*** Does the system have large prey or small prey? Large predators or small pred

Question

*** Does the system have large prey or small prey? Large predators or small predators? Explain the reasoning behind your decision. Again, process of elimination is not a valid explanation for this problem. Please help in my Differential Equations class ***

2. Consider the following systems of rate of change equations: System A System E ab da=0.3x-100 dt 15y (1-3, 17 dt 1020ab In both of these systems, a, b, r and y refer to the number different species at time t. In particular, in one of these systems the prey are large animals and the predators are 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, such as whales and krill. For each system, determine the following:

Explanation / Answer

SOLUTION:

We can use intuitive approach

Consider system A. There has to be a predator. If we suppose that the predator is a, then we know that decrease in b (one more prey eaten) must lead to increase in a. At the same time, increase in a should make y drop much faster, since there are more predators now.

Let's see the da/dt equation.

da/dt = 3x(1-a/10)-20ab

We see that, da/dt is a function of a and b. It is hard to say something about dependency between da/dt and a, but dependency between da/dt and b is quite simple: the greater the b, the greater becomes 20ab (since a and b are positive numbers). And therefore -20ab is more negative, and da/dt gets more negative. Conclusion: bigger b,means da/dt smaller or more negative, means less a

Check: This means that bigger b (more prey) means less a(less predators). Now, does this confirm our prediction that a is predator? No, because more prey (b) should mean easier lunch for predators, and therefore, more predators

We can see that a is prey and b is predator, because as there are more predators(b), a falls.

To check this, we use second equation:

db/dt = -5b+ab/20

Again, for now, we only talk about simple dependency here: one between db/dt and a. Bigger a (more prey) means b (predator) raises more quickly.

We use the same reasoning for system B.

Again, a is prey and b is predator (because greater a means greater 25ab and this leads to greater increase in b population (eq.2), and because greater y means greater ab/100 which leads to decrease in a 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 da/dt (change in number of humans) not to be significantly different in cases when b = i and b = i+1 (i is arbitrary number of piranhas)

We can see that system B has this characteristic:

da/dt = 0.3a - ab/100

Why? Because factor standing next to b is 1/100, and one more piranha cannot significantly lower the value of da/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 da/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 db/dt in both systems instead of da/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: db/dt = -5b ...)

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