Size- and Fecudnity-dependent Mortality Formulation Introduction & Background Methods Parameter Estimation Results & Discussion Size- and Fecudnity-dependent Mortality Formulation for Sailpin Sandfish(Arctoscopus japonicus) in the southwestern East Sea 안녕하십니까 수산학 실험식에 박사과정 1년 차 하승목입니다. 저는 오늘 1장의인 가입모델에 대해서 간단히 언급하고, 제가 연구한 도루묵의 자연사망계수에 대해 발표하겠습니다. Seungmok Ha Seungmok Ha JEJU National University

Contents Introduction & Background Methods Parameter Estimation Results & Discussion Contents Introduction & Background Methods 발표는 다음과 같은 순서로 먼저 인트로덕션에서 책의 1장 내용인 어류의 가입에 대해 설명드리고, 그 후 제가 연구한 도루묵의 자연사망률에 대한 백그라운드와 방법론, 관련 계수 추정 및 결과와 논의점에 대해 말씀드리겠습니다. Parameter Estimation Results & Discussion Seungmok Ha JEJU National University

Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment? means reaching a certain size or reproductive stage. With fisheries, recruitment usually refers to the age a fish can be caught. 먼저 1장에서 다루는 내용인 ‘가입’에 대해 말씀드리겠습니다. 가입이란 것은 특정 시간에 태어난 어류가 성장하여 어획의 대상이 되기 시작하면 어류가 어장으로 ‘가입 되었다’고 합니다. 이 가입량은 어획에 직접적인 요소로서 가입량을 추정하고 예측하는 것은 수산해양학의 가장 큰 관심사 중 하나입니다. 그런데 가입에서 성장 가능한 난의 생산력은 가입을 위한 기초자료를 제공하기에 1장에서는 가입량과 성장가능한 난의 수 사이의 관계에 대한 수학적인 모델들에 대해서 다루고 있습니다. Seungmok Ha JEJU National University

Mathematical model Questions and Problems in Science Modeling Introduction & Background Methods Parameter Estimation Results & Discussion Mathematical model Questions and Problems in Science 이에 먼저 수학모델에 대해 간략하게 설명드리겠습니다. 수학모델이라는 것은 과학적 질문을 모델링을 통하여 수학식으로 표현하고 해석하여 다시 생물학적인 해석으로 답을 찾는 것을 의미합니다. 예를 들면 추후에 자원량의 변동추세가 궁금하다던지, 자원보호를 위한 전략으로 어획 그물코 크기를 늘리는 것과, 산란장 조성의 전략 중 어떤 전략이 효과적인지 알고 싶다면 적절한 수학 모델을 세워 그 답을 정량적이 아닌 정성적으로 알 수 있습니다. Modeling Interpretation Mathematical Analysis Seungmok Ha JEJU National University

Mathematical model Discrete Difference Equations Continuous ODEs PDEs Introduction & Background Methods Parameter Estimation Results & Discussion Mathematical model Discrete Difference Equations Deterministic 수학적 모델은 크게 두가지, 결정론적 모델과 확률론적 모델이 있는데, 결정론적 모델은 식에 초깃값을 넣으면 결과가 딱 하나로 정해져서 산출되는 모델이고, 확률론적 모델은 식에 확률적인 요소를 집어넣어 그때 그때 결과가 달라지는 모델입니다. 결정론적 모델은 평균적인 결과라 생각하시면 되겠고, 확률론적 모델은 결과가 항상 다르기 때문에 여러 번(약 10000번 이상) 시뮬레이션 후 그 평균 추세를 주로 사용합니다. 이 결정론적 모델도 2가지로 나누어지는데 이산모델과 연속모델이 그것입니다. 이산모델은 이산방정식, 즉 우리가 자주 보는 방정식을 생각하시면 되고, 연속모델은 변수가 하나인 상미분방정식과 변수가 여러 개인 편미분 방정식이 있습니다. 이 모델을 선택하는 것은 정답은 없습니다. 우리가 구하고자 하는 결과나, 여러 특성에 의해 달라질 수 있습니다. 여기서는 짧은 연속적인 시간에서의 모델이기 떄문에 상미분 방정식이 주로 나옵니다. Continuous ODEs PDEs Mathematical Models Stochastic Seungmok Ha JEJU National University

