Global Stability for a Delayed Predator-Prey System with Stage Structure for the Predator

A delayed predator-prey system with stage structure for the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of equilibria of the system is discussed. The existence of Hopf bifurcation at the positive equilibrium is established. By using an iteration technique and comparison argument, respectively, sufficient conditions are derived for the global stability of the positive equilibrium and two boundary equilibria of the system. Numerical simulations are carried out to illustrate the theoretical results.


Introduction
Stage-structure is a natural phenomenon and represents, for example, the division of a population into immature and mature individuals.As is common, the dynamics-eating habits, susceptibility to predators, and so forth.are often quite different in these two subpopulations.Hence, it is of ecological importance to investigate the effects of such a subdivision on the interaction of species.In  where x t represents the density of the prey at time t; y 1 t and y 2 t represent the densities of the mature and the immature predator at time t, respectively.The parameters r, r 1 , r 2 , a 11 , a 12 , a 21 , a 22 , b, θ are positive constants in which r is the intrinsic growth of the prey, r 1 is the death rate of the mature predator population, r 2 is the death rate of the immature predator population, a 11 is the intra-specific competition rate of the prey population, a 12 is the capturing rate of the predator population, a 21 /a 12 is the conversion rate of nutrients into the reproduction of the predator, a 22 is the intra-specific competition rate of the mature predator, b is the birth rate of the immature predator, θ is the transformation rate from the immature predator individuals to mature predator individuals.The predation decreases the average growth rate of prey linearly with a certain time delay τ 1 ≥ 0, this assumption corresponds to the fact that predators cannot hunt prey when the predators are infant; predators have to mature for a duration of τ 1 units of time before they are capable of decreasing the average growth rate of the prey species; τ 2 ≥ 0 is the time delay due to gestation, the delay in time for prey biomass to increase predator number.In 3 , Gao et al. studied the global stability of the positive equilibrium and boundary equilibria of model 1.4 by constructing Liapunov functionals and comparison argument, respectively.We note that most of the predator-prey models with time delays studied in the literature are all of the Kolmogorov-type.In 9 , Wangersky and Cunningham proposed delayed predator-prey models that are not of the Kolmogorov-type.They considered the following delayed system: where the delay τ is a constant based on the assumption that the change rate of predators depends on the number of both the prey and the predators present at some previous time.

1.6
The meanings of the positive parameters r, r 1 , r 2 , a 11 , a 12 , a 21 , a 22 , b, θ are the same as those in system 1.4 .The meaning of time delay τ ≥ 0 is the same as in system 1.5 .
The initial conditions for system 1.6 take the form where . This paper is organized as follows.In the next section, we introduce some notations and state several lemmas which will be essential to our proofs.In Section 3, we discuss the local stability of a positive equilibrium and boundary equilibria of system 1.6 .The existence of Hopf bifurcation is studied.In Section 4, by means of an iterative technique and comparison argument, sufficient conditions are derived for the global stability of the positive equilibrium and boundary equilibria of system 1.6 .Some numerical examples are given to illustrate the results above.A brief discussion is given in Section 6 to end this work.

Preliminaries
In this section, we introduce some notations and state several results which will be useful in next section.Let R n be the cone of nonnegative vectors in R n .If x, y ∈ R n , we write x ≤ y x < y if x i ≤ y i x i < y i for 1 ≤ i ≤ n.Let e 1 , e 2 , . . ., e n denote the standard basis in R n .Suppose r ≥ 0 and let C C −r, 0 , R n be the Banach space of continuous functions mapping the interval −r, 0 into R n with supremum norm.If φ, ψ ∈ C, we write φ ≤ ψ φ < ψ when the indicated inequality holds at each point of −r, x, θ ∈ −r, 0 .Denote the space of functions of bounded variation on −r, 0 by BV −r, 0 .If t 0 ∈ R n , A ≥ 0, and x ∈ C −t 0 − r, t 0 A , R n , then for any t ∈ t 0 , t 0 A , we let x t ∈ C be defined by x t θ x t θ , −r ≤ θ ≤ 0. We now consider ẋ t f t, x t .

