1 Introduction

Surplus production models (Schaefer, 1957), also known as biomass dynamics models (Hilborn and Walters, 1992), can easily be applied to study the population dynamics according to catch data. This type of models only considers the total biomass and ignore age structure, including recruitment, growth, and differential vulnerability to fishing gear (Hilborn and Walters, 1992). They amalgamate individual growth, recruitment, and mortality into a single model component termed surplus production (Quinn and Dersio, 1999). Biomass dynamic models have been criticized for failing to account for the size/age structure of fish populations, leading to biased and unreliable maximum sustainable yields (MSY) estimates, and even causing some fisheries management failures (Hilborn and Walters, 1992; Mace, 2001; Pauly et al., 2002). However, they are still widely used for fish stocks that have limited data available (McAllister and Kirkwood, 1998; Prager, 2002; Chen et al., 2003; Hvingel and Kingsley, 2006; Stobberup and Erzini, 2006). Biomass dynamics models with equilibrium assumptions have fallen out of favor in the 1980s, giving way to dynamic or time series fitting methods. However, their parameters are constant in time, while the environment in which the population resides undergoes continual fluctuations (Hilborn and Walters, 1992; Haddon, 2011).

In a biomass dynamics model, the surplus production is determined by two parameters, i.e., carrying capacity and intrinsic growth rate. It is only possible to estimate either of these parameters accurately if the stock size is consistently at very high levels (approaching carrying capacity) or if the intrinsic growth rate is consistently at deficient levels (Morris and Mukherjee, 2006; Tinker et al., 2021). Auxiliary information other than the time series of catch and an index of abundance data can supplement fitting these models. For instance, habitat has been commonly used to estimate the carrying capacity for fish (Sharma and Hilborn, 2001; Jacobson et al., 2005; Ayllón et al., 2012), birds (Downs et al., 2008; Donovan et al., 2012), and mammals (Gregr et al., 2008; Donovan et al., 2012).

Carrying capacity is defined as the total abundance/biomass of a fish stock that can be supported by the resources in a habitat (de Kerckhove et al., 2008). The habitat of fish species like coho salmon (Oncorhynchus kisutch) (Sharma and Hilborn, 2001), yellowfin tuna (Thunnus albacores) (Yen et al., 2012), and neon flying squid (Ommastrephes bartramii) (Tian et al., 2009) fluctuates in response to environmental conditions. Hence, carrying capacity could vary seasonally, annually, and throughout the course of a fish's lifetime to reflect typical long-term environmental conditions (Jonsson and Jonsson, 2011). The intrinsic growth rate (the growth rate when density-dependence does not play a role) can be sensitive to subtle changes in the environment and is correlated with other life history parameters such as age and size at maturity, mortality, and recruitment (Jennings, 2000). Given that carrying capacity is a balance between density-independent intrinsic growth rate and density-dependent crowding effects (α) (Haddon, 2011; Mallet, 2012), we anticipate a decline in the carrying capacity, concomitant with an increase in mortality or a decrease in birth rates, aligning with the intrinsic growth rate. Furthermore, carrying capacity as a function of habitat is influenced by environmental conditions, rendering the intrinsic growth rate that is habitat-dependent.

The habitat quality for fish species can be assessed using habitat suitability indices (HSI). The HSI model, a valuable tool in ecology (USFWS, 1981), has been widely app lied to characterize fish habitat (Rubec et al., 1999; Vadas and Orth, 2001; Vinagre et al., 2006; Tian et al., 2009). The dynamics of fish populations are affected by their habitat, as their growth, survival, and reproduction are directly affected by abiotic and biotic resources (Hayes et al., 1996). Nonetheless, incorporation of HSI model in population dynamic models remains uncommon. Jacobson et al. (2005) developed an environmentally dependent surplus production (EDSP) model that relates habitat area indices to carrying capacity and demonstrated its benefit in studying climate effects on fishery surplus production. Such models are predicated on hypothesized species-habitat correlations as opposed to statements of proven cause-and-effect relationships. A good understanding of fish habitats and their interactions with fish population dynamics is thus critical (Hayes et al., 1996; Naiman and Latterell, 2005).

