Sink-limitation and the size-number trade-off of organs: production of organs using a fixed amount of reserves

To analyze the nature of size-number trade-off of organs, we develop models in which the effects of sink-limitation in the growth of organs and the loss of resources by maintenance respiration are taken into consideration. In these models, the resource absorption rate of an organ is proportional to either its absolute size or its surface area and either the initial size of an organ or the total initial size of the organs produced is fixed. In all models, organs are produced using a fixed amount of reserved resources and no additional resources become newly available for their growth. We theoretically show that size-number trade-offs are nonlinear if the resource absorption rate of an organ is proportional to the absolute size of the organ and the initial size of the individual organs is fixed or if the resource absorption rate of an organ is proportional to the surface area of the organ. In these nonlinear size-number trade-offs, the size of individual organs increases less rapidly than in linear trade-offs with a decrease in the number of organs and the total size of organs is an increasing function of the number of organs produced. This implies that increasing the number of organs produced is advantageous in terms of resource-use efficiency. In contrast, size-number trade-off is linear if the resource absorption rate of an organ is proportional to the absolute size of the organ and there is a linear trade-off between the initial size of organs and their number. To exemplify the effects of those size-number trade-offs on the life-history evolution, we calculate the optimal offspring sizes that maximize the number of offspring successfully being established. In the case of nonlinear size-number trade-offs, the optimal offspring sizes are smaller than the optimal offspring size in the case of linear size-number trade-offs, namely, that in the model of Smith and Fretwell (1974). Our optimal offspring size depends on the metabolism of organ development; the optimal offspring size decreases with an increase in maintenance respiration rate relative to the growth coefficient of organs.