Uncertainties in the estimates of natural mortality due to climate change and their impact on yield per recruit models

Abstract

Incorporating the uncertainties arising from climate change is essential for assessing fish stocks and, consequently, for the effective management of these stocks. This study addresses the uncertainties related to natural mortality (M) in fish caused by climate change, positing that these uncertainties are linked to the life strategies of the exploited species. To mathematically model these uncertainties, we employed fuzzy set theory to evaluate six commercially important fish species. Our results indicated a decrease in yield per recruit (Y/R) across all species, with a gradual decline observed between the current scenario and the most pessimistic scenario outlined in the 6th Intergovernmental Panel on Climate Change (IPCC) report. Furthermore, the model generated intervals for Y/R and their respective mortalities per fishery (F), which yielded biological reference points (BRPs) for fisheries management. Species that exhibit opportunistic life history traits displayed greater variability in the decline of Y/R. Given that these species are more vulnerable to risks associated with uncertainty, we recommend adopting more conservative scenarios when selecting BRPs for them. This study contributes to fisheries management by addressing the inherent uncertainties of the effects of climate change and refining inventory management based on the different life history characteristics of species.

Key words
Stock assessment models; Fish mortality; Fisheries management; Fuzzy logic

INTRODUCTION

The increase in the global average surface temperature of the planet, due to increased levels of greenhouse gas (GHG) emissions (mainly carbon dioxide) (Florides & Christodoulides 2009, Macedo & Madaleno 2023), can exert various effects on aquatic organisms (Albouy et al. 2020, Alfonso et al. 2020). The effects of climate change on fish and increases in temperature include biological changes in individuals that affect populations, which induce changes in abundance, migration patterns, geographical distribution and key features of fish life history along a latitudinal gradient (Jeppesen et al. 2014, Alfonso et al. 2020).

In general, fish species that inhabit low-latitude aquatic environments may have a reduced maximum body size (Griffiths 1997, Jeppesen et al. 2010), may grow faster, have a shorter life expectancy, abbreviated sexual maturation and a lower amount of energy allocated for reproduction than species that inhabit higher latitudes (Blanck & Lamouroux 2007). At the individual level, an increase in temperature implies physiological responses, with direct effects on the natural mortality (M) of fish (Freitas et al. 2013, Barros & Albernaz 2013, Jeppesen et al. 2014, Ojea et al. 2017, Andrade et al. 2019, Campos et al. 2018, 2023, Val 2019, Oremus et al. 2020, Alfonso et al. 2020). As a consequence, an increase in temperature, as a result of climate change, can cause changes in global fishery abundance and productivity (Cheung et al. 2010, Serpetti et al. 2017, Gaines et al. 2018, Cisneros-Mata et al. 2019, Free et al. 2019), thereby magnifying the negative effects of fishing pressure, which is currently above the sustainable limits for most global fish stocks (Cisneros-Mata et al. 2019).

Fishery stock assessment models identify the state of exploitation of natural stocks and estimate optimal levels for their sustainable exploitation. These models are the basis for the proposition of strategies to maintain the sustainability of stocks, thereby preventing fisheries from reaching excessive levels of intensity (Ovando et al. 2022). Most of these models require the biological data of the population, especially growth parameters and mortality rates. The mathematical models use, as basic premises, assumptions that methodologically or biologically are not always fulfilled, such as random samples, constant recruitment from year to year, distribution of individuals in discrete cohorts and constant mortality over time (King 1995, Sparre & Venema 1997), which can lead to biased estimates. In addition, these models do not incorporate the uncertainties that involve the assessment of fish stocks and can hinder decision-making regarding management strategies (Schrank 2007).

Natural mortality (M) is a parameter of stock valuation models that can present wide spatial and temporal variability, and the uncertainties surrounding its estimates can bias the estimates of fishing yield at a given time (Dutil & Lambert 2000, Quiroz et al. 2010, Kenchington 2014). The model generated by Campos et al. (2023) showed that the increase in temperature causes an increase in uncertainties in the estimates of M for freshwater and marine species, with different intensities according to the life strategy. Understanding how these uncertainties, around the estimates of M induced by temperature increase, are transferred to stock assessment models is fundamental for the foundation of robust fishery management strategies (Brodziak et al. 2011). The incorporation of these uncertainties brings robustness to fishery management when it is based on methodologies, such as fuzzy logic, that enable mathematical modeling of uncertainty (Zadeh 1965, Barros & Bassanezi 2010). Using fuzzy logic, intervals or fuzzy sets can be built for the parameters, which have different degrees of membership (Bitar et al. 2016).

In addition, the patterns found in the uncertainties of M when driven by global climate change can be used in more holistic approaches to the assessment and management of newly exploited fishery resources or those with limited biological data (King & McFarlane 2003, Campos et al. 2023). One way to address these information gaps is to characterize groupings of commercially exploited species based on known life history characteristics and use these groupings to characterize trends in fishing yield levels over various time scales and in different climate scenarios (King & McFarlane 2003). Conceptual management scenarios based on life history characteristics can therefore be used for the management of newly exploited species to proactively mitigate the unpredictable impacts of climate change on fish stocks.

