 > lυ 1, ỹ t 0 > lυ. If y 1 t, y t is a solution of system 11 satisfying y 1 t 0 = lυ 1, y t 0 = lυ, since the semiflow of system 11 is monotone and A ξ υ > 0, it follows from [14] that y i t is strictly increasing and y i t + as t +. Note that ỹ i t y i t for t > t 0. We have ỹ i t + as t +. This contradicts Lemma 3.1. The above assertion is thus proved. At this time, we are able to conclude from Lemma 3. that X 0 repels the positive solutions of 3 uniformly. As a consequence, there eists a ε > 0 such that each positive solution t, y 1 t, y t of 3 satisfies lim inf t + t ε and lies eventually outside the set Q 1 defined by Let Q 1 = {, y 1, y : > 0, 0 < y 1 ε, 0 < y ε }. { } Dε1 + mε 0 < ρ < min ε, r + r m a ε be fied, where r > a ε/1 + mε. Then y t > Dε/ on region Q defined by Q = {, y 1, y : > 0, y 1 ε, 0 < y ρ }. It follows that each positive solution t, y 1 t, y t of system 3 leaves Q eventually and lies eventually outside Q. Hence, y t ρ for t sufficiently large. In view of lim inf t + t ε, we see that for t sufficiently large, It follows that ẏ 1 t eρ y 1 t. lim inf y 1t eρ. t + Consequently, system 3 is permanent. This completes the proof. In the following, we show that there eists an orbitally asymptotically stable periodic orbit in system 3. Theorem 3. Let H1 hold. If p p 1 p 0 < 0, then system 3 has an orbitally asymptotically stable periodic solution. Proof. Letting z 1 =, z = y 1, z 3 = y, system 3 becomes z 1 = z 1 r + az 1 + a 1z 3, 1 mz 1 z = ez 3 z, z 3 = Dz r z 3 + a z 1 z 3 1 mz

8 Global dynamics of a predator-prey system with Holling type II functional response 49 If we write 1 as z = fz, the Jacobian matri of f at z is as follows: r + az 1 + a1z3 a 1 mz 1 0 1z 1 1 mz 1 Jz = 0 r 1 D e. a z 3 1 mz 1 D r + az1 1 mz 1 Denote E = {z 1, z, z 3 : z 1 < 0, z > 0, z 3 < 0}. Jz has nonpositive off-diagonal elements at each point of E. Thus, system 1 is competitive in E. Let z 1 =, z = y 1 and z 3 = y. It is obvious that z 1, z, z 3 is the unique equilibrium of system 1. Since p p 1 p 0 < 0 holds, the analysis above shows that z 1, z, z 3 is unstable and det Jz < 0. Moveover, since system 3 is permanent, there eists a compact subset B of E such that for each z 0 E, there eists a T z 0 > 0 such that zt, z 0 B for all t T z 0. Hence, by Theorem 1. of [15], system 1 has an orbitally asymptotically stable periodic solution. This completes the proof. In the following, we give one eample to illustrate the main results above. Eample 1. In system 3, we let a = 1, a 1 = 1, a = 4, e =, m = 1, r = 5, r 1 = 1, r =, D = 1. System 3 with above coefficients has a unique coeistence equilibrium E 1/3, 56/9, 8/9. Direct calculation shows that H1 holds and p 1 p p < 0. By Theorem 3, we see that system 3 has an orbitally asymptotically stable periodic solution. Numerical simulation illustrates the above result see Fig y y Fig. 1. The numerical solution of system 3 with a = 1, a 1 = 1, a = 4, e =, m = 1, r = 5, r 1 = 1, r =, D = 1, and φ 1, φ, φ 3 = 1, 1, 1. 4 Global stability In this section, we discuss the global stability of the coeistence equilibrium E and the predator-etinction equilibrium E 1 of system 3, respectively. The strategy of proofs is to construct suitable Lyapunov functions and use LaSalle invariance principle. Theorem 4. If a D+r 1 < a+rm[r D+r 1 ed], then the equilibrium E 1 r/a, 0, 0 of system 3 is globally asymptotically stable. Nonlinear Anal. Model. Control, 011, Vol. 16, No., 4 53

