Homework Help: Why are my problems written as sub x and sub 0

1. Oct 24, 2009

hominid

Why are my problems written as "sub x" and "sub 0"

1. The problem statement, all variables and given/known data

I am doing population tables in math where x represents age. L sub x represents survivor ship at age x. m sub x represents fecundity at age x.

My question is, why is x always "sub x"? There is a an equation $$R_0 = \sum_{x_first}^{x_last} L_x m_x=population-replacement-rate$$

What I don't understand is how can $$R_0$$ be variable. I see that when $$R_0=1$$ there is no growth. And my professor said that this allows us to simply say that $$r=0$$=no population growth. But I am thoroughly confused

Last edited: Oct 24, 2009
2. Oct 24, 2009

LCKurtz

Re: Why are my problems written as "sub x" and "sub 0"

I don't know what the term "survivorship at age x" means, but I have a question for you anyway. What is the index of summation? Is x meant to be the index of summation? It is common to use subscripts in that situation. Is it something like

$$R_0 = \sum_{x=1}^{100} L_x m_x$$

If so, then $R_0$ isn't a variable; it is constant.

3. Oct 24, 2009

hominid

Re: Why are my problems written as "sub x" and "sub 0"

What I said isn't very clear, sorry. I am doing "life tables" on the rates of population changes over time within age groups. e.g., for whatever reason there may be 100K people alive at age 50, and 75K alive at age 49 given the circumstances with their parents at birth, or other factors that affected that age group during their lives that lowered their age group's population vs another. Survivorship at age x $$l_x$$ is the percentage of people still alive from a sample of people at their given age.

My problem looks just like that without the "100" and x=1. Instead, it has on top, x last, and on bottom, x first. Meaning x first is the age of first reproduction, and x last is the age of last reproduction.

$$R_0 = \sum_{x_first}^{x_last} l_x m_x$$

What I don't understand is how could $$R_0$$ be a constant if the value for $$R_0$$ changes depending on the growth rate. That is, if the sum of $$l_x$$ and $$m_x$$ equal one then there is no growth, but if their sum equals 2.5 then we could say that each individual leaves an average of 2.5 offspring, thus population is growing.

Last edited: Oct 24, 2009
4. Oct 25, 2009

HallsofIvy

Re: Why are my problems written as "sub x" and "sub 0"

So R0= lxfirstmxfirst+ lxfirst+1mxfirst+1+ lxfirst+2mxfirst+ 2+ ... + lxlastmxlast[/math]. R0 is not necessarily a constant but it does NOT depend upon a specific vallue of "x" since it is a sum over all values of x.