Chilean jack mackerel (Trachurus murphyi), a straddling pelagic species, is widely distributed throughout the Southeast Pacific Ocean and from south-central Chile westwards to New Zealand, which is regarded as the 'jack mackerel belt' (Arcos et al., 2001; Cubillos et al., 2008; Cárdenas et al., 2009). T. murphyi has been a heavily exploited fishery resource, mainly targeted by purse seiners within the Exclusive Economic Zones (EEZ) and by international mid-water trawlers in the open waters (Li et al., 2013; Bertrand et al., 2016; Dragon et al., 2018). In Chilean waters, the habitat of Chilean jack mackerel located in the South Pacific subtropical gyre, surrounded by the northward Peru-Humboldt Current, the southwestward South Equatorial Current and the eastward Antarctic Circumpolar Current, and crossed by the eastward South Pacific Current at about 35˚S, which impacts the distribution of the main population (Yáñez et al., 2001; Ñiquen and Bouchon, 2004; Li et al., 2016). Following a dramatic increase in its annual catch since the 1970s, there was a peak catch of T. murphyi at about 4.96 million tons in 1995, followed by a significant decline until 2013. The stock started to recover as the South Pacific Regional Fisheries Management Organisation (SPRFMO) formally implemented total allowable catch (TAC) limits (SPRFMO, 2023). Multiple factors other than fishing may contribute to this abundance trajectory. For example, sea surface temperature (SST) has been widely recognized as an important environmental factor that contributes to the interannual and even multi-decade fluctuations in abundance (Espíndola et al., 2016). How the environment affects jack mackerel abundance, habitat, biological characteristics, and population genetic relationships has been the focus of a number of studies (Arcos et al., 2001; Cubillos et al., 2008; Vásquez, 2013; Li et al., 2013, 2016; Alegre et al., 2015). Comprehensive studies on how such factors affect stock assessment haven not been reported. Ignoring environmental changes may impact the accuracy of stock assessments and the effectiveness of management, especially in times of substantial decline in catches (Punt and Hilborn, 1997).

Simulation testing has often been conducted to investigate some concerns such as structural uncertainty and the impact of observation and process errors (Cao et al., 2014, 2020; Deroba et al., 2015; Perretti et al., 2020). In this study, we conducted a simulation-estimation experiment to evaluate the importance of considering the influence of environment and errors on the stock assessment of Chilean jack mackerel. Assessment models with various configurations are with respect to the assumption on the model parameters, carrying capacity and intrinsic growth rate (i.e., whether environmental impacts were considered). The primary objectives of this study were to determine the assessment performance based on different assumptions about environmental impacts using Chilean jack mackerel as an example, and to provide scientific guidance for future ecosystem-based stock assessment and management.

2 Materials and Methods 2.1 Fishery-Dependent and Fishery-Independent Data

Fishery-dependent and fishery-independent data were used in conditioning the operating model and developing the estimation model. The fishery-dependent data in terms of Chilean jack mackerel catch and catch per unit effort (CPUE) of high seas trawl fishery and coastal purse seine fishery during 1997 – 2014 (Table 1; Fig.1) were derived from Canales (2015), and Li and Zou (2015).

The fishery-independent data were the area of suitable habitat for Chilean jack mackerel estimated by HSI model (Li et al., 2016). HSI represents the capacity of a given habitat to support fish species, with minimum and maximum values of 0 and 1 denoting unsuitable and optimal habitat, respectively (USFWS, 1981). The HSI models with a combination of the arithmetic mean of suitability indices for SST, sea surface height (SSH), and chlorophyll-a concentration (Chl-a) developed by Li et al. (2016) were updated with additional data to provide the area of suitable habitat from 1997 to 2014. The estimated annual suitable habitat area was then rescaled between 0 and 1:

$ {H_t} = {A_t}/\max ({A_t}), $

(1)

where A_t is the annual suitable habitat area that HSI is no less than 0.5 in year t (Li et al., 2016), H_t is index of the suitable habitat area in year t.

