# Numerical Methods The logistic

1. Numerical Methods The logistic model for population growth of a single species assumes no natural predators, but limited resources (say rabbits in a field, or perhaps humans on a planet).  Under such circumstances, the population might be modeled by a single differential equation.

dP/dt=aP-bP^2 = P(a-bP) = aP(1-P/K),   (K= a/b)   P(0)=P0

The parameter a describes the reproductive rate and b takes into account the stress that limited resources places on the population.  Let a = 0.7 when time is measured in months.  Let b = 0.007.  The following exercises are about using Euler’s method by hand, without the use of a computer to automate things.  You can (should) use a calculator or computer to do the calculations, but you should convince yourself that you can do each step individually.  For any exercise asking for eight or more Euler steps, you should use Excel (or any other computational tool) to automate the process.  Include the spreadsheet

• Can there be an equilibrium population (where P(t) is not changing)? If so, what is it?
• Suppose P(0) = 10. Perform four iterations of Euler’s method on paper, using a time step of 1.0.  What is the population after those four months?
• For the same initial condition, perform eight iterations of Euler’s method, using a time-step of 0.50. You will still be solving out to four months.  You should do this in Excel or some other spreadsheet.  How does your result differ from that of part (b)?

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