Recruitment theory Vulnerable to fishery E1 E2 E3 L J1 J2 A1 A2 A3 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory Vulnerable to fishery 그럼 가입이론에 대해 설명드리겠습니다. 그 전에 일반적인 어류의 life cycle의 그림을 보시겠습니다. 그림에서 보시면 난을 E, L, J, A로 나타내었습니다. 가입은 앞에서 말씀드렸다시피 어류가 어획의 대상이 되는 집단이 되는 것을 의미합니다. 빨간 선을 기점으로 어류가 가입되게 됩니다. E1 E2 E3 L J1 J2 A1 A2 A3 Pre-recruits Post-recruits Life cycle life diagram including egg, larval, juvenile and adult stages. Eggs produced by adults of different ages can have different viabilities. Seungmok Ha JEJU National University

Recruitment theory 𝑅=𝑆𝐸 the number of recruits Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 𝑅=𝑆𝐸 the number of recruits the proportion surviving 가입량과 성장가능한 알의 수 사이의 관계는 다음과 같이 직관적인 식으로 나타낼 수 있습니다. 이 식에서 R은 가입량, E는 성장가능한 알의 수, S는 알에서부터 가입할 때까지 기간의 생존률으로 이 식의 의미는 ‘가입량은 성장 가능한 알 중 그 기간동안 살아남은 개체의 수이다’라는 것을 의미합니다. the number of viable eggs Seungmok Ha JEJU National University

Recruitment theory Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 이 생존률이 상수일 때 그래프를 보면 다음과 같이 알의 수와 가입량 사이에 비례관계, 즉 선형적인 관계가 있음을 알 수 있습니다. 각각의 그래프는 생존률 S가 각각 다른 상수일 때를 나타내었습니다. Density-independent model relating recruitment and egg production for three levels of the density-independent mortality rate. Seungmok Ha JEJU National University

Recruitment theory 𝑑𝑁 𝑑𝑡 =−𝜇𝑁 𝑁=𝐸 𝑅 𝑑𝑁 𝑁 =−𝜇 𝑡=0 𝑡 𝑟 𝑑𝑡 𝑅=𝐸 𝑒 −𝜇𝑡 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 𝑑𝑁 𝑑𝑡 =−𝜇𝑁 𝑁=𝐸 𝑅 𝑑𝑁 𝑁 =−𝜇 𝑡=0 𝑡 𝑟 𝑑𝑡 𝑅=𝐸 𝑒 −𝜇𝑡 the rate of change of a cohort the rate of change of a cohort the rate of change of a cohort 이 때 이 집단 개체 수에 대해 살펴보겠습니다. 여기서 미분이 나오는데 이 미분은 시간에 대한 개체수의 변화율이라고 보시면 됩니다. 우리가 보고자 하는 것이 알에서 부터 가입하기 까지의 연속적인 시간에서의 개체수의 변화를 보고자 하는 것이기 때문에 앞의 수학적 모델 중 연속모델이 사용되었습니다. 그러면 이 시간의 흐름에 따른 개체수의 변화는 개체 수에 비례하여 감소하는데 이것은 죽어나가는 것, 즉 사망하는 개체를 의미합니다. 즉 이 식은 시간의 흐름에 따라 개체수의 일정비율이 사망하고 있음을 나타낸 식입니다. 여기서 뮤는 자연사망률(밀도 독립적인)을 나타냅니다. 이 식을 풀면 다음과 같이 생존가능한 알에서 가입하는 기간까지 생존하여 가입되는 량을 알 수 있습니다. 이것은 ‘시간의 흐름에 따라 개체수의 일정비율이 사망하고 있다’는 가정에 의한 식이고 이 가정에 맞는 가입량과 생존가능한 알의 수(즉, 산란군) 사이의 관계입니다. where 𝑡= 𝑡 𝑟 − 𝑡 0 the survival fraction Seungmok Ha JEJU National University