2.1
We assume throughout this section that f : R × C → R n is continuous; f t, φ is continuously differentiable in φ; f t T, φ f t, φ for all t, φ ∈ R × C , and some T > 0. Then by 10 , there exists a unique solution of 2.1 through t 0 , φ for t 0 ∈ R, φ ∈ C .This solution will be denoted by x t, t 0 , φ if we consider the solution in R n , or by x t t 0 , φ if we work in the space C. Again by 10 , x t, t 0 , φ x t t 0 , φ is continuously differentiable in φ.In the following, the notation x t0 φ will be used as the condition of the initial data of 2.1 , by which we mean that we consider the solution x t of 2.1 which satisfies x t 0 θ φ θ , θ ∈ −r, 0 .To proceed further, we need the following results from 11, 12 .Let r r 1 , r 2 , . . ., r n ∈ R n , |r| max i {r i }, and define We write φ φ 1 , φ 2 , . . ., φ n for a generic point of C r .Let C r {φ ∈ C r : φ ≥ 0}.Due to the ecological applications, we choose C r as the state space of 2.1 in the following discussions.
Fix φ 0 ∈ C r arbitrarily.Then we set L t, • D φ 0 f t, φ 0 , and D φ 0 f t, φ 0 denotes the Frechet derivation of f with respect to φ 0 .It is convenient to have the standard representation of L L 1 , L 2 , . . ., L n as where η ij •, t is continuous from the left in −r j , 0 .We make the following assumptions for 2.1 .
h1 For all φ ∈ C r with φ i 0 0, L i t, φ ≥ 0 for t ∈ R. h3 For each j, for which r j > 0, there exist i such that for all t ∈ R and for positive constant ε sufficiently small, η ij −r j ε, t > 0.
The following result was established by Wang et   c θ e 2 .

2.10
The characteristic equation of system 1.6 at the equilibrium E 0 is of the form λ 2 a e θ λ a θ e − dθ − bλ b θ e e −λτ 0.

2.12
If a θ e < dθ b θ e , then it is easy to see that, for λ real,

2.13
Hence, f 1 λ 0 has a positive real root.Therefore, the equilibrium E 0 is unstable.If a θ e > dθ b θ e , when τ 0, it is easy to see that the equilibrium E 0 is stable.Therefore, if a θ e − dθ > b θ e , by Kuang and So 13, Lemma B , we see that the equilibrium E 0 is locally stable for all τ > 0.
If a θ e < dθ b θ e , the characteristic equation of the positive equilibrium E takes the form When τ 0, it is easy to see that the equilibrium E is stable.Therefore, if a θ e < dθ b θ e , by Kuang and So 13, Lemma B , we see that the equilibrium E is locally stable for all τ > 0.
Lemma 2.5.For system 2.8 , one has the following.
i If a θ e < dθ b θ e , then the positive equilibrium E of system 2.8 is globally stable.
ii If a θ e > dθ b θ e , the equilibrium E 0 of system 2.8 is globally stable.
Proof.We represent the right-hand side of 2.8 by f t, x t f 1 t, x t , f 2 t, x t and set L t, • D φ f t, φ .By direct calculation we have

2.15
We now claim that the hypotheses h1 -h4 hold for system 2.8 .It is easily seen that h1 and h4 hold for system 2.8 .We need only to verify that h2 and h3 hold.The matrix A t takes the form

2.16
Clearly, the matrix A t is irreducible for each t ∈ R.
Thus, the conditions of Lemma 2.1 are satisfied.Therefore, the positivity of solutions to system 2.8 follows.It is easy to see that system 2.8 is cooperative.By Lemma 2.4 we see that any solution starting from D C τ converges to some single equilibrium.However, system 2.8 has only two equilibria: E 0 and E .Note that if a θ e < dθ b θ e , the equilibrium E is locally stable.Hence, any solution starting from D converges to E .Using a similar argument one can show the global stability of the equilibrium E 0 when a θ e > dθ b θ e .This completes the proof.
By a similar argument one can show that all solutions of system 1.6 with initial conditions 1.7 are defined on 0, ∞ and remain positive for all t ≥ 0.

