Transformation system with time-dependent characteristics and population theory

Abstract

The theory of transformation systems can be applied to population models if allowance is made for certain particular relations which arise because the time is no longer merely an independent reference variable but also a defining property of the classes when the age of the elements has to be taken into account. It is easy to convert the differential formalism used in the theory of transformation systems [1] to the recurrence or discrete formalism often used in population theory in the form of Leslie matrices. We show that the use of this recurrence formalism may lead to artifacts of calculation because of the implicit approximations involved. It follows that in population theory the choice beween differential and recurrence formalisms is not a minor or irrelevant question; in practice, the former have a much wider range of validity than the latter. The question of stability properties in the differential or discrete formalisms, respectively, has often been put forward, for example by R.M. May [2] and P. Van den Driessche [3]. Here we emphasize particularly another aspect of the correspondence between the two formalisms, namely the partition of the objects studied into various equivalence classes. This problem of the equivalence classes needed to represent the properties of a sysem is treated, using the criteria regarding regrouping of classes established elsewhere (see Appendix II). It is shown that the type of reproduction of the organisms considered must be explicitly taken into account in population models.