Recruitment theory 𝜇= 𝜇 0 + 𝜇 1 𝑁 𝑑𝑁 𝑑𝑡 =− 𝜇 0 + 𝜇 1 𝑁 𝑁 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 1) Intracohort competition 𝜇= 𝜇 0 + 𝜇 1 𝑁 𝑑𝑁 𝑑𝑡 =− 𝜇 0 + 𝜇 1 𝑁 𝑁 the instantaneous rate of density-independent mortality the coefficient of density-dependent mortality 먹이와 같은 자원에 대한 종내경쟁에 의해서 밀도 종속적인 관계를 나타내는 버벌튼 홀트 모델에 대해 살펴보겠습니다. 0 : 밀도 독립적 순간사망률 1 : 밀도 종속적 순간사망률 Seungmok Ha JEJU National University

Recruitment theory 𝑅= 1 𝐸 𝑒 𝜇 0 𝑡 + 𝜇 1 𝜇 0 ( 𝑒 𝜇 0 −1) −1 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 1) Intracohort competition 𝑅= 1 𝐸 𝑒 𝜇 0 𝑡 + 𝜇 1 𝜇 0 ( 𝑒 𝜇 0 −1) −1 Beverton-Holt model : 𝑅= 𝛼 𝐸 +𝛽 −1 이것을 아까와 마찬가지로 풀면 가입량과 산란군 사이에 다음과 같은 관계를 나타내고, 이는 간단히 아래와 같이 나타낼 수 있습니다. Seungmok Ha JEJU National University

Recruitment theory 1) Intracohort competition Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 1) Intracohort competition 버벌튼 홀트 모델의 식을 그림으로 그려보면 다음과 같습니다. Beverton-Holt-type model relating recruitment and egg production for three levels of the parameter 𝛼. Seungmok Ha JEJU National University

Recruitment theory 𝜇= 𝜇 0 + 𝜇 2 𝑃 𝑑𝑁 𝑑𝑡 =− 𝜇 0 + 𝜇 2 𝑃 𝑁 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 2) Cannibalism by adults 𝜇= 𝜇 0 + 𝜇 2 𝑃 𝑑𝑁 𝑑𝑡 =− 𝜇 0 + 𝜇 2 𝑃 𝑁 the instantaneous rate of density-independent mortality the coefficient of 'stock-dependent' mortality 다음으로 성어에 의한 공식에 의해 밀도 종속적인 관계를 나타낸다고 보는 리커 모델에 대해 살펴보겠습니다. 0 : 밀도 독립적 순간사망률 2 : 자원 종속적 사망률 (상수) P : 공식을 하는 성어의 양 a measure of the cannibalistic component of the adult population Seungmok Ha JEJU National University

Recruitment theory 𝑅=𝐸 𝑒 − 𝜇 0 + 𝜇 2 𝑃 𝑡 𝑅=𝑘𝐸 𝑒 −𝛿𝐸 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 2) Cannibalism by adults 𝑅=𝐸 𝑒 − 𝜇 0 + 𝜇 2 𝑃 𝑡 Ricker model : 𝑅=𝑘𝐸 𝑒 −𝛿𝐸 이를 풀어보면면 가입량과 산란군 사이에 다음과 같은 관계를 나타내고, 이는 간단히 아래와 같이 나타낼 수 있습니다. Seungmok Ha JEJU National University

Recruitment theory 2) Cannibalism by adults Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 2) Cannibalism by adults 리커 모델의 식을 그림으로 그려보면 다음과 같습니다. Ricker-type model relating recruitment and egg production for three levels of the slope at the origin parameter. Seungmok Ha JEJU National University

Recruitment theory 𝑑𝑊 𝑑𝑡 =𝐺(𝑊) 𝑑𝑁 𝑑𝑡 =−𝜇 𝑊 𝑁 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 3) Size-dependent processes Consider a model for individual growth in weight: 𝑑𝑊 𝑑𝑡 =𝐺(𝑊) If the mortality rate is size-dependent then we have: 𝑑𝑁 𝑑𝑡 =−𝜇 𝑊 𝑁 a compensatory function for individual growth 마지막으로 크기종속적인 모델에 대해 설명 드리겠습니다. 이는 Biomass 같은 개체의 무게를 고려한 모델입니다. 이때 시간별 크기종속적인 사망률을 아래와 같은 식으로 나타내었습니다. Seungmok Ha JEJU National University

