Malthusian growth parameter is negative

Question

I'm trying to create python function that calculates the growth factor (R) according to the Malthusian growth model $$ X_{t+1} = R X_t $$

The table I have is table with two countries population when each country has different growing factor that is given: $$ \begin{array}{cc} R_1=0.962\\ R_2=0.997 \end{array} $$ and both have the same initial population: 100M

I want to build function that calculates R between specific periods(years and months) based on the population size of each contries every year.

My population table looks like this:

>>> year    pop1        pop2
0   0   1.000000e+08    1.000000e+08
1   1   9.620000e+07    9.970000e+07
2   2   9.254440e+07    9.940090e+07
3   3   8.902771e+07    9.910270e+07
4   4   8.564466e+07    9.880539e+07

And this is my function:

def find_r(df,init_pop,period,time_unit):
    res=[]
    
    if time_unit=='years':
        pop1=df.loc[df['year']==period].iloc[:,1]
        pop2=df.loc[df['year']==period].iloc[:,2]
        R1=pop1/init_pop
        res.append(R1)
        R2=pop2/init_pop
        res.append(R2)
    
    elif time_unit=='months':
        
        if 0<period<=12:
            
            pop1=((((df.loc[data['year']==1].iloc[:,1]).values)-((data.loc[data['year']==0].iloc[:,1]).values))/12)*period
            pop2=((((data.loc[data['year']==1].iloc[:,2]).values)-((data.loc[data['year']==0].iloc[:,2]).values))/12)*period
            R1=pop1/init_pop
            res.append(R1)
            R2=pop2/init_pop
            res.append(R2)
        else:
            print('to be continued')
            
    
    
    return(pd.DataFrame(res))

The problem: When I check for eyars for example and put year 1, seems like it works correct and I get the correct R (0.962 aand 0.997), but when I run it for months (that are less than 1 year) I get negative results and that suprised me because I thought that if population shrinking I'll get less than 1 and for growing population I'll get greater than 1 but not negativ values.

For that reason I'm afriad the way I calculate it is wrong.

elaboratio nregard my calculation for months: I reduce the inital population from the 1 year population. divide by 12 and then multiply by number of months -> to get the number of people after certain month

Then I take this number and divide it by the initial pop to get the R.

I have also tried to add the initial pop to pop1 calculation:

pop1=((((df.loc[data['year']==1].iloc[:,1]).values)-((data.loc[data['year']==0].iloc[:,1]).values))/12)+init_pop*period
            pop2=((((data.loc[data['year']==1].iloc[:,2]).values)-((data.loc[data['year']==0].iloc[:,2]).values))/12)+init_pop*period

but then I have gotten numbers like 6.55 ect. so I belive is also wrong,considering the trend of population shrinking.

Where am I wrong? is it the function that is bad or my logic?

Answer 1

The problem occurs because you subtract the wrong values.

What you currently do is $$ pop_1 = \left[pop(year=1)-pop(year=0)\right] \frac{period}{12} \\ R_1 = \frac{pop_1}{init\_pop} $$

What you need to do is $$ R_1 = \frac{pop(year=1)}{pop(year=0)}\frac{period}{12} $$