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

<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>