Recruitment theory 𝑑𝑁 𝑑𝑊 =− 𝜇 𝑊 𝐺(𝑊) 𝑁 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 3) Size-dependent processes 𝑑𝑁 𝑑𝑊 =− 𝜇 𝑊 𝐺(𝑊) 𝑁 𝑁 𝑊 1 =𝑁( 𝑊 0 ) 𝑒 − 𝜇(𝑊) 𝐺(𝑊) 𝑑𝑊 보충 및 과보충 모델 크기 종속적 모델 the number in the population surviving to weight (size) 𝑊 1 Seungmok Ha JEJU National University

Recruitment theory 𝑑𝑊 𝑊 =− 𝐺 ∗ 𝜇 𝑑𝑁 1+ 𝑁 𝐾 𝑁 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 3) Size-dependent processes Assume that that 𝐺= 𝐺 ∗ /(1+𝑁/𝐾) where 𝐺 ∗ is the maximum growth rate, 𝑑𝑊 𝑊 =− 𝐺 ∗ 𝜇 𝑑𝑁 1+ 𝑁 𝐾 𝑁 log 𝑒 𝑊 1 𝑊 0 =− 𝐺 ∗ 𝜇 log 𝑒 (𝐾+𝐸) 𝑁 1 𝐾+ 𝑁 1 𝐸 보충 및 과보충 모델 크기 종속적 모델 Seungmok Ha JEJU National University

Recruitment theory 𝑅=𝑁( 𝑊 1 )= 𝐴𝐸 1+(1−𝐴) 𝐸 𝐾 Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 3) Size-dependent processes Cushing-Shepherd model : 𝑅=𝑁( 𝑊 1 )= 𝐴𝐸 1+(1−𝐴) 𝐸 𝐾 where 𝐴=exp⁡ − 𝜇 𝐺 ∗ ln 𝑊 1 𝑊 0 보충 및 과보충 모델 크기 종속적 모델 Seungmok Ha JEJU National University

Recruitment theory 3) Size-dependent processes Introduction & Background Methods Parameter Estimation Results & Discussion Recruitment theory 3) Size-dependent processes 보충 및 과보충 모델 크기 종속적 모델 Cushing-Shepherd-type model relating recruitment and egg production for three levels of the density-dependent parameter K. Seungmok Ha JEJU National University

Sailpin Sandfish… sailfin sandfish = 도루묵 (출처 : 국립수산과학원) Introduction & Background Methods Parameter Estimation Results & Discussion Sailpin Sandfish… sailfin sandfish = 도루묵 (출처 : 국립수산과학원) 셀린 샌드피쉬, 즉 도루묵은 거의 강원도지방에서 잡히는 물고기로 다음 (클릭!) 그림과 같이 생긴 물고기입니다. 도루묵은 (클릭!) 찌개, 구이, 조림 등으로 많이 먹는데 그림에서 보시듯이 다른 생선들에 알이 큽니다. 그리고 이 알이 정말 맛있습니다. 이 맛있는 물고기인 도루묵이 (클릭!) Seungmok Ha JEJU National University

Sailpin Sandfish… Nomenclature : Arctoscopus japonicus Introduction & Background Methods Parameter Estimation Results & Discussion Sailpin Sandfish… Nomenclature : Arctoscopus japonicus Max reported age : 6 years Max length : 30 cm Spawning season : November ~ January Distribution : North Pacific - Aleutian Islands (one record) to Okhotsk and East sea Fishing areas : Eastern coast of the Korean peninsula (출처 : 국립수산과학원 차형기박사) This fish is sailfin sandfish. It is an important species of Korean fisheries industry. Sailfin sandfish’s Nomenclature is Arctoscopus japonicus. Sandfish's life expectancy is six years, and females grow faster than males. It’s max length is 30cm (female). Sailfin sandfish spawn only once each spawning season. Spawning grounds of the stock are located along the east coast of the Korean Peninsula. In the spawning season from November to January, and main spawning season is December This stock is distributed at depths from 100m to 300m in waters from Gangwon-Do of the Korean Peninsula to Toyama Prefecture in Japan In Korea, sandfish fishing areas are distributed along the east coast, from Gangwon-Do to the northern Gyeongsangbuk-Do, Seungmok Ha JEJU National University