Local Stability
In this section, we discuss the local stability of each equilibria and the existence of Hopf bifurcation of system 1.6 .

3.2
We now study the local stability of each of the nonnegative equilibrium of system 1.6 .Let E x, y 1 , y 2 be any arbitrary equilibrium.Then the characteristic equation of system 1.6 at the equilibrium E is given by The characteristic equation of system 1.6 at the equilibrium E 0 0, 0, 0 reduces Clearly, λ r is a positive real root.Hence, E 0 0, 0, 0 is always unstable.The characteristic equation of system 1.6 at the equilibrium E 1 r/a 11 , 0, 0 reduces Clearly, λ −r is a negative real root of 3.5 .All other roots are give by the roots of equation

3.7
If θ r 2 r 1 − a 21 r/a 11 < bθ, then it is easy to see that for λ real, Hence, f λ 0 has a positive real root.Therefore, the equilibrium E 1 r/a 11 , 0, 0 is unstable.If θ r 2 r 1 − a 21 r/a 11 > bθ, when τ 0, it is easy to see that the equilibrium E 1 r/a 11 , 0, 0 is stable.Therefore, if θ r 2 r 1 − a 21 r/a 11 > bθ, by Kuang and So 13, Lemma B , we see that the equilibrium E 1 r/a 11 , 0, 0 is locally stable for all τ > 0.

3.9
Clearly, if r > a 12 y 1 , λ r − a 12 y 1 is a positive real root.Hence, if r > a 12 y 1 , then E 2 0, y 1 , y 2 is unstable.Noting that if r < a 12 y 1 , then 3.9 only has negative real root, and E 2 0, y 1 , y 2 is stable.
The characteristic equation of system 1.6 at the positive equilibria where

3.14
Squaring and adding the two equations of 3.14 , it follows that where

3.16
It is easy to show that

3.21
This will signify that there exists at least one eigenvalue with positive real part for τ > τ 0 .Moreover, the conditions for the existence of a Hopf bifurcation 10 are then satisfied yielding a periodic solution.To this end, differentiating equation 3.11 with respect τ, we obtain that dλ τ dτ

3.23
If Therefore, system 1.6 undergoes a Hopf bifurcation.We therefore obtain the following results.Theorem 3.1.For system 1.6 , let τ 0 be defined as in 3.18 , one has the following.
i The positive equilibrium E 0 of system 1.6 is always unstable.
iii Let H1 hold.If r > a 12 y 1 , the equilibrium E 2 0, y 1 , y 2 of system 1.6 is unstable; if r < a 12 y 1 , E 2 is stable for all τ ≥ 0.

Global Stability
In this section, we are concerned with the global stability of the equilibria E 1 , E 2 , E * of system 1.6 .The strategy of proofs is to use an iteration technique and comparison arguments, respectively.Theorem 4.1.Let (H2) hold.Then the positive equilibrium E * of system 1.6 is globally asymptotically stable provided that H3 a 11 a 22 > a 12 a 21 .
Proof.Let x t , y 1 t , y 2 t be any positive solution of system 1.6 with initial conditions 1.7 .Let The strategy of the proof is to use an iteration technique.
We derive from the first equation of system 1.6 that ẋ t ≤ x t r − a 11 x t .

4.2
A standard comparison argument shows that Hence, for ε > 0 sufficiently small there exists a T 1 > 0 such that if t > T 1 , x t ≤ M x 1 ε.We derive from the second and the third equations of system 1.6 that for t > T 1 τ, Consider the following auxiliary equations: Since H2 holds, by Lemma 2.5 it follows from 4.5 that

4.6
By comparison, we obtain that

4.7
Since these inequalities are true for arbitrary ε > 0 sufficiently small, it follows that 1 , where

4.8
Hence, for ε > 0 sufficiently small, there is a

Discrete Dynamics in Nature and Society 13
For ε > 0 sufficiently small, we derive from the first equation of system 1.6 that, for t > T 2 , 4.9 By comparison it follows that Since this is true for arbitrary ε > 0 sufficiently small, we conclude that V ≥ N x 1 , where Therefore, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, we derive from the second and the third equations of system 1.6 that, for t > T 3 τ, Consider the following auxiliary equations: Since H2 and H3 hold, by Lemma 2.5 it follows from 4.13 that lim