Several studies have examined the uncertainties related to natural fish mortality (Quiroz et al. 2010, Gaertner 2015, Legault & Palmer 2016, Chen et al. 2018, Campos et al. 2023) as well as the effects of climate change on global fish production (Cheung et al. 2010, Serpetti et al. 2017, Gaines et al. 2018, Cisneros-Mata et al. 2019, Free et al. 2019). However, there are no reports on how yield per recruit models relate these two factors. Therefore, this study aimed to evaluate how uncertainties in natural mortality due to global climate change affect yield per recruit models. Taking into account the uncertainty in M estimates presented by Campos et al. (2023) and the scenarios predicted by the 6^th Intergovernmental Panel on Climate Change report (IPCC 2014), specifically the representative concentration pathways (RCPs) 2.6, 4.5, 6.0, and 8.5, we hypothesize that opportunistic and freshwater species will exhibit greater uncertainty in $Y/R$ estimates. Natural mortality was chosen to be fuzzified due to its relationship with temperature, a variable straightly related to climate change (Figure 1).

Figure 1
Conceptual figure to show the fuzzification procedure of biological parameters, based on RCP scenarios, to identify uncertainties in Y/R estimates.

MATERIALS AND METHODS

Data collection

Marine and freshwater fish species with different life strategies were selected and then grouped into three categories: equilibrium strategist, seasonal, or opportunistic, according to the classification of Winemiller (1989) and Winemiller & Rose (1992), with all species being important for commercial fishing. The freshwater species were Cichla temensis (equilibrium strategist), Prochilodus nigricans (seasonal strategist) and Triportheus angulatus. Although Triportheus angulatus is not considered a strictly opportunistic species, it exhibits several key traits aligning with this life strategy. These include a small body size of 20 to 24 cm (Malabarba 2004), low fecundity, single spawning (Araújo et al. 2012), and an absence of parental care. Additionally, its population dynamics are characterized by high growth (k) and high natural mortality (M) rates, low theoretical maximum ($L_{\infty }$) and first maturation ($L_{50}$) lengths, and a short lifespan ($T_{\max }$) (Prestes et al. 2010, Araújo et al. 2012). The species from marine environments were Cynoscion acoupa (equilibrium strategist), Lutjanus synagris (seasonal strategist) and Sardinella brasiliensis (opportunistic strategist).

The population parameters of each species, which served as input data for the estimation of the fuzzy sets of yield per recruit ($Y/R$) models, were extracted from the literature. For each species, parameters from a single study were used to avoid possible errors resulting from methodological bias. Some parameters were not available, but it was possible to estimate them (Table I). Fuzzy sets associated with the natural mortality (M) parameter were used for each IPCC scenario: RCPs 2.6, 4.5, 6.0 and 8.5 (IPCC 2014), estimated in Campos et al. (2023). Although the 6^th IPCC report introduced the shared socioeconomic pathways (SSPs), this study utilized the representative concentration pathways (RCPs) from 5^th report (IPCC 2014). This methodological choice ensures consistency and direct comparability with historical climate impact data and previous fishery assessment models, which extensively rely on the RCP framework. Nevertheless, the RCPs evaluated in this study map directly onto the updated SSP framework based on their equivalent radiative forcing targets for the year 2100: the mitigation scenario RCP 2.6 aligns with the sustainable scenario SSP1-2.6; the stabilization scenarios RCP 4.5 and RCP 6.0 correspond to the intermediate SSP2-4.5 and SSP4-6.0, respectively; and the pessimistic RCP 8.5 mirrors the fossil-fueled development scenario SSP5-8.5 (IPCC 2021). Consequently, the patterns of uncertainty regarding natural mortality and yield per recruit demonstrated here remain directly applicable to the current SSP-based climate impact literature.

DATA ANALYSIS

Age of fishery recruitment (${\mathit{𝑻}}_{\mathit{𝒓}}$)/age of first catch (${\mathit{𝑻}}_{\mathit{𝒄}}$)

Recruitment ($T_r$) and first capture ($T_c$) ages were computed using an adaptation of the Von Bertalanffy growth equation (von Bertalanffy 1938, King 1995, Sparre & Venema 1997):

$$T_r=t_0-\left(\frac{1}{k}\right)\cdot \ln (1-\frac{L_r}{L_{\infty }})$$

$$T_c=t_0-\left(\frac{1}{k}\right)\cdot \ln (1-\frac{L_c}{L_{\infty }})$$

Where: k: individual growth coefficient; $L_{\infty }$: theoretical maximum length; $L_c$: first capture length; $L_r$: average length of recruitment; $t_0$: theoretical age at zero length.

