Rat and Cat Markov Chain

In summary, the problem involves Rat and Cat moving between two rooms using different paths, with their motions governed by transition matrices. If they are in the same room, Cat eats Rat. The question asks for the probability of Rat surviving and the average time he will survive. The solution involves denoting the states (i,j) and creating a transition matrix based on the number of possible states. Once this is done, the rest of the problem can be solved.
  • #1
Andrusko
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Homework Statement


Rat and Cat move between room 1 and 2 using different paths. Their motions are governed by their respective transition matrices:

[0.9, 0.1 ; 0.2, 0.8] [0.6, 0.4 ; 0.3, 0.7]

(semi colon is a new line in the matrix, like in matlab)

If they are ever in the same room, cat eats rat. What is the probability that rat will survive? How long on average will he survive? Hint: denote the state (i,j) where i is the location of the rat and j is the location of the cat.

Homework Equations



No idea what is even relevant to the problem.

The Attempt at a Solution



Well, I haven't got the foggiest idea where to start. I think that the Markov chains are independent of one another and haven't got a clue how to deal with that, because I've never seen a problem like this one before.

The hint sort of suggests to me that you only need one transition matrix, so my only idea is that you multiply the matrices together.

Any help appreciated.
 
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  • #2
If you denote the states (i,j), how many states are possible?

What would the size of your transition matrix be, given this number of states?

If you are in state (i,j), what would the probability of a transition to (k,l) be?

Once you have created the correct transition matrix, the rest of the problem should follow.
 

1. What is a Rat and Cat Markov Chain?

A Rat and Cat Markov Chain is a mathematical model used to study the behavior of two interacting animals, a rat and a cat. It is based on the principles of Markov chains, which describe how a system changes over time based on probabilities.

2. How is a Rat and Cat Markov Chain useful in scientific research?

A Rat and Cat Markov Chain can be used to study the dynamic relationship between two animals and how their behavior affects each other. It can also be used to predict the future behavior of the two animals based on their current state.

3. What are the key assumptions in a Rat and Cat Markov Chain?

The key assumptions in a Rat and Cat Markov Chain include: 1) the behavior of the rat and cat are dependent on each other, 2) the system is in a steady state, and 3) the transition probabilities between states are constant over time.

4. How is a Rat and Cat Markov Chain different from a traditional Markov Chain?

A Rat and Cat Markov Chain differs from a traditional Markov Chain in that it involves two interacting animals rather than just one. This adds a level of complexity as the behavior of one animal can affect the behavior of the other.

5. What are some potential applications of a Rat and Cat Markov Chain?

A Rat and Cat Markov Chain can be applied in various fields such as animal behavior research, ecology, and predator-prey dynamics. It can also be used to study the spread of diseases between animal populations and to predict the outcome of different management strategies in pest control.

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