2.2 Description of Habitat Based Surplus Production Model

The difference equation for the classic surplus production model is expressed by:

$ B_{t+1}=B_t+r B_t\left(1-\frac{B_t}{K}\right)-C_t, $

(2)

where B_t is the biomass in year t, r is the intrinsic growth rate, K is the carrying capacity or virgin biomass, and C_t is the annual catch in year t (Hilborn and Walters, 1992).

There is a direct association, typically a linear one, between the habitat suitability index and the carrying capacity of the habitat for a particular species (Oldham et al., 2000). We assumed that K varied annually and was proportion to the area of suitable habitat:

$ K_t=\lambda H_t . $

(3)

In addition, K is directly proportional to r, rather than independent, through the density-independent crowding effect parameter, that is K = r/α (Mallet, 2012). When given λα = β, the intrinsic rate of growth would be:

$ r_t=\beta H_t . $

(4)

Thus, the surplus production model parameterized with the habitat index can be obtained through

$ B_{t+1}=B_t+\beta H_t B_t\left(1-\frac{B_t}{\lambda H_t}\right)-C_t . $

(5)

Harvest rate (E_t) and the biological reference points (BRPs), i.e., biomass (B_MSY) and fishing mortality (F_MSY) at maximum sustained yield (MSY) can be calculated as follow:

$ E_t =\frac{C_t}{B_t}, $

(6)

$ F_{\mathrm{MSY}, t} =\frac{r_t}{2}=\frac{\beta H_t}{2} , $

(7)

$ B_{\mathrm{MSY}, t} =\frac{K_t}{2}=\frac{\lambda H_t}{2}. $

(8)

We assumed that error of CPUE, I_ti, was multiplicative and lognormal distributed with constant variance (Haddon, 2011):

$ \begin{aligned} \hat{I}_{t i} & =q_i \frac{B_t+B_{t+1}}{2} \exp ^{\varepsilon_{i i}} \\ \varepsilon_{t i} & \sim N\left(0, \sigma_i^2\right) \end{aligned}, $

(9)

$ \sigma_i^2 =\sum\limits_t \frac{\left(\ln I_{t i}-\ln \hat{I}_{t i}\right)^2}{n}, $

(10)

where i is 1 for the purse seine fishery and 2 for the high seas trawl fishery, ${\hat I_{ti}}$is the time series for predicted CPUE of fishery i in year t, ε_ti is the observation error of CPUE for fishery i in year t, σ_i is the standard deviation of ε_ti, q_i is the catchability coefficient of fishery i, and n is the number of years of CPUE time series. Log-likelihood approach was applied to estimate model parameters, and the objective function expressed as (Haddon, 2011):

$ {\text{Sum}}L = \sum\limits_i {L{L_i} = \sum\limits_i {(- \frac{n}{2}(\ln (2{\rm{ \mathsf{ π}}}) + 2\ln ({{\hat \sigma }_i}) + 1))} }, $

(11)

where SumL refers to sum of log-likelihood, LL_i is the loglikelihood for fishery i.

2.3 Operating Model Development and 'True' Population Dynamics Simulation

We developed a habitat based surplus production model as an operating model to simulate the 'true' population dynamics of Chilean jack mackerel. The flowchart Ⅰ shows how the accuracy of the operating model was conditioned by iterative fitting to reduce errors in the CPUE data (Fig.2), with detailed procedures for each step summarized in the following.

Step 1: Substitute data into the operating model

The data included actual catches, observed CPUEs and original parameters. The initial values of B₁₉₉₇, β and λ were set to 9212 thousand tons (SPRFMO, 2015), 0.30 (Zou et al., 2016) and 25000 thousand tons, respectively, in which the value of K was five times the maximum historical catch (Punt and Hilborn, 2001).

