Markov chain model for demersal fish catch analysis in Indonesia

As an archipelagic country, Indonesia has considerable potential fishery resources. One of the fish resources that has high economic value is demersal fish. Demersal fish is a fish with a habitat in the muddy seabed. Demersal fish scattered throughout the Indonesian seas. Demersal fish production in each Indonesia’s Fisheries Management Area (FMA) varies each year. In this paper we have discussed the Markov chain model for demersal fish yield analysis throughout all Indonesia’s Fisheries Management Area. Data of demersal fish catch in every FMA in 2005-2014 was obtained from Directorate of Capture Fisheries. From this data a transition probability matrix is determined by the number of transitions from the catch that lie below the median or above the median. The Markov chain model of demersal fish catch data was an ergodic Markov chain model, so that the limiting probability of the Markov chain model can be determined. The predictive value of demersal fishing yields was obtained by calculating the combination of limiting probability with average catch results below the median and above the median. The results showed that for 2018 and long-term demersal fishing results in most of FMA were below the median value.


Introduction
As an archipelagic country, Indonesia has the potential of marine and fishery resources that can improve the Indonesian economy. The Government of Indonesia has been divided the Fisheries Management Area (FMA) into 11 regions. Each FMA has provided fish catches that vary according to geographical location. One of the fish resources that boost Indonesia's economy is demersal fish. Demersal fish is a fish with a habitat in the muddy seabed. Although demersal fish does not become the target of Indonesian exports, but this fish is very popular with the community because it tastes good and the price is quite affordable. Some types of demersal fish are black pomfret, silver pomfret, halibut, baramundi, red snapper, jack travelie, hairtail, giant cat fish and others. During the period of 2005-2014 demersal fish production was quite varied and contributed an average of 130 thousand tons each year. Most demersal fish were caught in the Java sea and Karimata straits, Natuna sea and South China Sea [1]. The study conducted related to the results of fish catch, namely the prediction of potential areas of pelagic fishing in Mamuju district [5]. To predict a value in the future, various methods can be used, such as the Markov chain or times series. Markov chain method has been used in stock price prediction [2], in predicted fishing results in India [3], predicted dominant fish catch areas in India [6]. In addition to the study of catch areas, research on the prediction of fish catch is also necessary because it will have an impact on Indonesian exports. In this study the Markov chain method will be used to analyze the demersal fishing results in Indonesia. The objective is to know the predictions of demersal fishing results across Indonesia's FMA. This is expected to be useful for the government (Marine Affairs and Fisheries) in taking policy for future anticipation if the catch is decreased.

Markov Chain Model
The Discrete-time Markov chain is a stochastic process with the nature that the future state depends only on the present state of being free from the past.

Stationary Distribution
A Markov chain is ergodic if irreducible, positive recurrent, and aperiodic. Ergodic's Markov chains have limiting probabilities, lim →ஶ ‫‬ = ߨ , ݅, ݆ = 0, 1, 2, … which is free from the initial state i, ൣߨ ൧ is called the stationary distribution (steady state) of the Markov chain [4].

Methodology
Demersal fishing results has changed every period and its catching area. Discrete time markov chain can model problems with data that changes with time.
Demersal fishery data obtained from the Directorate of Fisheries Fishing Ministry of Marine Affairs and Fisheries Indonesia, the data is compiled by year and FMA. Assume that the number of Markov chain states were 2, i.e., state 1; the number of catches below the median value, and state 2; the number of catches above the median value. For each FMA it has been done the same stepsfigure: a. Determine the state "1" and state "2" of each demersal fish catch data based on the value of the captured median. b. Calculate the number of transitions from state "1" to state "2" and vice versa    From the data was determined median for each FMA. The median value of each FMA was presented in Table 2.    Because ∑ ‫‬ ஶ ୀଵ = ∞, then state "1" and state "2" were recurrent state. So it was said the Markov chain is irreducible and recurrent. The same is true for the matrices A, B, C, D, and E. The next step, determine the periodicity of the Markov chain with the transition matrix P, A, B, C, D, E, we obtained ݀ሺ0ሻ = ݃ܿ݀ሼ1,2,3, … ሽ = 1 , then state "0" was aperiodic ݀ሺ1ሻ = ݃ܿ݀ሼ1,2,3, … ሽ = 1, the state "1" was aperiodic. it could be concluded that the Markov chain was aperiodic. For ݊ → ∞, we get Because ߤ = 2,6 < ∞ and ߤ ଵ = 1,63 < ∞, then the Markov chain is said to be positive recurrent.
Since the Markov chains were irreducible, positively recurrent and aperiodic, it was said to be ergodic Markov chains. Since the Markov chain was ergodic, then a stationary distribution was = ሾ0,3846 0,6154ሿ. This means for a Markov chain with a transition probability matrix P, the probability of demersal fish quantities above the median value was 0,6154. In the same way, a steady state distribution for the transition probability matrix A, B, C, D, and E are: ܽ ଵ = average catch above median ߨ = the probability of catch is below median ߨ ଵ = the probability of catch is above median The predictions of demersal fish yields throughout FMA can be seen in the table 3.

Conclusion
From the Markov model analysis, it was concluded that the demersal fish catch in some FMA was below the median value. This should be anticipated by the government with various efforts to make demersal fish production is always increasing.