Exponential Extinction of Nicholson’s Blowflies System with Nonlinear Density-Dependent Mortality Terms

and Applied Analysis 3 aij , bij , cik, γik : R → 0, ∞ are all continuous functions bounded above and below by positive constants, and τik t ≥ 0 are bounded continuous functions, ri max1≤j≤l{supt∈Rτij t } > 0, and i, j 1, 2, . . . , n, k 1, 2, . . . , l. Furthermore, in the case Dij t,N aij t − bij t e−N , to guarantee the meaning of mortality terms we assume that aij t > bij t for t ∈ R and i, j 1, 2, . . . , n. The main purpose of this paper is to establish the conditions ensuring the exponential extinction of system 1.5 . For convenience, we introduce some notations. Throughout this paper, given a bounded continuous function g defined on R, let g and g− be defined as g− inf t∈R g t , g sup t∈R g t . 1.7 Let R R be the set of all nonnegative real vectors, we will use x x1, . . . , xn T ∈ R to denote a column vector, in which the symbol T denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| |x1|, . . . , |xn| T and define ||x|| max1≤i≤n|xi|. Denote C ∏n i 1C −ri, 0 , R and C ∏n i 1C −ri, 0 , R as Banach spaces equipped with the supremum norm defined by ||φ|| sup−ri≤t≤0max1≤i≤n|φi t | for all φ t φ1 t , . . . , φn t T ∈ C or ∈ C . If xi t is defined on t0 − ri, ν with t0, ν ∈ R and i 1, . . . , n, then we define xt ∈ C as xt x1 t , . . . x t T where x t θ xi t θ for all θ ∈ −ri, 0 and i 1, . . . , n. The initial conditions associated with system 1.5 are of the form: Nt0 φ, φ ( φ1, . . . , φn )T ∈ C , φi 0 > 0, i 1, . . . , n. 1.8 We writeNt t0, φ N t; t0, φ for a solution of the initial value problem 1.5 and 1.8 . Also, let t0, η φ be the maximal right-interval of existence of Nt t0, φ . Definition 1.1. The system 1.5 with initial conditions 1.8 is said to be exponentially extinct, if there are positive constants M and κ such that |Ni t; t0, φ | ≤ Me−κ t−t0 , i 1, 2 . . . , n. Denote it as Ni t; t0, φ O e−κ t−t0 , i 1, 2, . . . , n. The remaining part of this paper is organized as follows. In Sections 2 and 3, we shall derive some sufficient conditions for checking the extinction of system 1.5 . In Section 4, we shall give two examples and numerical simulations to illustrate our results obtained in the previous sections. 2. Extinction of Nicholson’s Blowflies System with Dij t,N aij t N/ bij t N i, j 1, 2, . . . , n Theorem 2.1. Suppose that there exists positive constant K1 such that aii b ii K1 > n ∑ j 1,j / i a ij b− ij l ∑ j 1 c ij γ− ij eK1 , i 1, 2, . . . , n. 2.1 Let E1 { φ | φ ∈ C , φ 0 > 0, 0 ≤ φi t < K1, ∀t ∈ −ri, 0 , i 1, 2, . . . , n } . 2.2 4 Abstract and Applied Analysis Moreover, assume N t; t0, φ is the solution of 1.5 with φ ∈ E and Dij t,N aij t N/ bij t N i, j 1, 2, . . . ,n . Then, 0 ≤ Ni ( t; t0, φ ) < K1, ∀t ∈ [ t0, η ( φ )) , i 1, 2, . . . , n, η ( φ ) ∞. 2.3 Proof. Set N t N t; t0, φ for all t ∈ t0, η φ . In view of φ ∈ C , using Theorem 5.2.1 in 17, p. 81 , we have Nt t0, φ ∈ C for all t ∈ t0, η φ . Assume, by way of contradiction, that 2.3 does not hold. Then, there exist t1 ∈ t0, η φ and i ∈ {1, 2, . . . , n} such that Ni t1 K1, 0 ≤ Nj t < K1 ∀t ∈ [ t0 − rj , t1 ) , j 1, 2, . . . , n. 2.4 Calculating the derivative of Ni t , together with 2.1 and the fact that supu≥0ue −u 1/e and a t N/ b t N ≤ a t N/b t for all t ∈ R,N ≥ 0, 1.5 and 2.4 imply that 0 ≤ N ′ i t1 −Dii t1,Ni t1 n ∑ j 1,j / i Dij ( t1,Nj t1 )