9 50 X. Tian, R. Xu Proof. Let t, y 1 t, y t be any positive solution of system 3. Denote 0 = r/a. Define a V 1 t = 0 0 ln 0 + ky 1 + a 1 y, m 0 where k = a 1 D/. Calculating the derivative of V 1 t along positive solutions of system 3, we obtain that d dt V 1t = a 1 [ m 0 r a a ] 1y 1 + m + k [ [ ] ey y 1 + a1 Dy 1 r y + a ] y m On substituting r = a 0 into 14, it follows that d dt V 1t = aa 0 + a 1a y 1 + m m a 1 a y 1 + m1 + m 0 a1 ed + We derive from 15 that a 1 a 0 y 1 + m1 + m 0 + = aa 1 + m a 1a y 1 + m a 1 r y 1 + m m 0 + a 1 [ ed r ] y. 15 d dt V 1t = a a a+rm 0 a 1 + a + rm { a a + rm [ r ed ]} y. 16 Hence, if a < a + rm[r ed], then V 1t 0. By Theorem in [16], solutions limit M, the largest invariant subset of {V 1t = 0}. Clearly, we see from 16 that V 1t = 0 if and only if = 0, y = 0. Noting that M is invariant, for each element in M, we have t = 0, y t = 0. It therefore follows from the third equation of system 3 that 0 = ẏ t = Dy 1 t, which yields y 1 t = 0. Hence, V 1t = 0 if and only if, y 1, y = 0, 0, 0. Accordingly, the global asymptotic stability of E 1 follows from LaSalle invariance principle. This completes the proof. Theorem 5. Let H1 hold. Then the coeistence equilibrium E, y 1, y of system 3 is globally stable provided that H > r a. Here, > 0 is the persistency constant for satisfying lim inf t + t.

10 Global dynamics of a predator-prey system with Holling type II functional response 51 Proof. Let t, y 1 t, y t be any positive solution of system 3. Since > r/a, it is seen that there is a T > 0 such that t > r/a for all t T and also that > r/a. Clearly, p > 0. By calculations, we derive that p 1 p p 0 a 1 y [ = 1 + m r D + r1 + ed [ + a r ed r + D + r1 + r + a 1 y ] 1 + m + a r + ed [ a r + ed + Accordingly, by Theorem 1, E is locally asymptotically stable. Define a V t = 1 + m ln + a 1 y y y ln y y a 1 y ] 1 + m a 1 y 1 + m + k y 1 y1 y1 ln y 1 y1 ] > 0., 17 where k = a 1 D/. Calculating the derivative of V t along positive solutions of system 3, it follows that d dt V t = [ a m r a a ] 1y 1 + m + k 1 y [ey ] 1 y 1 y 1 + a 1 1 y y [ Dy 1 r y + a ] y m On substituting r = a + a 1 y/1 + m into 18, we derive that [ d dt V a t = m r a r a + a 1 y ] 1 + m a m a1 y 1 + m + a 1eD y a 1eD y y 1 y 1 + a 1 Dy1 a 1 Dy a 1 r y + a 1 r y + a 1a y y 1 + m a 1a y 1 + m a [ = r a m ] + a 1a y m + a 1a 1 + m y + a 1eD y a 1 r y a 1eD y y 1 a 1 Dy y 1 y 1 + a 1 Dy 1 + a 1 r y a 1a y 1 + m. 19 y 1 y Nonlinear Anal. Model. Control, 011, Vol. 16, No., 4 53

11 5 X. Tian, R. Xu Noting that ey = y 1, d dt V a t = 1 + m ed D+r 1 r + a 1+m [ r a + ] + a 1eDy + a 1a y m + a 1a y 1 + m a 1a y 1 + m. a [ = r a m ] + a 1eDy + a 1a y 1 + m 1 + m 1 + m 1 + m 1 + m = 0, 19 can be rewritten as y 1y y y y y 1 1 y1 y y 1y y y y y 1 1 y1 y. 0 Since the arithmetic mean is greater than or equal to the geometric mean, it is clear that y 1y y y y y 1 1 y1 y 0, 1 + m 1 + m 1 + m 1 + m 0, and the equality hold only for =, y 1 = y1, y = y. If t > r/a for t T, we derive that a [ r a m ] 0, with equality if and only if =. This implies that if t > r/a for t T, V t 0, with equality if and only if =, y 1 = y1, y = y. The largest compact invariant subset in M = {, y 1, y V, y 1, y = 0} is the singleton {E }. Therefore, by LaSalle invariance principle, the coeistence equilibrium E is globally asymptotically stable. This completes the proof. Remark. From Theorem 5, we see that if H1 holds and lim inf t + t r/a, then the coeistence equilibrium E is globally stable. We now give a sufficient condition for this inequality. Denote δ = min{d+r 1 a 1 D/k, r ke/a 1 }. By Lemma 1, one can show that solutions of system 3 have an ultimately upper bound M = a r+δ /4aδ. Hence, we derive from the first equation of system 3 that ẋt tr a 1 M at, which yields lim inf t t = r a 1 M/a = [4arδ a 1 a r + δ ]/4a δ. Obviously, we need only to choose parameters satisfying arδ > a 1 a r + δ. 5 Conclusion In this paper, we have studied the global dynamics of a predator-prey model with Holling type II functional response and stage structure. The global stability of the predator-etinction equilibrium E 1 and the coeistence equilibrium E of system 3 has been established by using the Lyapunov LaSalle type theorem. By Theorem 5, we see that if H1 and H hold, the coeistence equilibrium E is globally asymptotically stable. Biologically, these indicate that when the intrinsic growth rate of the prey, the birth rate of new predators and the rate of immature predator becoming mature predator are large enough, and the death rate of the immature predator and mature predator are small enough, then the prey and the predator population coeist, and the ecological system is therefore permanent.

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