Step 2: Calculate the final predicted CPUEs

Running the operating model once could address the missing CPUE in the high seas trawl fishery in 1999 and 2000. We iteratively updated the value of parameters until the relative bias (RB) between the observed and predicted CPUEs was less than 2% (the mean square error is less than 0.001; Table 1):

$ R B_{t i}=\frac{ { predicted } \ CPUE_{t i}- { observed } \ CPUE_{t i}}{ { observed } \ CPUE_{t i}} \times 100 \% . $

(12)

Step 3: Obtain the 'true' population dynamics

The model would again be run to determine the 'true' population dynamics using the final predicted CPUE as the 'true' time series of abundance index data.

2.4 Stock Assessment Model Scenarios

Four assessment scenarios regarding the relationships between the area of suitable habitat and the two key model parameters K and r were developed in the simulation: ⅰ) 'S' model, neither K nor r is dependent on H_t, i.e., the traditional surplus production model; ⅱ) 'K' model, only K is dependent on H_t; ⅲ) 'r' model, only r is dependent on H_t; and ⅳ) 'Joint K-r' model, both K and r are dependent on H_t. This last estimation scenario matches the assumptions on K and r specified in the operating model. For the other three scenarios, there were different model misspecifications regarding K and r to serve the purpose of examining the in-fluence of environment. The fishery data and initial values of the parameters used in each simulation scenario were set identically.

2.5 Stochasticity

In addition to considering model structure and using deterministic data for the scenarios, we evaluated the effects of observation and process errors on the stock assessment. Lognormally distributed observation errors were added to the actual catch and 'true' CPUE to generate the 'observed' catch and abundance index. Annual biomass was also subjected to lognormally distributed random errors. The coefficient of variance (CVs) of annual catch, CPUE, and biomass was set to 0.05, 0.1, and 0.05, respectively. For each of the four stochastic simulation scenarios, 100 sets of 'observed' catch and CPUE data were generated with process

errors included in the annual biomass (Fig.2Ⅱ):

$ \tau^2=\ln \left(C V^2+1\right), $

(13)

$ \begin{array}{l} {B_{t + 1}} = ({B_t} + r{B_t}(1 - \frac{{{B_t}}}{K}) - {C_t}){\exp ^{{\mu _t}}} \hfill \\ {\mu _t} \sim N(0, {\tau ^2}) \hfill \end{array}, $

(14)

where μ_t is the process error of biomass in year t, τ is the standard deviation of μ_t.

We used relative error (RE) and relative bias to measure the disparity between the operating model and assessment models (Guan et al., 2013; Cao et al., 2014):

$ R E_x=\frac{1}{2} \sum\limits_{t=1997}^{2014} \sqrt{\left(\frac{ { Scen }_{x, t}- { Simu }_{x, t}}{ { Simu }_{x, t}}\right)^2}, $

(15)

where x refers to biomass, exploitation rate and reference points, Scen_t is the estimated quantity of year t in the stock assessment model, Simu_t is the simulated value in year t from the operating model. For each of the 100 simulation runs in each year, RB and boxplot were used to quantify the influence of errors and model uncertainty:

$ R B_{x, t}=\frac{ { Scen }_{x, t}- { Simu }_{x, t}}{{Sim}_{x, t}} \times 100 \%. $

(16)

The data collected were statistically analyzed using software packages, e.g., MS Excel and R.

3 Results 3.1 Simulated 'True' Population Dynamics

CPUEs of two fisheries were simulated in this study, one for purse seine fishery and the other for high seas trawl fishery, which had trends consistent with the 'true' biomass (Figs.1, 3). The simulated 'true' biomass and catch rate were relatively high in 1997 and then declined but maintained stable during the next five years, after which they continuously decreased and dropped to the lowest level in 2011. And then Chilean jack mackerel stock showed signs of recovery with the biomass increasing to 1303 thousand tons in 2014, and approaching the levels at 2009 (Fig.3). Estimated 'true' B_MSY followed the trend of habitat area and varied between 10486 and 12400 thousand tons, 3 to 16 times the level of the biomass, which indicated that Chilean jack mackerel stock was overfished.

Harvest rate derived from the simulation fluctuated sharply. It showed a downward trend during the period of 1997 – 2000, and then increased sharply until 2009. Afterwards, harvest rate dropped and reached the lowest value in 2013. The 'true' F_MSY changed yearly and was less than the annual harvest rate except in 2013, i.e., overfishing did not occur only in the penultimate year.

