In this lab we will investigate the use of direction fields to evaluate solutions to first order differential equations.
For this lab, we will study these qualitative analysis tools via the example of population growth using the Gompertz equation as a model. This equation is used to model the growth of tumor cells, and gives an expression of the dependence of tumor size on time.
The differential equation governing the growth is:
dV/dt=βV ln(k/V)
Here β is the growth rate k is a reflection of the carrying capacity of the tumor.
Let β=2; k=100; and V range from 0 to 100
Plot the graph of dV/dt vs. V. Note you may have to use the vectorization operator “.” to evaluate the function of the differential equation.
Determine the equilibrium points (by-hand or MATLAB) and plot them on the same graph as part (1) using red circles. Use a legend to indicate the equilibrium points and the original plot.
Using the gradient() function, plot the derivative of dV/dt versus V on the same plot. That is, plot d^2 V/dt vs. V.
Determine the stability of each equilibrium point and the region of attraction. Write these answers as comments in your code. Comment on the stability as it related to the graph.
From this graph, are we able to tell how long the tumor will take to grow to V=80 units if the initial size is V(0)=10 units? Why? Write your answers as comments in the code.
We will now use the direction field tool to examine the solution.
Download the file from the course website in the MATLAB Resources folder. You may have to right-click and save the file to your computer. IMPORTANT! Make sure it is in the current directory you are working in.
OR
Access the dfield tools at http://math.rice.edu/~dfield/dfpp.html
Using the same parameters as above, plot the direction field of this differential equation. Set the time and variable limits appropriately.
Compare your results from part (4) to the direction field obtained to check: the stability of each equilibrium point and the region of attraction. Include these as comments in your code.
Estimate how long it will take for V=80 if V(0)=10. Hint: You can go to: WINDOW DFIELD Keyboard Input to enter the initial conditions.
Repeat the direction field plot with B=5 and estimate again how long it takes for V=80 if V(0)=10. Comment on your results as they compare to part (9).
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