Sailpin Sandfish… Changes in annual catch of sandfish in Korea Introduction & Background Methods Parameter Estimation Results & Discussion Sailpin Sandfish… Catch (mt) 도루묵 이 맛있는 물고기인 도루묵이 (클릭!) 과거 어획량 그래프를 보면 1970년대와 80년대 중반에 많이 잡히다가 어획량이 급격히 감소한 것을 볼 수 있습니다. 최근에는 계속 4000톤 미만으로 저평형 상태에 머물러 있습니다. 그래서 이 도루묵 자원을 지속적으로 잘 이용 하기 위해서는 자원회복이 필요한 실정이며. (클릭!) 실제로도 국립수산과학원에서 자원회복사업을 하고 있습니다. (클릭!) year Changes in annual catch of sandfish in Korea (Biomass Estimation of Sailfin Sandfish, Arctoscopus Japonicus, in Korean Waters - Lee, S.I. et al.) Seungmok Ha JEJU National University

Introduction & Background Methods Parameter Estimation Results & Discussion Introduction & Background Methods Parameter Estimation Results & Discussion Natural Mortality occur through predation or non-predation events such as senescence and disease a parameter in most fish stock assessment models 자연 사망률은 노화와 질병과 같은 포식 또는 비 포식 이벤트를 통해 발생할 수 있다. 대부분의 물고기 자원평가 모델의 매개 변수입니다. (Research on the population dynamics and fisheries resources through mathematical analysis, NFRDI) Seungmok Ha JEJU National University

Estimate size-specific natural mortality rates of sailfin sandfish Introduction & Background Methods Parameter Estimation Results & Discussion Size-dependent mortality formulation for Pacific cod – Jung et al. (2008) This paper is Size-dependent mortality formulation for Pacific cod. In this research, using similary methods in this paper, we estimate size-specific natural mortality rates of sandfish. Estimate size-specific natural mortality rates of sailfin sandfish Seungmok Ha JEJU National University

Contents Introduction & Background Methods Parameter Estimation Results & Discussion Contents Introduction & Background Methods Next, I introduce methods of our research. Parameter Estimation Results & Discussion Seungmok Ha JEJU National University

Main hypothesis ‘Bigger is Better’ Introduction & Background Methods Parameter Estimation Results & Discussion Main hypothesis ‘Bigger is Better’ Natural mortality rate is inversely proportional to the body length of fish The main hypothesis is “bigger is better” hypothesis. This hypothesis means large fish’s survival rate is more than In other words, the natural mortality rate is inversely proportional to the body length of fish. So, we setting this equation. M is natural mortality rate, L is length of sandfish. This L derived from von Bertalanffy growth equation. So, we can calculate this L. Q is inversely proportional constant. q is Constant Length derived from Von Bertalanffy growth function Seungmok Ha JEJU National University

Using a Mathematical Model (Leslie matrix) Introduction & Background Methods Parameter Estimation Results & Discussion Q) How to estimate ‘q’ ??? Using a Mathematical Model (Leslie matrix) Now, for calculated M, we must know this q. So, our problem is “how to estimate tis q?” For this, we use a Mathematical Model. Specially Lesile matrix. Seungmok Ha JEJU National University

1. The purpose of the research Introduction & Background Methods Parameter Estimation Results & Discussion Q) Why Leslie matrix??? 1. The purpose of the research 2. Ecological characteristics of the sandfish There is many mathematical model. why we choose Leslie matrix? The answer of this quastion are these two things. First, The purpose of the research, And second, because Ecological characteristics of the sandfish. Sandfish spawn only once each spawning season. So, sandfish population change largely in this season. So, we choose discrete model. Seungmok Ha JEJU National University

Q) Why Leslie matrix??? E1 E2 E3 E4 E5 N0 N1 N2 N3 N4 N5 N6 Introduction & Background Methods Parameter Estimation Results & Discussion Q) Why Leslie matrix??? There is many mathematical model. why we choose Leslie matrix? The answer of this quastion are these two things. First, The purpose of the research, And second, because Ecological characteristics of the sandfish. Sandfish spawn only once each spawning season. So, sandfish population change largely in this season. So, we choose discrete model. E1 E2 E3 E4 E5 N0 N1 N2 N3 N4 N5 N6 Seungmok Ha JEJU National University

