Dynamics of a stage-structured single population model with state-dependent delay

Abstract

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−τ′(z(t))z˙(t) $1-\tau'(z(t))\dot{z}(t)$, 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.

Keywords

• Stage structure
• State-dependent delay
• Correction term
• Stability
• Permanence