Predator-Prey

Reuel · 2011-02-15 11:23:50

I am trying to understand how to 'read" a set of equations and it isn't really making sense. Here is an example of the sort of problem I am trying to learn about:

"In one of these systems, the prey are animals very large in size and the predators are very small animals. 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. Determine which system is which and provide a justification for your answer."

Systems:

and

Apparently we are not only concerned with the evolution of the predators and the prey within some habitat but somehow it is supposed to depend on the size of the two? The book gives elephants and mosquitoes as an example of large and small predators and prey... but the idea of elephants eating mosquitoes is silly. That, however, is the objective.

How is this determined?

Bob · 2011-02-16 07:15:38

hi Reuel,

This has me completely puzzled. LATER EDIT. GRADUALLY GOT THIS SORTED SO READ MY OTHER POSTS.

x and y should represent the prey and predator (not necessarily that way round) and the differential equations tell you by how much the populations go up or down as time passes.

You don't say which is which out of x and y so I've tried some values both ways round (eg x = 1000 prey and y = 1 predator) and both populations always drop drastically. But the predators should increase in number if there's plenty of food! I cannot find start values using x as predator either no matter what values I start with.

So I wondered if I was wrong about the modelling and had a look on Wiki.

http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation

You will see that the idea is correct but the equations you have posted don't match this model.

So I am at a bit of a loss as to any advice I can give you apart from to check you have the right equations.

What should be happening is something like this.

Lots of prey and few predators => d(prey)/dt is negative and d(predator)/dt is positive. (ie. prey numbers drop and predators become more numerous because there is plenty to eat)

Few prey and lots of predators => d(prey)/dt is negative and d(predator)/dt is uncertain. (ie. predators starve because of lack of food so prey numbers can then recover)

Post again if you can give any more details of the problem .... where it came from ...... any worked examples .....

Bob

Last edited by Bob (2011-02-16 09:17:02)

Bob · 2011-02-16 07:54:37

hi again,

Still puzzling away. I've found some values that make sense for the second pair. See snapshot from Excel below.

All I've done is put the formulas into Excel and experiment with some start values.

Still cannot achieve the same for the first set of equations.

Bob

Bob · 2011-02-16 08:34:14

Oh hi, it's me again!

I think I'm getting to grips with this.

Take the second set of equations. and assume that x represents the prey and y the predators.

I have found that when x = 8 and y = 100, dx/dt = -5.6 whilst dy/dt = + 11500.

Remember these are the gradients on the graph so this is telling us that x goes down a small amount whilst y goes up a lot.

So try x = 7 and y = 110 and we get -5.6 and + 8800. So there is some consistency there.

I think this is the system where the prey is large and the predators are small. A single death of a prey animal can support a large number of small prey; let's say a dead elephant and lots of insects that feed off the body.

So that suggests the other set is for a large predator and small prey; say, an anteater and some ants.

So we need to find a larger value for x and smaller value for y that leads to sensible changes over time. I have not found these values yet.

Bob

Bob · 2011-02-16 09:15:54

Yes, that seems to be it.

x = 400, y = 6 is about right.

Bob

Reuel · 2011-02-17 08:04:21

Hey bob,

I saw your replies earlier and wasn't sure what to say so I went and asked my instructor about these types of problems. What he said is something along these lines:

In an ecosystem we care about the evolutions of the two species in question as the species interact with one another and in any predator-prey interaction, the predators will benefit while the prey will not. The two systems, given above, give two different scenarios. By looking and what each derivative concerns itself with and how the derivative will change with respect to the other variable, one may deduce which variable is which and then to the size of the predators as well as the prey.

For example, in the first system,

we can see that x with respect to time is going to be harmed greatly by an increase in xy interactions (the -20xy represents x and y interactions) because it is negative. The key here is looking to the portion of the derivative that has to do with both terms x and y. Therefore, the more that x and y interact the more it hurts x and x is therefore the prey because whenever a predator and its prey interact, within this scenario, we can assume that the predator will eat the prey.

Likewise, y is therefore going to be the predator which can be seen by the positive xy/20 where every time the two interact the predator population increases. We can also tell by this system that the prey must be very small for when they increase in number the predators benefit only by factors of 1/20 while, for the prey, the increase of predators affects them by a factor of 20 and so the predators must be large animals, such as wolves and field mice or something.

As for the second system,

It is now obvious that the predators must still be y for as y increases it hurts the change in x more and more. However, now the interaction of the two in the derivative of x is divided by 100 and so the predators must be very small in this system. In the prey system, dy/dt, the factor of 25 benefits the predator population. Ergo, the predators must be very small and the prey very large.

You may furthermore see, too, that if the predators were to go extinct, the prey would approach a carrying capacity of 15 units. The same can be said for if the prey go extinct, the predators will then diminish in count until their number is zero.

Bob · 2011-02-17 10:32:07

hi Reuel

Thanks for clearing that up.

Sorry it took me a while to begin to come to an answer on this; it's years since I met one of these.

But I claim some success.

In post #4 I wrote:

Take the second set of equations. assume that x represents the prey and y the predators.

then

I think this is the system where the prey is large and the predators are small. A single death of a prey animal can support a large number of small prey; let's say a dead elephant and lots of insects that feed off the body.

and finally

So that suggests the other set is for a large predator and small prey; say, an anteater and some ants.

Bob

Last edited by Bob (2011-02-17 10:32:38)

Reuel · 2011-02-17 11:15:16

No way, man. You did a great job. I really appreciate all of your effort. This is a great place to come get help with textbook problems. Thanks so much for your help.

Bob · 2011-02-17 20:41:49

hi Reuel,

You're welcome.

Bob

Math Is Fun Forum

#1 2011-02-15 11:23:50

Predator-Prey

#2 2011-02-16 07:15:38

Re: Predator-Prey

#3 2011-02-16 07:54:37

Re: Predator-Prey

#4 2011-02-16 08:34:14

Re: Predator-Prey

#5 2011-02-16 09:15:54

Re: Predator-Prey

#6 2011-02-17 08:04:21

Re: Predator-Prey

#7 2011-02-17 10:32:07

Re: Predator-Prey

#8 2011-02-17 11:15:16

Re: Predator-Prey

#9 2011-02-17 20:41:49

Re: Predator-Prey

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