# Question on Lotka-Volterra model and Lanchaster combat model

1. Aug 13, 2007

### murshid_islam

i have 2 questions

1. what does a, b, m, n represent in the following Lotka-Volterra model:

$$\frac{dx}{dt} = (a - by)x$$

$$\frac{dy}{dt} = (-m + nx)y$$

here,
x(t) = number of krills at time t
y(t) = number of whales at time t

2. what does a, b represent in the following Lanchaster combat model:

$$\frac{dx}{dt} = -ay$$
$$\frac{dy}{dt} = -bx$$

here,
x(t) = the number of tanks in operation in time t
y(t) = the number of anti-tanks in operation in time t

2. Aug 13, 2007

### HallsofIvy

Staff Emeritus
This looks an awful lot like homework to me! Any reason I shouldn't move it?

Well, what do you think? Suppose all except a were 0. What would be the result? What if all except b were 0? c? d?
What effect on x and y would increasing or decreasing each of a, b, c, d while leaving the others alone?

Same idea as above.

3. Aug 14, 2007

### murshid_islam

well, it's not homework.

if all except a were 0, then the number of whales is not increasing or decreasing. and the number of krills is increasing and that increment is proportional to the number of krills at time t.

don't have any idea about that.

4. Aug 14, 2007

### arildno

Let's take the Lanchester combat model as our starting point.

Both tanks and anti-tanks begin at some population level, and neither of those populations can reproduce.

Furthermore, we assume that the only way a tank (or anti-tank!) can die/be removed from the population is by being hit by an anti-tank (or tank).
We could have said that tanks and anti-tanks could die due to non-combative effects like rusting, but we don't in this model.

Now that we have this clear, it should be evident to you that the more anti-tanks you have, the more tanks will die off, since they are bombarded by the anti-tanks all the time.
Now, the simplest way to model this mathematically is to say that the rate of decrease of the tanks is PROPORTIONAL to the number of anti-tanks present.

That is, the ratio between the rate of tank decrease and the anti-tank population equals a constant, in this case "a".

What can the magnitude of "a" mean?
Clearly, if anti-tanks are bad at actually hitting a tank, or not very effective when they DO hit, then that means the tank population is not so adversely affected by the bombardment than if the anti-tanks had been very dangerous and effective.

Thus, "a" is a measure of the effectivity of the anti-tank of destroying a tank.
The larger "a" is, the more dangerous is the anti-tank for the tank population.

Get it?

5. Aug 14, 2007

### murshid_islam

thanks a lot arildno.
i will try to apply similar ideas to the lotka-volterra model too and see how far i can get and return tomorrow if i have any problems.

6. Aug 14, 2007

### HallsofIvy

Staff Emeritus
Although you didn't say it, I will assume that a, b, m, and n are positive. My point was: if all but a are 0, you have dx/dt= ax which gives an exponential solution. a represents the rate at which the increase in the population of the krill depends on the present number of krill. If all but b are 0, you have dx/dt= -bt. b is the rate at which the decrease in krill population is proportional to the number of whales. If all but m are 0, dy/dt= -my. m is the rate at which the decrease in the whale population is proportional to the whale population. Do you see what the kirll population increases if there are no whales but the whale population decreases if there are no krill? Finally, if all but n are 0, dy/dt= nx. n is the rate at which the whale population increases proportional to the krill population. Do you see why the whale population increases proportional to the krill population but the krill population decreases proportional to the whale population?

Okay, again, although you didn't say it, I assume that both a and b are positive. dx/dt= -ay. The larger a is the faster x will decrease for the same y. Isn't it obvious that a is the rate at which the anti-tanks destroy tanks? Similarly, isn't it obvious that b is the rate at which the tanks destroy anti-tanks?

Last edited: Aug 14, 2007
7. Aug 14, 2007

### arildno

Actually, Halls, b is the rate of anti-tanks destroyed by tanks per unit tanks..