4.14
By comparison, we obtain that

4.15
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that 1 , where

4.16
Therefore, for ε > 0 sufficiently small, there exists a 1 − ε.For ε > 0 sufficiently small, it follows from the first equation of system 1.6 that, for t > T 4 , 4.17 A comparison argument yields Since this is true for arbitrary ε > 0, we conclude that U ≤ M x 2 , where Hence, for ε > 0 sufficiently small, there exists a T 5 ≥ T 4 such that if t > T 5 , x t ≤ M x 2 ε.Again, we derive from the second and the third equations of system 1.6 that for t > T 5 τ,

4.20
Since H2 and H3 hold, by Lemma 2.5, a comparison argument shows that

4.21
Since these inequalities are true for arbitrary ε > 0 sufficiently small, we conclude that 2 , where

4.22
Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, it follows from the first equation of system 1.6 that, for t > T 6 , ẋ t ≥ x t r − a 11 x t − a 12 M y 1 2 ε .

4.23
By comparison we obtain that Since this is true for arbitrary ε > 0 sufficiently small, we conclude that V ≥ N x 2 , where Therefore, for ε > 0 sufficiently small, there is a T 7 ≥ T 6 such that if t > T 7 , x t ≥ N x 2 − ε.For ε > 0 sufficiently small, we derive from the second and the third equations of system 1.6 that for t > T 7 τ,

4.26
Since H2 and H3 hold, by Lemma 2.5 and by comparison, it follows from 4.26 that

4.27
Since these two inequalities hold for arbitrary ε > 0 sufficiently small, we conclude that where

4.28
Continuing this process, we derive six sequences

4.29
It is readily seen that

4.33
We therefore obtain from 4.30 and 4.33 that It follows from 4.30 , 4.33 and 4.34 that

4.50
Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, it follows from the first equation of system 1.6 that for t > T 1 , ẋ t ≤ x t r − a 11 x t − a 12 y 1 − ε .

4.51
Since θ r 2 a 22 r a 12 r 1 < a 12 θb and ε > 0 is sufficiently small, by comparison we derive that lim sup t → ∞ x t ≤ 0.

4.52
We therefore have lim t → ∞ x t 0. Hence, for ε > 0 sufficiently small, there is a For ε > 0 sufficiently small, it follows from the first equation of system 1. 6

4.55
Noting that if H1 holds and θ r 2 a 22 r a 12 r 1 < a 12 θb, the equilibrium E 2 of system 1.6 is locally stable.We therefore conclude that E 2 is globally stable.The proof is complete.

Numerical Examples
In this section, we give some examples to illustrate the main results.By Theorem 4.2 we see that the equilibrium E 1 0.5, 0, 0 of system 1.6 is globally stable.Numerical simulation illustrates our result see Figure 1 .

