Dynamic Analysis of a Model for Spruce Budworm Populations with Delay

A class of delayed spruce budworm population model is considered. Compared with previous studies, both autonomous and nonautonomous delayed spruce budworm population models are considered. By using the inequality techniques, continuation theorem, and the construction of suitable Lyapunov functional, we establish a set of easily verifiable sufficient conditions on the permanence, existence, and global attractivity of positive periodic solutions for the considered system. Finally, an example and its numerical simulation are given to illustrate our main results.


Introduction
As is well known, since the spruce budworm population site model [1] has been proposed and was accepted by numerous scholars, during the last decade, spruce budworm population models have been extensively investigated both in theory and applications, such as for protection of spruce trees and development of a strategy for spruce budworm population control [1][2][3][4][5][6][7][8][9][10][11][12][13]. For example, in [2], the authors considered the following standard structured partial differential model of the budworm population: where Nðx, bÞ is the size distribution of the budworm population and EðbÞ and P b are the mortality rate and the predation rate.
On the other hand, in mathematical modeling of real world problems, the growth rate of a natural species will not often respond immediately to changes in its own population or that of an interacting species but will rather do so after a time lag [14]. Recently, some delayed mathematical models have been proposed in the study of spruce budworm population models [2,[11][12][13], and some research results were obtained. For example, in [11], the authors further ana-lyzed system (1) and proposed the following delayed spruce budworm population model: where mðtÞ = Ð ∞ τ Nðt, aÞda is the mature population density at time t,τ is the maturation time, D is the average mortality rate of the mature budworms,d is the average death rate of the immature population, p = pðmðtÞÞ is a predation rate function for the matured population, and b = bðmðtÞÞ is the birth function. In [11], the authors nondimensionalize system (2) and obtained the following delayed nondimensional spruce budworm population model: where A = Dγ/β is related to the death of the matured population, B = q 1 γe −dτ /β and C = α 1 γ are related to birth and survival of the immature population, τ =τβ/γ is a time delay, and q 1 is the birth rate-related parameter, and the meaning of other parameters of model (3) is given in [11]. the authors in [13] have studied the dynamic behaviors of system (3) and obtained some sufficient conditions on the local stability of the positive equilibrium and Hopf bifurcation occurrence.
On the other hand, the autonomous systems (2) and (3) irrespective of the environmental changes have some limitations in mathematical modeling of ecological systems. Moreover, to the best of our knowledge, no study has been conducted to date for dynamics on the nonautonomous population model with stage structure for spruce budworm. Hence, based on the above models and analysis, in this paper, we study the following delayed nonautonomous population model with stage structure for spruce budworm: The interaction between the spruce budworm and the forests is one of the important themes in mathematical ecology due to the protection of spruce and balsam fir trees. In addition, the main problems in spruce budworm population models are the boundedness, permanence, extinction of the population, and the existence of the periodic solution and global attractivity of the system. Hence, in this paper, our main purpose is to establish some sufficient conditions on the above mentioned dynamical behaviors for systems (3) and (4).

Preliminaries
In system (4), yðtÞ denote the density of the spruce budworm population and τ is a time delay. In this study, for system (4), we introduce the following basic assumption: (H 1 ) τ > 0 and AðtÞ, BðtÞ, CðtÞ are all continuously positive ω-periodic functions on ½0, ω The following is the initial condition for system (4): where ϕðtÞ is nonnegative continuous functions defined on ½−τ, 0Þ and satisfying ϕð0Þ > 0. For a ω-periodic continuous function f ðtÞ defined on Now, we present some useful lemmas.

Boundedness, Extinction, and Periodic Solution
Theorem 4. Assume that the assumption (H 1 ) holds, then for any positive solution yðtÞ of system (4), there exists a constant M such that where Proof. From the equation of system (4) and for t > τ, we have Then applying the following inequality [16,18], we have By Lemma 1, we get Finally, there exists T 0 > 0 such that for t > T 0 . For system (3), we have the following result. Proof. From the equation of system (4) and for t > τ, we have Note the following equation: By Lemma 2, we derive By comparison, there exists T 3 > 0 such that yðtÞ ⟶ 0 for t ≥ T 3 .
For system (3), we have the following result.
Then species Y of autonomous system (3) is extinct, that is, Theorem 8. Suppose that assumption (H 1 ) holds, then system (4) has at least one positive ω-periodic solution.
Proof. Let yðtÞ = exp fuðtÞg, then system (4) can be rewritten as Let, X = Z = fu ∈ CðR, RÞ: uðt + ωÞ = uðtÞg be Banach spaces equipped with the norm k·k, where kuk = max t∈½0,ω juðtÞj. Thus, we have for any u ∈ X, it is easy to see that Γðu, ·Þ ∈ Cðℝ, ℝÞω-periodic. Let We easily see Ker L = ℝ, Im P = Ker L, Therefore, operator L is a Fredholm mapping of index zero. Furthermore, denoting by L −1 P : Im L ⟶ DðLÞ ∩ ker P the inverse of Lj DðLÞ∩Ker P , we have Thus, we have

Journal of Function Spaces
and ðI − QÞÑu ∈ ImL, for all u ∈ X.
As in [18], we can easily show that for any open bounded set Ω ∈ X,Ñ is L-compact on Ω. For the operator equation Suppose that u ∈ X is a solution of (28), for some λ ∈ ð0 , 1Þ, then there exist ξ, η ∈ ½0, ω such that It follows from (28) and (29) that From (30), we obtain By (11) and (32), we further have Next, from (31), we further obtain Let G > max fG 1 , jG 2 jg be a fixed constant and define Ω = fu ∈ X : kuk < Gg. Then (33) and (34) imply that there is no λ ∈ ð0, 1Þ and u ∈ ∂Ω such that Lu = λÑu.
If QÑðGÞ ≥ 0, from (11) and (26), it follows that which implies that a contradiction to the choice of G. Thus, QÑðGÞ < 0. Define continuous function Gðu, μÞ by setting It follows from (35) that G u, μ ð Þ≠ 0 for all u ∈ ∂Ω ∩ Ker L: ð41Þ Using the homotopy invariance theorem, we obtain It then follows from the continuation theorem (Lemma 3) that Lu =Ñu has a solution  Journal of Function Spaces