A spatially and size-structured population model with unbounded birth process

Abstract

<p style='text-indent:20px;'>In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.</p>

Controllability under positivity constraints of a size‐structured population model with delayed birth process

“…In the realm of structured population equations, linear semigroup methods have proven to be effective tools for examining the linear stability and bifurcation phenomena of solutions (see, e.g., [9,11,16]). In particular, the well-posedness and asymptotic behavior of such models have been investigated in several works [9,[16][17][18][19]. Age-dependent models with delay in the birth process were investigated in [18].…”

“…In the latter, the author applied Perron‐Frobenius techniques introduced in [20] and the theory of positive semigroups to establish stability criteria. Recently, using the feedback theory of $$$ {L}^p $$$‐well‐posed and regular linear systems, the authors in [9, 19] established the well‐posedness and stability results of structured population systems with unbounded birth process. Unlike the asymptotic behavior, which is well understood, there are no works addressing the controllability properties of age/size‐structured population models with a delayed birth process, to the best of our knowledge.…”

Controllability under positivity constraints of a size-structured population model with delayed birth process

“…In the studies of structured population equations, linear semigroup methods have been successfully developed to investigate the linear stability and bifurcation phenomena of solutions (see, e.g., 13,8,9 ). In particular, the well-posedness and asymptotic behavior of such models have been investigated in several works 14,15,13,8,16 . Age-dependent models with delay in the birth process were investigated in 15 .…”

“…In the latter, the author applied Perron-Frobenius techniques introduced in 17 and the theory of positive semigroups to establish stability criteria. Recently, using the feedback theory of 𝐿 𝑝 -well-posed and regular linear systems, the authors in 16,8 established the well-posedness and stability results of structured population systems with unbounded birth process. Unlike the asymptotic behavior, which is well understood, there are no works addressing the controllability properties of age/size-structured population models with a delayed birth process, to the best of our knowledge.…”