Abstract
We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by \begin{document} $C_0-{\rm{semigroup}}$ \end{document} framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.
Highlights
In this paper we study the following system of a hierarchical size structured population model ∂u(s, t) + (γ(s)u(s, t)) = −μ(s)u(s, t), s ∈ [0, m], t > 0, ∂t ∂s m0 u(0, t) =β(s, σ, Q(s, t + σ))u(s, t + σ)dσds, t > 0,0 −τ u(s, t) = u0(s, t), s ∈ [0, m], t ∈ [−τ, 0], (1.1)where the function u = u(s, t) denotes the density of individuals of size s ∈ [0, m] at time t ∈ [0, ∞)
The function γ = γ(s) represents the growth rate of an individual of size s, while μ = μ(s) is the mortality rate of an individual of size s. β ∈ C([0, m] × [−τ, 0) × C1[−τ, ∞)) should be understood as the probability that an individual of size s reproduces after a time lag −τ starting from conception, here τ > 0 is a constant denoting the maximal time delay the birth process takes
Such a form of the delayed birth process was initialed by Pizzera [30] by taking into account the time which the birth process may take, and it has been adopted in many papers, such as in papers [18, 19, 37, 38], to discuss the long-time behavior of the corresponding nonlinear systems and explore how the time lag influences the asymptotic behaviors of the considered systems
Summary
In this paper we study the following system of a hierarchical size structured population model. Β ∈ C([0, m] × [−τ, 0) × C1[−τ, ∞)) should be understood as the probability that an individual of size s reproduces after a time lag −τ starting from conception, here τ > 0 is a constant denoting the maximal time delay the birth process takes Such a form of the delayed birth process was initialed by Pizzera [30] by taking into account the time which the birth process may take, and it has been adopted in many papers, such as in papers [18, 19, 37, 38], to discuss the long-time behavior of the corresponding nonlinear systems and explore how the time lag influences the asymptotic behaviors of the considered systems.
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