3.2 Deterministic Model Scenarios

The CPUEs for the purse seine and trawl fisheries obtained from the four assessment scenarios were broadly similar, which were lower than the 'true' CPUE in the first a few years but closely matched after 2005 (Fig.1). K-r assessment model matched the operating model, so estimates such as biomass, harvest rate and reference points were identical with the true estimates when using the actual catch data and simulated CPUE data without errors. However, the outputs of the other three scenarios with model structure misspecification presented some biases (Fig.4). REs in the S model were higher than those in the other two misspecification models, suggesting that estimates from the model were more biased. On the contrary, r model was more accurate than other models because REs for biomass, B_MSY, harvest rate and F_MSY were 2%, 55%, 2% and 4%, respectively, which were the lowest among all the scenarios besides the K-r model (Table 2). REs for B_MSY were significantly biased, especially in the S and K models, which indicated that B_MSY was sensitive to ignoring the habitat effect on carrying capacity and intrinsic growth rate in the assessment.

RBs of biomass, harvest rate and reference points and model parameters (K and r) were shown in Fig.5. The RBs for biomass were almost positive in all mismatched scenarios, with variations of 3% – 7% in the S model, 2% – 5% in the K model, and less than 1% in the r model, so that annual biomass was overestimated in most cases, apart from 1999 – 2000 in the r model. On the other hand, the opposite was true for the harvest rate in the r model. The absolute value of RBs for harvest rate achieved the maximum in the S model, followed by the K and r models, with seemingly minor differences between the operating model and the three misspecified models. B_MSY was underestimated with negative RB values, in which it ranged from −29% to −41% in the S model, fixed at −33% in K model and ranged from −13% to −2% in the r model in all years but 2000 (Fig.5). Based on RBs value 0.41% for the r model and the same trend of fluctuating RBs between −4% and 14% for the S and K model, F_MSY values were overestimated in some years and underestimated in the other years (e.g., 2007 – 2009 and 2013 in the K model scenario; Fig.5).

Estimates of model parameters, biomass, harvest rate and reference points were sensitive to model structures or ignoring the impacts of habitat in the assessment. The absolute value of RBs for these estimates in the S model were higher than those in the K and r models, suggesting that estimates from the S model were more biased, K model moderate and r model relatively robust when habitat effects were entirely or partially excluded from the assessment.

3.3 Stochastic Model Scenarios

Observation and process errors were added in the stochastic scenarios to evaluate their impacts on the stock assessment and compare the performance of assessment models. Under the circumstance of stochastic scenarios, REs increased dramatically even in the K-r model, although they appeared to decrease for B_MSY in the other three models (Table 2). REs for F_MSY in the r and K-r model were greater than those in the S and K model, which indicated that F_MSY as well as the intrinsic growth rate were sensitive to habitat changes when observation and process errors were added to the assessment. In general, the REs declined from the S model to K-r model, demonstrating that K-r model was more reliable than the mismatched model scenarios.

The variations after the inclusion of errors to the data were reflected in the RBs for each year, with notably larger values of median RBs for biomass and harvest rate. Biomass was overestimated as suggested by the positive median RBs in all scenarios, ranging from 10% to 17% for the S model, 10% to 15% for the K model, and 7% to 11% for the r model as well as K-r model, as was also found for the absolute value of median RBs for harvest rate. As for B_MSY, it was underestimated in the S and K models with median RBs of −36% to −25% and fixed at −28%, respectively. However, it was overestimated in the r and K-r models, with the former being more than 0 except for 1997 and 2007 – 2013 and the latter fixed at 5%. The median RBs for F_MSY were quite similar at −8% in the r and K-r models, and the errors did not cause significant bias in the other two models (i.e., the median RBs were centered around zero); however, overall F_MSY tended to be underestimated (Fig.6).