Q) Why Leslie matrix??? Leslie matrix model Discrete Difference Introduction & Background Methods Parameter Estimation Results & Discussion Q) Why Leslie matrix??? Leslie matrix model Discrete Difference Equations Now, for calculated M, we must know this q. So, our problem is “how to estimate tis q?” For this, we use a Mathematical Model. Specially Lesile matrix. Deterministic Continuous ODEs PDEs Mathematical Models Stochastic Seungmok Ha JEJU National University

Methods Leslie matrix model Survival rate Number of individuals Introduction & Background Methods Parameter Estimation Results & Discussion Methods Leslie matrix model we can write this model in equation form N0 is hatching stage summation of each total number of effective fecundity. Other stages can expressed multiply with survival rate and prior stage N, before time step. Survival rate Number of individuals Parameter description Seungmok Ha JEJU National University

Methods Leslie matrix model Effective fecundity Introduction & Background Methods Parameter Estimation Results & Discussion Methods Leslie matrix model Effective fecundity Number of surviving individuals after 1 year we can write this model in equation form N0 is hatching stage summation of each total number of effective fecundity. Other stages can expressed multiply with survival rate and prior stage N, before time step. Number of surviving individuals after 1 year Parameter description Seungmok Ha JEJU National University

Methods Leslie matrix model Introduction & Background Methods Parameter Estimation Results & Discussion Methods Leslie matrix model Total number of hatched eggs from 6-age female we can write this model in equation form N0 is hatching stage summation of each total number of effective fecundity. Other stages can expressed multiply with survival rate and prior stage N, before time step. Number of surviving individuals after 1 year Parameter description Seungmok Ha JEJU National University

Methods Leslie matrix model Effective fecundity Leslie matrix Introduction & Background Methods Parameter Estimation Results & Discussion Methods Leslie matrix model Effective fecundity Leslie matrix Now, I introduce about leslie matrix. leslie matrix’s form s like this. Small n’s are Number of individuals in each age f’s are Effective fecundity of each age stage And s’s are Survival rates. This matrix is Leslie matrix. Number o individuals Survival rate Parameter description Seungmok Ha JEJU National University

Methods Parameter of Leslie matrix Introduction & Background Methods Parameter Estimation Results & Discussion Methods Parameter of Leslie matrix   In leslie matrix, there are two kind of parameters. Effective fecundity and Survival rate. Effective fecundity is multiply with these factors. I will explain this equation later. Survival rate equation gives this. this M is natural mortality.   Seungmok Ha JEJU National University

Methods Introduction & Background Methods Parameter Estimation Results & Discussion Methods   S can expressed using cumulates of M. where cumulates of M’s form like this. In cumulates of M, K, Loo t0 are parameter of growth equation, and those are constants. So, cumulates of M is equation on q Seungmok Ha JEJU National University

Methods : Equation on ‘q’ In other words, Introduction & Background Methods Parameter Estimation Results & Discussion Methods   Therefore S is equation on q, IN other words, all of leslie matrix was used to derive constant q. : Equation on ‘q’ In other words, the Leslie matrix was used to derive ‘q’!! Seungmok Ha JEJU National University

To solve the Leslie matrix… Introduction & Background Methods Parameter Estimation Results & Discussion To solve the Leslie matrix… Hypothesis   To calculate this q, we needs some hypothesis. Important hypothesis is first thing. 엠브리아닉 Seungmok Ha JEJU National University

To solve the Leslie matrix… Introduction & Background Methods Parameter Estimation Results & Discussion To solve the Leslie matrix… Hypothesis The sandfish population is at long-term equilibrium status   This hypothesis means the Strictly dominant eigenvalue of Leslie matrix is 1. Than we can be calculated q, using this hypothesis, In other words, we can estimates natural mortality rates of sailfin sandfish : Equation on ‘q’ Seungmok Ha JEJU National University

To solve the Leslie matrix… Introduction & Background Methods Parameter Estimation Results & Discussion To solve the Leslie matrix… Characteristic equation of Leslie matrix This is Characteristic equation form of Leslie matrix, If lambda satisfies P(lambda) is 1, than that lambda is called eigenvalue of Leslie matrix. Using some mathematical theorem, we know, Leslie matrix has just one Strictly dominant positive. Theorem. 1 Seungmok Ha JEJU National University

