Positive Global Solutions for a General Model of Size-dependent Population Dynamics

We are concerned with size-structured population models of general type with the growth rate depending on the individual’s size and time. In [3], the local existence and uniqueness of the solution have been investigated. In this paper, we discuss the positivity and global existence of the solution as well as L∞ solutions. As is explained in [3], the model in our mind is the population dynamics of plants in forests or plantations. In this case, the growth rate may be influenced by the environment such as the light, temperature, and nutrients. These must change with time. It is also reasonable to think that the growth rate varies with the size because the size is important to capture the light to grow. From these points, it is natural to consider the growth rate depending on the size and time. From the mathematical point of view, our results are the generalizations of G. Webb’s results [5, Theorems 2.3, 2.4, 2.5, and 4.3] in the age-dependent case. Besides, we investigate L∞ solutions. Our results also have a close relation to the results of A. Calsina and J. Saldaña [1], where they treated a nonlinear growth rate depending on the size and the total population at each time, whereas the aging and birth functions have the special form of the Gurtin-MacCamy type (see below). For other related works, we refer to [4], where a finite number of structure variables are treated.