4 Discussion

In this study, we have investigated the impact of suitable habitat area on biomass dynamics and estimates of reference points of Chilean jack mackerel using a surplus production model. The model parameters, i.e., carrying capacity and intrinsic growth rate, were linked to suitable habitat area. The hypothesis is that biomass as well as reference points vary with environmental conditions since the suitable habitat area estimated by the HSI model depends on environmental factors. The results indicated that ignoring habitat changes resulted in significant biases in the estimates of annual biomass, harvest rate and MSY-based reference points, which were amplified by the inclusion of observation and process errors (Fig.5). In comparison to other parameters and biological reference points, B_MSY with environmental influences neglected and F_MSY with observation and process errors neglected were more sensitive. Scenarios that fully considered environmental factors, with or without observation and process errors, would have brought the assessment results more in line with the 'true' population status of Chilean jack mackerel (Table 2).

Climate-driven changes in ocean conditions have direct non-lethal effects on fish growth and reproduction, in particular for pelagic fishes (Klyashtorin, 1998; Punt et al., 2021). The life history characteristics of these fishes include rapid growth, early maturity, short lifespans and small body sizes (Blaxter and Hunter, 1982; Tsikliras et al., 2019). Recruitment that determines the stock abundance depends on a large extent of conditions in the environment and/or reproductive stocks at different temporal and spatial scales (Tourre et al., 2007). Environmental change would inevitably alter habitat suitability and, in turn, population distribution and abundance patterns (Brander, 2010). For example, both chub mackerel (Scomber japonicus) and Pacific saury (Cololabis saira) in the Western Pacific experience negative effects of reduced suitable habitat area during El Niño events (Li et al., 2014; Chang et al., 2018). In the case of Chilean jack mackerel, a significant positive correlation was found between HSI and PDO indices (Feng et al., 2022), and seasonal variation in suitable habitat in latitude was consistent with the 15℃ isotherm (Li et al., 2016). Cubillos et al. (2008) suggested that suitable spawning conditions for Chilean jack mackerel were associated with SST, which had been noted as one of the major abiotic factors affecting almost all biological processes. Espíndola et al. (2016) found that incorporating SST into the stock-recruitment relationship of Chilean jack mackerel might be a promising way to examine the effects of fishing and environment conditions simultaneously. The development of an ecosystem-based model for Chilean jack mackerel to determine the extent of environmental drivers on the stock dynamics will enhance our comprehension of this population (Dragon et al., 2018).

Incorporation of various forms of auxiliary information into the surplus production model is straightforward and only involves an additional criterion onto the goodness of fit for the time series alone (Hilborn and Walters, 1992). Previous researches developed an environmental dependent surplus model to access the status of some pelagic species, but they only focused on water temperature or its proxy (Jacobson et al., 2005; Wang et al., 2016, 2018). It is well known that not just temperature, but many other environmental variables have an impact on population dynamics (Lehodey et al., 2006; Maunder and Watters, 2003). Therefore, the inclusion of a single environmental factor in the assessment model is not likely to accurately reflect the actual fishery resource status. Despite being directly based on multiple factors, the current model is only able to comprehend the composite effect of the environment on catches (Yi et al., 2017). As it combines multiple ecologically relevant environmental variables, the HSI model has the natural advantage of being able to be incorporated into the assessment model (Tanaka and Chen, 2016). Tanaka et al. (2019) used HSI to inform the recruitment dynamics of the American lobster in the size-structure model, demonstrating the potential to improve population assessment. In this study, we used the HSI model of suitable habitat area as an indicator of intrinsic growth rate and carrying capacity to investigate the suitability of different scenarios considering environmental factors to assess the resource status of Chilean jack mackerel. Our analysis found that bias typically worsened as the difference between parameters such as biomass or harvest rate and 'true' stock dynamics increases. The four assessment scenarios, from i to iv, the REs (between estimates using simulation and operating model) for each parameter and biological reference point showed a decreasing trend (Table 2), which indicated that the population dynamics of Chilean jack mackerel were influenced by environmental conditions in addition to fishing activities. With habitat-based models, dynamic BRPs that help to adjust annual regulations for fisheries management were obtained. However, resource status could not be predicted since the future proportion of area with favorable environmental conditions is unknown.

