Consider the following systems of rate of change equations: In both of these sys
ID: 3109889 • Letter: C
Question
Consider the following systems of rate of change equations: In both of these systems, x and y refer to the number of two different species at time t. In particular, in one of these systems the prey are large animals and the predators arc small animals, such as piranhas and humans. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey. a) For both systems of differential equations, what does x represent? The predator or the prey? Explain. b) Which system represents the large prey and small predator? Explain. c) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. d) Use your results from part c) to sketch in the long-term behavior of solutions with initial conditions anywhere in the first quadrant of the phase plane. For example, describe the long-term behavior of solutions if the initial condition is in such and such region of the first quadrant. Provide a sketch of your analysis in the x-y plane and write a paragraph summarizing your conclusions and any conjectures that you have about the long-term outcome for the two populations depending on the initial conditions.Explanation / Answer
We can use intuitive approach
Consider system A. There has to be a predator. If we suppose that the predator is x, then we know that decrease in y (one more prey eaten) must lead to increase in x. At the same time, increase in x should make y drop much faster, since there are more predators now.
Let's see the dx/dt equation.
dx/dt = 3x(1-x/10)-20xy
We see that, dx/dt is a function of x and y. It is hard to say something about dependency between dx/dt and x, but dependency between dx/dt and y is quite simple: the greater the y, the greater becomes 20xy (since x and y are positive numbers). And therefore -20xy is more negative, and dx/dt gets more negative. Conclusion: bigger y,means dx/dt smaller or more negative, means less x
Check: This means that bigger y (more prey) means less x(less predators). Now, does this confirm our prediction that x is predator? No, because more prey (y) should mean easier lunch for predators, and therefore, more predators
We can see that x is prey and y is predator, because as there are more predators(y), x falls.
To check this, we use second equation:
dy/dt = -5y+xy/20
Again, for now, we only talk about simple dependency here: one between dy/dt and x. Bigger x (more prey) means y (predator) raises more quickly.
We use the same reasoning for system B.
Again, x is prey and y is predator (because greater x means greater 25xy and this leads to greater increase in y population (eq.2), and because greater y means greater xy/100 which leads to decrease in x population)
After that, we want to know which system has small predators and which has large ones.
To conclude something about that we need to know that in case of piranhas, one piranha more or less doesn't affect the human population very much.
So we would expect dx/dt (change in number of humans) not to be significantly different in cases when y = i and y = i+1 (i is arbitrary number of piranhas)
We can see that system B has this characteristic:
dx/dt = 0.3x - xy/100
Why? Because factor standing next to y is 1/100, and one more piranha cannot significantly lower the value of dx/dt and therefore cannot contribute much to lowering of human population.
In system A, we can see that predator is big one, since one predator more or less means a great change in dx/dt.
I hope you managed to follow this text. And i also hope that i didnt make any mistake in the process :)
You should try to understand this, not to learn it by heart because there is benefit if you truly understand the task.
Other methods of solving task could be...maybe to try to consider dy/dt in both systems instead of dx/dt, and then see which prey is small and which is big. Big predator should mean that there cannot be too many predators in one place or they will eat each others meal (system A has this characteristic: dy/dy = -5y ...)