Introduction -
Systems of differential equations can be used to model many mechanical systems. Some researchers have also tried to use these same methods to examine other less mechanistic systems, such as modeling population growth and interactions in biology. A little contemplation will probably convince you that this is realty one of those Hard Problems that can give scientists fits: There are so many different variables involved, many of which are difficult to measure (assuming that we even know what all of them are), that finding an accurate model for population growth seems moderately hopeless. This doesn't mean that attempting to model these systems is a waste of time: we can start with fairly simple models and then work our way up to more comprehensive ones. Along the way, we can capture some basic properties of population dynamics by examining these models. This lab will take you through some basic ideas to model population using some very simple equations and systems using several "species" from historical fiction.
Night of the living zombies - An exploration of the Predator-Prey system
Questions:
Explain why x y makes sense as an "interaction term". Why might this make more sense than x+ y or just x or y? (There could. of course, be far more complicated models for this, like x2y or y√x, but this one is a good simple first approximation.)
From the general form of these equations (see above), find all equilibrium points in terms of the parameters a. b. p, q.
For each of those equilibrium points, find the eigenvalues (again, in terms of a, b, p, q) for the local linearization (Jacobian) at those points. Clearly state what those eigenvalues tell you about the solutions near those points. If that behavior depends on the value of the parameters, show this clearly. (You may assume that a. b. p. q > 0. so you don't need to consider negative values of these or equilibrium points outside of the first quadrant and positive axes.)
In particular, what sort of long-term coexistence is possible far the Montagues and the zombies? How will their populations relate to each other over time?
Use the Phase Field widget at the end of this lab to examine the Impact of the values of the parameters on the shapes of the solution curves. In particular, briefly explain the impact of the following situations on the solutions and why those impacts do or do not make sense physically: (For simplicity. stick with all parameters between 0 and 1.)
- Montague growth rate Increases?
- Zombie decay rate increases?
- Interactions become more lethal to the Montagues?
- Interactions become more beneficial to the zombies?
The nightmare is over... or is it?
Questions:
Explain why x2 makes sense as a "limiting term" for the population. Why might this make more sense than x by itself? (There could. of course, be far more complicated models for this, but this one is a good simple first approximation.)
Find the equilibrium points for this system. There are 3 of these that always exist, regardless of the parameter values (assuming all parameter values are >0) plus one that sometimes exists, depending on the parameters. Find a general formula for the 3 that always exists, in terms of the parameters and briefly explain which physical parameters impact the location of each one. Explain geometrically what determines whether there is a fourth equilibrium point. (I suggest you turn on the "nuliclines" option in the Phase Field widget and look at where the green and orange nuliclines intersect as you vary the parameters.) If you were assigned to use mathematics commands to do your lab, find a formula for that fourth equilibrium point and explain what is required mathematically for it to exist.
Find general formulas for the eigenvalues for the local linearization at the first 3 equilibrium points (the ones that always exist). For each of them, clearly snow (with examples) what this tells you about the behavior of the system at those points. (Hint: For some of them. that behavior will change, depending on the values of the parameters. so pay attention to when each one is positive. negative. or complex.)
The fourth equilibrium point might be considered the most socially "hopeful" solution. (Why?) You can totally analyze this system algebraically (especially if you know how to use Mathematica to do the grunt-work), but honestly, the best way to see what's going on here is to examine how the eigenvalues change at the equilibrium points as you move the nullclines. The nullclines that really matter here are the ones that are out in the 1st quadrant. Consider all the different ways these two nullclines could be related (i.e., intersecting, not intersecting, X nullclines steeper. Y nullclines steeper, and combinations of these) and, for each case, show an example and clearly classify the eigenvalues of the local linearization at each equilibrium point. (Hint: you can move the nullclines without changing their slope by varying a and p.) For each of these configurations clearly explain what the long-term behavior of the system would be.
Given your work in the previous question, under what conditions can the Montagues and the Capulets live in peaceful coexistence? (You do not have to give restrictions on the variables, but rather explain what must be true about the nullclines for this to happen.)
It has been suggested that the Montagues and the Capulets can live together in harmony if bq>cr (assuming there is a fourth equilibrium point and it is in the first quadrant). Does this agree with your conclusion from above? (Show why or why not...) Explain what this inequality implies about the physical situation. (Hint: Try to figure out the meaning of each side in terms of what the different parameters do.)
Attachment:- DCQ Lab.rar