Towards Age Determination of Southern King Crab (<i>Lithodes santolla</i>) Off Southern Chile Using Flexible Mixture Modeling

• Main Authors: Javier E. Contreras-Reyes, Freddy O. López Quintero, Alejandro A. Yáñez
• Format: Article
• Language: English
• Published: MDPI AG 2018-12-01
• Series: Journal of Marine Science and Engineering
• Subjects: finite mixture; skew-<i>t</i> distribution; southern king crab; length-frequency data; von Bertalanffy
• Online Access: https://www.mdpi.com/2077-1312/6/4/157

This study addresses the problem of age determination of the southern king crab (<i>Lithodes santolla</i>). Given that recapture is difficult for this species and, thus, age cannot be directly determined with the help of the annual marks on the shell, the von Bertalanffy growth function (vBGF) cannot be used to directly model length-frequency data (LFD). To determine age classes, some researchers have proposed using the MIX algorithm that consists of sampling realization of a finite mixture of normal (FMN) distributions for each LFD. However, normality assumption in age-length data has been questioned in several works related to fish growth analysis. For this study, we considered the biological information of the southern king crab for the period 2007&#8315;2015 and localization between <inline-formula> <math display="inline"> <semantics> <msup> <mn>50</mn> <mo>∘</mo> </msup> </semantics> </math> </inline-formula>06<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>&#8242;</mo> </msup> </semantics> </math> </inline-formula>&#8315;<inline-formula> <math display="inline"> <semantics> <msup> <mn>53</mn> <mo>∘</mo> </msup> </semantics> </math> </inline-formula>15<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>&#8242;</mo> </msup> </semantics> </math> </inline-formula> S and <inline-formula> <math display="inline"> <semantics> <msup> <mn>76</mn> <mo>∘</mo> </msup> </semantics> </math> </inline-formula>36<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>&#8242;</mo> </msup> </semantics> </math> </inline-formula>&#8315;<inline-formula> <math display="inline"> <semantics> <msup> <mn>72</mn> <mo>∘</mo> </msup> </semantics> </math> </inline-formula>18<inline-formula> <math display="inline"> <semantics> <msup> <mrow></mrow> <mo>&#8242;</mo> </msup> </semantics> </math> </inline-formula> W. We assumed that LFD could be modelled by the novel class of finite mixture of skew-<i>t</i> (FMST). Assigned age classes were used to estimate the vBGF parameters. The estimated vBGF parameters were <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mo>&#8734;</mo> </msub> <mo>=</mo> <mn>176.756</mn> </mrow> </semantics> </math> </inline-formula> cm, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>0.151</mn> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi>year</mi> </mrow> <mrow> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>&#8722;</mo> <mn>1.678</mn> </mrow> </semantics> </math> </inline-formula> year for males, and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mo>&#8734;</mo> </msub> <mo>=</mo> <mn>134.799</mn> </mrow> </semantics> </math> </inline-formula> cm, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>0.220</mn> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <msup> <mrow> <mi>year</mi> </mrow> <mrow> <mo>&#8722;</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>&#8722;</mo> <mn>1.302</mn> </mrow> </semantics> </math> </inline-formula> year for females. This study concludes that (a) FMST modal decomposition can detect a group of younger individuals at age 2, given that those individuals have LFD with a left heavy-tail and asymmetry; (b) FMST produces a better representation of LFD than the FMN model; (c) males have bigger <inline-formula> <math display="inline"> <semantics> <msub> <mi>L</mi> <mo>&#8734;</mo> </msub> </semantics> </math> </inline-formula> but grow slower than females; and (d) as expected, a high correlation exists among the vBGF estimates.