Abstract
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.
Highlights
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]
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
Which is an ω-periodic solution to equation (22)
Summary
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]. 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. In [11], the authors further analyzed system (1) and proposed the following delayed spruce budworm population model: dmðtÞ = −DmðtÞ − pðmðtÞÞmðtÞ + e−d~~τbðmðt − ~τÞÞ, ð2Þ dt 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. Based on the above models and analysis, in this paper, we study the following delayed nonautonomous population model with stage structure for spruce budworm: y_ðtÞ =. In this paper, our main purpose is to establish some sufficient conditions on the above mentioned dynamical behaviors for systems (3) and (4)
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