Reduction to Ordinary Differential Equations

Systems containing coupled partial differential equations and integro-differential equations, such as (2.1), present significant challenges to mathematical analysis. Much progress on these difficult equations is presented in [MD]. Following Cushing [Cu2] with only minor differences, we assume that (2.1) defines a unique solution S t) and p t,l) for / 0 and introduce the moment functions  

In [Cu2], the factors and / of A(t) and L t) are omitted. We introduce them to make for cleaner expressions. Ignoring a scaling factor, A(t) is the total surface area of the population, L t) is the total length of the population, and P t) is the total number of individuals in the population at time t. 

The immediate goal is to obtain differential equations for these new variables. Multiplying (2.1b) by (l/lb) and integrating from to infinity results in 

The net result is that we can trade (2.1) for the following system of ordinary differential equations  

In [Cu2] a is called the reproductive efficiency of the organism, since it is a ratio of the fraction of energy derived from uptake that is allocated to reproduction to the conversion factor relating food units to weight for reproduction (w/ is the amount of nutrient needed to produce one offspring). For similar reasons, 0 is called the growth efficiency of the organism. 

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