To solve the Leslie matrix… Introduction & Background Methods Parameter Estimation Results & Discussion To solve the Leslie matrix… Characteristic equation of Leslie matrix Like these theorems. Theorem. 2 Seungmok Ha JEJU National University

To solve the Leslie matrix… Introduction & Background Methods Parameter Estimation Results & Discussion To solve the Leslie matrix… Characteristic equation of Leslie matrix Now, Leslie matrix has just one Strictly dominant positive, 1. So we should solve this equation. this is also equation on q. We can solve this problem using mathematical computer program. : Equation on ‘q’ Seungmok Ha JEJU National University

Contents Introduction & Background Methods Parameter Estimation Results & Discussion Contents Introduction & Background Methods Now, I tell you about parameter estimation. Parameter Estimation Results & Discussion Seungmok Ha JEJU National University

Parameter estimation Von Bertalanffy growth equation Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Von Bertalanffy growth equation First, growth equation. This is Von Bertalanffy growth equation’ form. Normally, in this equation, estimates 3 factors. L, Loo and T0. But this methods can’t explain earlier stage well. For example, 0yr fish’s length was so large. Here, we setting t0 like this. L0 is length of sandfish when hatching period. We know L0 in this paper. Data using in 2004~2007 in korea, and using nonlinear regression. Age divided by monthly , And, Using the average length for each age, because data bias. Contrast in Reproductive Style Between Two Species of Sandfishes (Family Trichodontidae) - Oklyama - Otolith data collected from eastern coastal areas of Korea in 2004~2007 : non-linear regression - Age divided by monthly (the main spawning season : December) - Using the average length for each age. Seungmok Ha JEJU National University

Parameter estimation Von Bertalanffy growth equation Parameter Female Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Von Bertalanffy growth equation Results of growth equation Parameter estimation. These points are average length for each age, and This curve is growth curve. The parameter of growth equation are these. So, We can get growth equation. Parameter Female 27.171194 0.4688451 -0.10457 Seungmok Ha JEJU National University

Parameter estimation Effective fecundity Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Effective fecundity Effective fecundity consists these factor. Followings are these factors means. E is Fecundity, P is The probability that an average female sandfish matures to participate in spawning R is sex ratio and Sh is Embryonic survival rate. E and P can get form parameter estimation, R and Sh are assumption. 머튜러스, 엠브리아닉     Seungmok Ha JEJU National University

Parameter estimation Fecundity Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Fecundity - Data from the paper : non-linear regression (Age, Growth and Maturity of Sandfish, Arctoscopus japonicas in the Eastern Sea of Korea - Soo Ha Choi et al.) First, Fecundity. The Data from in this paper. Fork length (cm) No. of individuals Ovary weight (g) No. of eggs 16 3 12.3 622 17 10 15.5 859 18 6 16.6 961 19 5 17.5 1,010 20 8 19.6 1,068 21 11 23.1 1,327 22 28.6 1,642 Seungmok Ha JEJU National University

Parameter estimation Fecundity Parameter Female 0.548386813 2.547786 Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Fecundity This graph is Fecundity Results The parameter of Fecundity these. Now, We can get Fecundity equation . Parameter Female 0.548386813 2.547786 Seungmok Ha JEJU National University

Parameter estimation Maturity Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Maturity Second, Maturity. Data using in spawning season in korea, This data is just two cases. Attending spawning or not. So we using logistic regression = 파티시피케이션 Total of 963 female and 792 male sandfish samples collected during the spawning season (November ~ January, 2004 ~ 2007) : logistic regression Participation / Absent Seungmok Ha JEJU National University

Parameter estimation Maturity Parameter Female -7.303786 0.4583563 Introduction & Background Methods Parameter Estimation Results & Discussion Parameter estimation Maturity This graph is Fecundity Results The parameter of Fecundity these. Now, We can get Fecundity equation . Parameter Female -7.303786 0.4583563 15.935cm   Seungmok Ha JEJU National University

Contents Introduction & Background Methods Parameter Estimation Results & Discussion Contents Introduction & Background Methods Finally Results & Discussion. Parameter Estimation Results & Discussion Seungmok Ha JEJU National University

