Abstract
In this paper, a novel stage-structured single population model with state-dependent maturity delay is formulated and analyzed. The delay is related to the size of population and taken as a non-decreasing differentiable bounded function. The model is quite different from previous state-dependent delay models in the sense that a correction term, 1-tau'(z(t))dot{z}(t), is included in the maturity rate. Firstly, positivity and boundedness of solutions are proved without additional conditions. Secondly, existence of all equilibria and uniqueness of a positive equilibrium are discussed. Thirdly, local stabilities of the equilibria are obtained. Finally, permanence of the system is analyzed, and explicit bounds for the eventual behaviors of the immature and mature populations are established.
Highlights
In a natural ecosystem, the individual members of the population have a life history that takes them through two or more stages, especially, with regard to mammalian populations, which usually exhibit two distinct stages: immature and mature stages [1,2,3,4]
4 Stability of equilibria we study the linearized stability of the two equilibria E0 and E∗ by linearizing system (2.2)
With the state-dependent maturity delay, the changes in the number of mature individuals depend on reproduction and death and the changing definition of maturity, which is in line with the correction term 1 – τ (z(t))z(t)
Summary
The individual members of the population have a life history that takes them through two or more stages, especially, with regard to mammalian populations, which usually exhibit two distinct stages: immature and mature stages [1,2,3,4]. In 1990, the authors of [14] developed and analyzed the following stage structure model of population growth with a constant maturity time delay:. Where the state-dependent time delay τ (z(t)) is taken to be an increasing differentiable bounded function of the total population z(t) = x(t) + y(t). (A1) Parameters α, γ , β are all positive constants; (A2) The state-dependent maturity time delay τ (z) is an increasing differentiable bounded function of the total population z = x + y, where τ (z) ≥ 0, τ (z) ≤ 0, and 0 < τm ≤ τ (z) ≤ τM with τ (0) = τm and τ (+∞) = τM.
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