STABILITY AND HOPF BIFURCATION ANALYSIS ON A SPRUCE-BUDWORM MODEL WITH DELAY

In this paper, the dynamics of a spruce-budworm model with delay is investigated. We show that there exists Hopf bifurcation at the positive equilibrium as the delay increases. Some sufficient conditions for the existence of Hopf bifurcation are obtained by investigating the associated characteristic equation. By using the theory of normal form and center manifold, explicit expression for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are presented.


Introduction
The outbreaks of insect pests have significant impacts on the forest ecosystems. Eastern spruce budworm is a destructive insect living in the spruce and fir forest of North America. Every 35-40 years there is an outbreak of these insects, resulting in serious defoliation. Fortunately, trees are hardly killed on account of defoliation, but they need 7 to 10 years to replace their foliage. However, the periodical outbreaks of these insects lasting for about 10 years cause billions of dollars loss to forest industry. Therefore, how to control the growth of budworm and protect spruce and fir forest is of great importance. Birds preying on budworms always have a flexible diet, that is, they search for other food resources when the density of budworm under a certain threshold. C.S. Holling [6] believed that the switching functional response of predators facilitate the budworm outbreaks. To understand the dynamics of spruce budworm population, Ludwing et al [9] proposed a model separating the timescales of slow spruce regrowth versus the fast population dynamics of budworm larve and their predators. For the food limited dynamics of budworms in the absence of predators the so-called logistic model is applied for the budworm population density where r and K are the intrinsic rate and the carrying capacity of the population, respectively. May [11] further reduced the model proposed in [9] to the following two differential equations where the variables N and S are the spruce budworm population density and average leaf area per tree, respectively. Refer to [11] for the biological meaning of other parameters. Following [11], Rasmusse et al [14] studied a relaxation oscillation of this system by using singular perturbation theory. Liu et al [8] and Muratori [12] pointed out that the existence of such relaxation implies that the coexistence of predators and prey periodically alternated. Taking into account time delay, Wang and Han [18] proposed a system with distributed delay based on system (1.2). By using geometric singular perturbation theory, they illustrated the existence of the relaxation oscillation and transition of the solution trajectory. Motivated by [18] and taking into the consideration the fact that spruce leaf needs some time to grow, we apply the following system to describe the dynamics of interactions between a predator and a prey specie with a delay in the prey population S(t) at time t. A number of nonlinear model equations with time delay [2-4, 17, 18] have been investigated in recent decades. Let z = ρ P β N , y = Kρ P β S,t = ρt,τ = ρτ . Dropping the bar for conciseness, then (1.3) can be rewritten as , Q = Kδ ρ and f (z, y) = γz 1 − z y − γz 2 α 2 y 2 +z 2 . It has been shown in [14] that system (1.4) has exactly one equilibrium when α 2 ≥ 1 27 and can has one, two or three equilibrium points when 0 < α 2 < 1 27 . In this paper, we will analysis the effect of delay τ on the stability of the positive equilibrium of the system and investigate the Hopf bifurcation by normal form and center manifold theory. Here we will focus on the influence of time delay on bifurcations when α 2 ≥ 1 27 , however the case when 0 < α 2 < 1 27 will be considered in the future.

Stability of equilibrium and existence of Hopf bifurcations
In this section, we focus on the investigation of local stability and Hopf bifurcation criteria of the positive equilibrium E(z 0 , y 0 ) for system (1.4). Letting z 1 = z(t)−z 0 , z 2 = y(t) − y 0 , we rewrite system (1.4) by Taylor series expression at E(z 0 , y 0 ) as follows: The characteristic equation of the delay system takes the form When the time delay τ = 0, the characteristic equation (2.3) becomes Clearly, the roots of (2.4) must have negative real parts when Separating the real and imaginary parts, we get which leads to the following polynomial equation For the case when we can see easily that equation (2.7) has only one positive root Substituting the value of ω 0 in (2.6) yields Hence the characteristic equation (2.3) has a pair of purely imaginary roots ±iω 0 when τ = τ j . Let λ(τ ) = r(τ ) + iω(τ ) be a root of (2.3) such that r(τ j ) = 0 and ω(τ j ) = ω 0 . Substituting λ(τ ) = r(τ ) + iω(τ ) into (2.3) and differentiating the resulting expression with respect to τ , we get . (2.8) Inserting τ = τ j into (2.8) and careful computing, one has which implies that the transversality condition holds, so Hopf bifurcation occurs at τ = τ 0 .

Direction and stability of Hopf bifurcations
In the previous section, we have obtained the conditions which guarantee that spruce-budworm model with delay (1.3) undergo Hopf bifurcation at some critical values of τ . We now apply the center manifold theory and normal form method proposed by Hassard et al [5] to study the stability of the bifurcated periodic solutions and the direction of these Hopf bifurcations.
Using the same notations as in literature [5,16,20], we now compute the coordinates to describe the center manifold C 0 at µ = 0. Let x t be the solution of (3.1) with µ = 0 and z(t) =< q * , x t >, On the center manifold C 0 , we have

It is clear that the solution of above equation is
For θ = 0, we see from (3.19) that Substituting (3.21) into (3.22), we have Similarly, we have Thus, g 12 is determined by the parameters and delay. The following quantities can be derived accordingly According to the results in Hassard [5], one knows that the sign of µ 2 determines the direction of the Hopf bifurcation, the sign of β 2 determines the stability of the bifurcating periodic solutions and the sign of T 2 determines the monotonicity of the period of the bifurcating periodic solutions. Thus we have the following conclusions. (1) The Hopf bifurcation is supercritical (subcritical) if µ 2 > 0 (µ 2 < 0); (2) The bifurcating periodic solutions are stable (unstable) if β 2 < 0 (β 2 > 0); (3) The periodic of bifurcating periodic solutions increase (decrease) if T 2 > 0 (T 2 < 0).

Numerical Simulations
In this section, we perform some numerical simulations of system (1.4) to testify the analytical results proved in the previous sections.
(i) Take τ 0 = 1.51. According the previous analysis, the equilibrium should be stable. Starting from the initial value (z(0), y(0)) = (0.0395335, 0.7351731), we get Figure 1. The simulation results coincide exactly with the our conclusion.
(ii) Take τ 0 = 1.57. According the previous analysis, the equilibrium should be unstable and there exists a periodic solution bifurcating from the equilibrium. Starting from the same initial value as (i), we get Figure 2. It implies that the Hopf bifurcation associated with the critical value τ 0 ≈ 1.5422 is supercritical, the bifurcating periodic solution is stable, and the equilibrium E is unstable. All these results are consistent with our analysis.

Conclusions
In this paper, we have discussed the dynamics of a spruce-budworm model with delay, which is based on system (1.2). Firstly, we studied the effects of the discrete time delay τ on the stability of positive equilibrium of system (1.4). Next, we investigated the existence of Hopf bifurcation, the bifurcating direction and stability of the bifurcating periodic solutions by the normal form and center manifold theorems. Just as we have pointed out in Introduction that system (1.3) possibly has one, two or three equilibrium for the case when α 2 < 1 27 , which implies that it may have more abundant dynamics. In fact, the dynamics of a system might be impacted by other factors. It has been shown that Allee effect exists in many natural populations, such as birds, insects, mammals, marine invertebrates and plants [7,15,21]. Allee effect has attracted a significant amount of attention from both mathematicians and ecologists in recent decades. Many interesting results on Allee effect have been obtained in various model equations [1,10,13,19]. The impact of Allee effect on the system investigated in this paper will be considered in our next work.