Well-posedness and asynchronous exponential growth of an age-weight structured fish population model with nonautonomous past

We study the fish population model integrating age and weight structures, introduced in E. Sánchez, M. L. Hbid, R. Bravo de la Parra. (J. Evol. Equ. 14:603–616, 2014). We reformulate the model in the nonautonomous past setting, and then as a boundary perturbation problem with unbounded operators in the boundary. Using semigroup theory of linear operators in Banach spaces, and via the theory of time-invariant regular system with feedback, we prove the existence and uniqueness of a classical solution with a form of a variation of parameters formula. We give an explicit criterion of the uniform exponential-stability by determining the characteristic equation of the model. We also prove the asynchronous exponential growth behaviour for the system unconditionally with the system parameters, and we give the associated projection in an explicit form. The results in this paper represent an improvement of the ones given in loc. cit. for the well-posedness and asymptotic behaviour.