The carrying capacity of an environment is determined not only by the abundance and distribution of limited resources, but also by the methods of individuals competing for the resources (Ayllón et al., 2012). The relative role of intrinsic density-dependent factors (e.g., interand intraspecific competition, predation) that can maintain relatively stable and regular fluctuations in populations, and extrinsic, density-independent factors (environmental change) associated with irregular changes in population size is critical for population dynamics regulation (Lorenzen and Enberg, 2002; DingsØr et al., 2007). Mortality, growth and reproduction in marine fish populations are susceptible to density-independent effects and the action of density-dependent processes (Lorenzen and Camp, 2019). These processes do not occur uniformly throughout the life cycle of fishes, and density-dependent regulation determined by the size of the habitat may occur both early and late in the life history (Andersen et al., 2017). This study showed that habitat area is positively correlated with both K and densityindependent r (Figs.4, 7), implying that the higher abundance requires more space to be occupied, thus favoring the growth of Chilean jack mackerel.

Process errors in this study refer to the underlying stochasticity in the population dynamics resulting from factors other than the environment variability that cannot be describe by a surplus production model, while observation errors are certainly unavoidable when collecting data (Newman, et al., 2006). The results suggest that ⅰ) when the assessment model structure matches the operating model, observation and process error leads to an overestimation of biomass and B_MSY, as opposed to harvest rate and F_MSY; ⅱ) when the assessment model structure is mismatched, the bias of each estimate other than B_MSY is significantly increased; ⅲ) the combined effect of observation and process error is greater than the environment acting alone (Table 2). Therefore, all types of errors need to be recognized and appropriately quantified before models are utilized (Maunder and Watters, 2003). In any scenario, good data quality can help mitigate certain biases, e.g., estimates of life history parameters can be improved by biological studies prior to their input (Zhang et al., 2015).

It is generally recognized that ignoring uncertainty associated with estimates of current stock size and potential productivity in assessment may lead to a biased view of stock status (Francis and Shotton, 1997). This is most likely to happen when the estimated biomass or harvest rate are close to the reference points. If harvest rate is underestimated and biomass is overestimated, the assessments might be overly optimistic, leading to mismanagement and stock collapse (Bouch et al., 2021). Such misjudgment can further lead to risks in decision-making, which exist objectively (Hilborn et al., 1993; Sethi, 2010). The conclusion above coincides with the final-year assessment results of the S and K models in this study. However, in the scenario where only observations and process uncertainty were studied, the interpretation for the status of Chilean jack mackerel stock was consistent with the 'true' one (Figs.3, 7). Without comparing assessment results under high and low errors, the risks in management will increase with random error (Zou et al., 2016). For this reason, it needs to be minimized through precautionary principle and ecosystembased management measures that can achieve sustainable use of fisheries resources (Kirkwood and Smith, 1996).

Combining climate-driven changes in suitable habitats with key parameters in assessment models can provide potential improvements in stock assessments. However, it should be noted that even in previous studies the habitatK relationship was developed as a linear relationship, though it is arbitrary. Future researches should simulate multiple different types for this relation, which is important for exploring the development of habitat-based assessment models. Additionally, the effects of errors should also be fully considered in stock assessments of Chilean jack mackerel. Our approach could be extended in the future to other similar pelagic fish stocks affected by environmental changes.

Acknowledgement

This work was supported by the National Key Research and Development Program of China (No. 2019YFD0901 404).

Author Contributions

All authors contributed to the study conception and design. Data collection was performed by Xiaorong Zou. Modeling and data analysis performed by Yangming Cao. The draft of the manuscript was written by Yangming Cao and Gang Li, and supervised by Xinjun Chen. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data Availability

All data generated and analyzed during this study are included in this published article and its additional files.

Declarations

Ethics Approval and Consent to Participate

This article does not contain any studies with human participants or animals performed by any of the authors.

Consent for Publication

Informed consent for publication was obtained from all participants.

Conflict of Interests

The authors declare that they have no conflict of interests. Xinjun Chen is one of the Editorial Board Members, but he was not involved in the journal's review of, or decision related to, this manuscript.