Question on Lotka-Volterra model and Lanchaster combat model

  • Thread starter murshid_islam
  • Start date
  • Tags
    Model
In summary: 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
  • #1
murshid_islam
457
19
i have 2 questions

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

[tex]\frac{dx}{dt} = (a - by)x[/tex]

[tex]\frac{dy}{dt} = (-m + nx)y[/tex]

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:

[tex]\frac{dx}{dt} = -ay[/tex]
[tex]\frac{dy}{dt} = -bx[/tex]

here,
x(t) = the number of tanks in operation in time t
y(t) = the number of anti-tanks in operation in time t
 
Physics news on Phys.org
  • #2
This looks an awful lot like homework to me! Any reason I shouldn't move it?

murshid_islam said:
i have 2 questions

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

[tex]\frac{dx}{dt} = (a - by)x[/tex]

[tex]\frac{dy}{dt} = (-m + nx)y[/tex]
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?

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:

[tex]\frac{dx}{dt} = -ay[/tex]
[tex]\frac{dy}{dt} = -bx[/tex]

here,
x(t) = the number of tanks in operation in time t
y(t) = the number of anti-tanks in operation in time t
Same idea as above.
 
  • #3
HallsofIvy said:
This looks an awful lot like homework to me! Any reason I shouldn't move it?
well, it's not homework.

HallsofIvy said:
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?
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.

HallsofIvy said:
What effect on x and y would increasing or decreasing each of a, b, c, d while leaving the others alone?
don't have any idea about that.
 
  • #4
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
arildno said:
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?
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
murshid_islam said:
i have 2 questions

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

[tex]\frac{dx}{dt} = (a - by)x[/tex]

[tex]\frac{dy}{dt} = (-m + nx)y[/tex]

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


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

[tex]\frac{dx}{dt} = -ay[/tex]
[tex]\frac{dy}{dt} = -bx[/tex]

here,
x(t) = the number of tanks in operation in time t
y(t) = the number of anti-tanks in operation in time t
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 by a moderator:
  • #7
Actually, Halls, b is the rate of anti-tanks destroyed by tanks per unit tanks..
 

1. What is the Lotka-Volterra model?

The Lotka-Volterra model, also known as the predator-prey model, is a mathematical model used to describe the interactions between two species in an ecosystem. It is based on the idea that as the population of a predator species increases, it will consume more of its prey, leading to a decrease in prey population. This, in turn, will cause a decrease in the predator population due to a lack of food. The model is used to understand the dynamics of predator-prey relationships in nature.

2. What is the Lanchester combat model?

The Lanchester combat model is a mathematical model used to study the dynamics of warfare. It was developed by Frederick Lanchester in 1916 and is based on the idea that the outcome of a battle depends on the relative strengths of the opposing forces. The model takes into account factors such as weapons technology, tactics, and morale to predict the outcome of a battle.

3. How do the Lotka-Volterra and Lanchester combat models differ?

The Lotka-Volterra model focuses on the dynamics of predator-prey relationships, while the Lanchester combat model focuses on the dynamics of warfare. The Lotka-Volterra model assumes a constant relationship between the two species, while the Lanchester combat model takes into account factors such as tactics and morale that can affect the outcome of a battle.

4. What are the limitations of the Lotka-Volterra and Lanchester combat models?

The Lotka-Volterra model is a simplified representation of predator-prey interactions and does not take into account other factors that may affect the relationship between the two species. Similarly, the Lanchester combat model has limitations in its assumptions about warfare and may not accurately predict the outcome of battles in real-world situations.

5. How are the Lotka-Volterra and Lanchester combat models used in research?

The Lotka-Volterra model is widely used in ecology and population biology to study predator-prey relationships and the effects of other factors on these interactions. The Lanchester combat model is used in military and defense research to analyze and predict the outcomes of battles and wars. Both models have also been applied in other fields, such as economics and politics, to study competitive interactions between individuals or groups.

Similar threads

  • Differential Equations
Replies
18
Views
2K
Replies
4
Views
1K
  • Differential Equations
Replies
1
Views
2K
  • Differential Equations
Replies
2
Views
2K
  • Differential Equations
Replies
8
Views
2K
  • Differential Equations
Replies
1
Views
1K
Replies
1
Views
1K
  • Differential Equations
Replies
7
Views
2K
Replies
6
Views
2K
  • Special and General Relativity
Replies
10
Views
1K
Back
Top