Natural mortality rates Introduction & Background Methods Parameter Estimation Results & Discussion Results Leslie matrix (q = 21.1086871800944) L= Now all parameter are estimated. Using this parameter, we can get, constant q, This is result of leslie matrix. And these are survival rate and natural mortality rate at each age. Age 1 2 3 4 5 6 Survival rates 0.013395 0.222053 0.329653 0.384594 0.414941 0.432545 0.443061 Natural mortality rates 4.312888 1.504841 1.109716 0.955568 0.879619 0.838069 0.814048 Seungmok Ha JEJU National University

Results L= Leslie matrix Natural mortality rates = 1.016977 Introduction & Background Methods Parameter Estimation Results & Discussion Results Leslie matrix (q = 21.1086871800944) L= Now all parameter are estimated. Using this parameter, we can get, constant q, This is result of leslie matrix. And these are survival rate and natural mortality rate at each age. Natural mortality rates = 1.016977 Seungmok Ha JEJU National University

Introduction & Background Methods Parameter Estimation Results & Discussion Results Sensitiveness of q with respect to varying embryonic survival rate (Sh) In this works, we assumed Sh is 0.9. So, we needs sensitive analysis about Sh. This graph is Sensitiveness of q with respect to varying embryonic survival rate (Sh) In this graph, If sh is more than 50% , we know what q doesn’t change much 베어링 Seungmok Ha JEJU National University

Embryonic survival rates Natural mortality rates Introduction & Background Methods Parameter Estimation Results & Discussion Results Sensitiveness of natural mortality rates with respect to varying embryonic survival rates Embryonic survival rates Natural mortality rates 1% 0.445922 10% 0.732268 50% 0.939783 70% 0.983884 90% 1.016977 100% 1.030889 And This graph is Sensitiveness of natural mortality rates with respect to varying embryonic survival rates In this graph, If sh is more than 50% , we know what Natural mortality rates doesn’t change much, too. Seungmok Ha JEJU National University

Discussion Comparison with past studies Introduction & Background Methods Parameter Estimation Results & Discussion Discussion Comparison with past studies Author Paper Natural mortality rates Kyuji WATANABE, Hideki SUGIYAMA, et al. Estimating and monitoring the stock size of sandfish Arctoscopus japonicus in the northern Sea of Japan 0.16 Jae Hyeong YANG, Sung Il LEE, Hyung Kee CHA, et al. Biomass estimation of sailfin sandfish, Arctoscopus japonicus, in Korean waters 0.482 Web-page (Japan) http://fishbase.sinica.edu.tw/PopDyn/KeyfactsSummary_3.php?ID=463&GenusName=Arctoscopus&SpeciesName=japonicus&vStockCode=477&fc=365&var_tm=2.5 0.45 Our research 1.016977 This table is Comparison with our research and past studies. Seungmok Ha JEJU National University

Discussion Comparison with past studies Introduction & Background Methods Parameter Estimation Results & Discussion Discussion Comparison with past studies Our estimate of M is higher than past studies, probably because …. Growth equation estimation Long-term equilibrium status hypothesis Our estimate of M is higher than past studies, probably because …. Firstly, Differences in the growth equation estimation. Mostly, Using 장 & 머레이 mehod when Natural mortality estimation. This method has growth equation parameters. So, the results are different. And limitation of Long-term equilibrium status hypothesis Seungmok Ha JEJU National University

Discussion Limitation and Problems of this study Introduction & Background Methods Parameter Estimation Results & Discussion Discussion Limitation and Problems of this study Results under the assumption that the Long-term equilibrium status 2) This methods for isochronal fish species 한계점 및 문제점은 이 결과가 장기적 평형상태를 가정한 결과 입니다. 그리고 도루묵과 같이 1년에 1번 산란하는 종에 대해서만 적용할 수 있습니다. Seungmok Ha JEJU National University

Discussion Future works Introduction & Background Methods Parameter Estimation Results & Discussion Discussion Future works Validation of the model results based on length frequencies from commercial catch data Consideration of density-dependency and fishing mortality Implications for fisheries management Future works. 어획데이터의 길이 빈도를 기반으로 하여 모델 결과를 검증 밀도종속적인 어획사망률 고려 어업관리정책과 연관 Seungmok Ha JEJU National University

Thank you Introduction & Background Methods Parameter Estimation Results & Discussion Thank you Thank you for listening. Seungmok Ha JEJU National University