Discussion
In this paper, we considered a delayed predator-prey model with stage structure for the predator.By using the iteration technique and comparison argument, respectively, sufficient conditions were established for the global stability of the positive equilibrium and two boundary equilibria of system 1.6 .By Theorems 4.1, 4.2 and 4.3, we see that: i If H2 holds and a 11 a 22 > a 12 a 21 , system 1.6 is permanent.ii If r 1 θ r 2 > bθ a 21 r/a 11 θ r 2 , the prey species is persistent but the predator becomes extinct.iii If H1 holds and θ r 2 a 22 r a 12 r 1 < a 12 θb, the predator species is persistent but the prey species becomes extinct.
1 , Chen et al. introduced the following stagestructured single-species population model: into the mature.If the birth rate of model 1.1 obeys the Malthus rule, that is, B t aN m , the death rates of the immature and mature populations are logistic, that is, Based on the idea above, many authors studied different kinds of stage-structured models, and a significant body of work has been carried out see, for example, 2-8 .In 3 , Gao et al. considered the following predator-prey model with stage structure: i t and N m t denote the immature and mature population densities at time t, respectively; B t is the birth rate of the immature population at time t; D i t and D m t are the death rates of the immature and mature at time t; W t represents the transformation rate of the immature into the mature; α is the probability of the successful transformation of the immature ẋ t x t r − a 11 x t − a 12 y 1 t − τ 1 , ẏ1 t y 1 t −r 1 a 21 x t − τ 2 − a 22 y 1 t θy 2 t , ẏ2 t αby 1 t − θ r 2 y 2 t , 1.4 Motivated by the work ofGao et al. 3 and Wangersky and Cunningham 9 , in the present paper, we consider the following predator-prey model with stage structure and time delay: ẋ t x t r − a 11 x t − a 12 y 1 t , ẏ1 t a 21 x t − τ y 1 t − τ − r 1 y 1 t − a 22 y 2 1 t θy 2 t , ẏ2 t by 1 t − θ r 2 y 2 t .
al. 12 .If φ and ψ are distinct elements of C r with φ ≤ ψ and t 0 , t 0 σ with n|r| < σ ≤ ∞ is the intersection of the maximal intervals of existence of x t, t 0 , φ and x t, t 0 , ψ , then Definition 2.3.A square matrix A is said to be a reducible matrix if and only if for some permutation matrix P the matrix P T AP is block upper triangular.If a square matrix is not reducible, it is said to be an irreducible matrix.System 2.1 is called irreducible if the Jacobian matrix ∂f i /∂x j is irreducible.Lemma 2.4 Smith 11 .If 2.1 is cooperative and irreducible in D, where D is an open subset of C, and the solution with positive initial data is bounded, then the trajectory of 2.1 tends to some single equilibrium.We now consider the following delay differential system: ẏ1 t −ay 1 t by 1 t − τ − cy 2 1 t θy 2 t , ẏ2 t dy 1 t − θ e y 2 t , 2.8 with initial conditions 11 a 22 a 21 a 12 x * y * 1 a 11 bθx * r 2 θ 2a 11 a 21 x * 2 − a 12 a 21 −a 11 a 22 x * y * ≥ y * 1 and a 11 a 22 > a 12 a 21 , it follows from 4.31 that n

:
Noting that if a 11 a 22 > a 12 a 21 , then p 0 − q 0 > 2 θ r 2 a 11 a 21 x * 2 > 0. By Theorem 3.1, the positive equilibrium E * is locally stable.We therefore conclude that E * is globally stable.The proof is complete.If r 1 θ r 2 > bθ a 21 r/a 11 θ r 2 , the equilibrium E 1 of system 1.6 is globally asymptotically stable.The temporal solution found by numerical integration of system 1.6 with r 2, r 1 2, r 2 1, a 11 4, a 12 3, a 21 2, a 22 4, b 1, θ 2, τ 1 and initial value is 0.6, 0.2, 0.2 .By Theorem 3.1, if r 1 θ r 2 > bθ a 21 r/a 11 θ r 2 , the boundary equilibrium E 1 is locally stable.We therefore conclude that E 1 is globally stable in this case.This completes the proof.Let H1 hold.If θ r 2 a 22 r a 12 r 1 < a 12 θb, then the equilibrium E 2 of system 1.6 is globally asymptotically stable.Proof.Let x t , y 1 t , y 2 t be any positive solution of system 1.6 with initial conditions 1.7 .We derive from the second and the third equations of system 1.6 that, for t > T 1 τ, that for t > T 2 τ ẏ1 t ≤ −r 1 y 1 t a 21 εy 1 t − τ − a 22 y 2 . System 1.6 with above coefficients has a unique positive equilibrium E * 1.5, 2, 4 .It is easy to show that a 11 a 22 − a 12 a 21 1 > 0. By Theorem 4.1 we see that the positive equilibrium E * 1.5, 2, 4 of system 1.6 is globally stable.Numerical simulation illustrates our result see Figure3.b 4, and θ 1. System 1.6 with above coefficients has a unique positive equilibrium E * 0.6, 1.