ABSTRACT

The objective of this paper is to find the stationary distribution of a certain class of Markov chains arising in a biological population involved in a specific type of evolutionary conflict, known as Parker’s model. In a population of such players, the result of repeated, infrequent, attempted invasions using strategies from, is a Markov chain. The stationary distributions of this class of chains, for are derived in terms of previously known integer sequences. The asymptotic distribution (for) is derived.

3. The Stationary Distribution

Now the dominant eigenvalue is m, and we derive a recurrence relation for the corresponding left eigenvector, the stationary distribution, where we set the right-most element equal to 1. It is straightforward to demonstrate that the final three elements of the eigenvector are, and 1.

Observe that

where, throughout, is the identity matrix, is a k element column vector and is a k element row vector.

We then have

.

Also

and so

Now consider

We have

so is the required eigenvector.

We now have a recurrence relation for the which is

This is valid for using and

.

Suppose we write for the sum of the elements of so that is the stationary distribution of the Markov chain. We have immediately that

with,. This is sequence A001040 [5] specified as, where our where the sequence is initiated with and.

We can extract individual elements of the stationary distributions. Suppose that is the i’th element of. Then we have that

for, , with initial values and, and and for, we have and. The sequence for is A058307, and is A058279 in [5].

Table 1 gives some values of.

4. The Asymptotic Eigenvector

Having derived recurrence relations for the elements of the eigenvectors we now consider the limit as. We begin with a simple Lemma.

Lemma

Suppose we have a recurrence relation of the form where the is not dependent on the, and. Suppose we have two sequences, and satisfying the recurrence relationship but initiated by different values i.e. by and respectively. Then as where is a constant, which depends on the initial values.

Proof

Since

We have

Now the denominator increases at a rate greater than

as one can see easily by considering the liniting case, so the above expression tends to zero as and so as.

Comment Much weaker conditions are necessary than those stated above for as.

We can apply the lemma immediately to the elements of the stationary distribution, expressed in the integer form. The ratios for 0 and 1 elements for are 1, , 1.307692, 1.308824, 1.308789, 1308789 illustrating the speed with which convergence takes place. We have no expression for the asymptotic value but for m = 200 the ratio is approximately 1.3087893731. The ratios for 1 and 2 elements for are 1, 0.5, 0.529412, 0.528090, 0.528131, 0.0.528130 and for m = 200 approximately 0.5281297672.

In the absence of a simple way of evaluating the limiting ratios discussed above analytically we adopt a different method to derive the asymptotic stationary distribution, again expressed in integers. Suppose this is given by, and define. We have that. Thus, and

.

In a similar way we can obtain, and, and so on. It is clear that the signs alternate. For ease we introduce the following notation; we write when so that the sequences and for consist of positive integers. Similarly we write

when so the sequences

and consist of positive integers. Thus we have, , , , and so on, while, , and so on.

The theorem below gives recurrence relations for, , and, in terms of A058279 and A058307 [5].

Theorem

For we have where

with ([5] A058279)

and where with and ([5] A058307). Further and.

Proof

We have that and , and we have already shown that, , , , which satisfy the formula given in the statement of the theorem.

We prove that if the formula holds for for then it holds for for, and thence for for. Thence by induction.

Note first that

Suppose then that the formula for holds for. Then since we have and substituting for the expressions given in the statement of the theorem we have

Now clearly we also have so we have

Table 2 gives the first 15 elements for the eigenvector for. Some idea of the speed of convergence can be gained by observing that these values agree with the elements of the eigenvector for except in the final 2 decimal places.

5. Conclusion

We have derived the stationary distribution of the frequencies of the available strategies in a population in which mutations occur infrequently, for Parke’s model when the reward is 2⁺ and for integer valued strategies. These relate to certain known integer sequences. This work provides a base for further investigations for other values of the reward, and more complex invasion processes.

6. Discussion

Parker’s model, which is also known as the Scotch Auction, is often used in the conflict theory literature as an example of a simple model in which there is no ESS (evolutionarily stable strategy). The implication of this is that there is no population assembly which is resistant to invasion. Of course if such a contests actually occurs it is important to ask what will happen in the population. This is the question which is addressed in [3], and which generates the class of matrices considered here. The stationary distribution then corresponds to the frequency with which one would observe a population to be playing a specific strategy, except if one happened on a population in transition.

The class of cases discusses above arises from Parker’s model when we consider a fixed reward value, and when the value of m, the range of possible strategies, is allowed to vary. It would be of interest to examine

other possible values of V as m varies. For example, for, the “Markov matrices” have 1’s for , 0’s for and diagonal elements to make the row sums m. It is hoped to treat these models in a subsequent paper.

We observe from the numerical values that the most frequent strategy value played is 1, that the distribution is uni-modal and that the strategies are played over 90% of the time; asymptotically approximately 0.90667 which agrees to five decimal places to the value for, while the mean value is asymptotically approximately 1.1207 which agrees to five decimal places to the value for. These latter figures confirm the rapidity of the convergence.

REFERENCES