The parameter $t_0$ was considered zero since this parameter has no biological connotation, being a mathematical correction for the adjustment of the growth curve (Sparre & Venema 1997). For the first capture length ($L_c$) and the average recruitment length ($L_r$), $L_c=L_r$ was assumed based on the smallest length class that is fully represented in the capture data (King 1995, Sparre & Venema 1997).

Asymptotic maximum weight (${\mathit{𝑾}}_{\mathit{\infty }}$)

The asymptotic maximum weight ($W_{\infty }$) was estimated according to the following equation (Sparre & Venema 1997):

$$W_{\infty }=a\cdot L_{\infty }^b$$

Where $L_{\infty }$ is the theoretical maximum length, and a and b are weight-to-length ratio parameters.

Total mortality (Z)/fishing mortality (F)

Since total mortality (Z) is the sum of natural mortality (M) and fishing mortality (F) (Sparre & Venema 1997), it follows that:

$$Z=M+F$$

However, the total mortality was assumed as a fuzzy number ($\tilde{Z}$), due to the natural mortality (M) being replaced by the fuzzy number ($\tilde{M}$), and the fishing mortality (F) was maintained in its crisp form, where possible values for F from 0 to 6 (depending on the species) were adopted.

Elements of fuzzy set theory

In this section, we present some definitions from fuzzy set theory (Zadeh 1965, Pedrycz & Gomide 1998, Barros & Bassanezi 2010) that were used in the construction of the Beverton-Holt mathematical model to obtain the (fuzzy) yield-per-recruit calculation.

Fuzzy sets constitute a natural generalization of classical set theory by allowing the membership of an element in a set to be expressed in a gradual manner, as an alternative to classical mathematics, which treats membership in a binary way (Zadeh 1965). This feature makes them particularly suitable for modeling phenomena that are characterized by uncertainty. Instead of classifying an element only as belonging or not belonging to a set, fuzzy set theory allows degrees of membership in the set, with values in the interval $[0,1]$. This representation adds greater fidelity to the complexity of many real-world systems, such as those involving the exploitation of natural fishery resources by humans.

Fuzzy sets and natural mortality

Let X be a universal set (the set of interest for the application). In this paper, we assume $X=\mathbb{ℝ}$, the set of real numbers. A fuzzy subset A of X is characterized by a membership function ${\mu }_A:X\to [0,1]$ that assigns to each element $x\in X$ a value ${\mu }_A(x)$, indicating the degree to which x belongs to or is associated with the set A. Formally, a fuzzy subset A of X can be described as follows:

$$A=\{(x,{\mu }_A(x))\mid x\in X\}$$

By defining natural mortality (M) as a fuzzy set, the model is able to process linguistic terms (e.g., "moderately high mortality") and transform them into operable mathematical values. In addition, this variable depends on factors that are not binary. Variables such as water temperature, prey availability, and senescence vary along "degrees" and influence the range of values that M may assume for each species. In general, natural mortality is treated as a fuzzy set due to the imprecise and biologically variable nature of this parameter (Maunder et al. 2023), especially given the limited availability of data and the risk of introducing false precision through a single-point estimate. Finally, this approach permits ecological variability and epistemic uncertainty to be incorporated in a formal manner, while maintaining consistency with fish stock dynamics.

Concept of $\mathit{𝜶}$-level

Let A be a fuzzy subset of X and $\alpha \in [0,1]$. The $\alpha$-level of A is the classic subset $A^{\alpha }$ of X defined as follows:

$$A^{\alpha }=\{x\in X:{\mu }_A(x)\ge \alpha \},\mathrm{\text{for}}0<\alpha <1$$

When $\alpha =0$, $A^0$ is defined topologically as the closure of the support set $\mathrm{\text{supp}}(A)=\{x\in X:{\mu }_A(x)>0\}$, that is, the smallest closed set containing $\mathrm{\text{supp}}(A)$ (Barros et al. 2017).

$\alpha$-levels are considered central tools for the analysis of fuzzy sets. They function as a conservatism filter: a high $\alpha$ (close to 1) represents a scenario of low uncertainty, in which the stock manager makes decisions based only on what is "almost certain"; a low $\alpha$ (close to 0) represents a scenario of stress or precaution, in which the interval is wide and includes extreme mortality values (such as a predation surge). Projecting maximum yield per recruit using a low $\alpha$-level provides a means of evaluating the resilience of the stock under severe uncertainty. Methodologically, the use of fuzzy sets allows natural mortality to be treated through the construction of plausible values for M, while the use of $\alpha$-levels makes it possible to propagate uncertainty throughout the Beverton-Holt equation, enabling the construction of a possibility interval for yield per recruit with corresponding degrees of membership.

Zadeh’s extension principle

Zadeh’s extension principle makes it possible to extend concepts from classical mathematics to the fuzzy setting. For example, let $f:X\to Z$ be a classical function. The extension principle allows the image of a fuzzy subset A of X to be computed by f to a function $\hat{f}$ for the proper treatment of fuzzy subsets of X. In this case, the image $\hat{f}(A)$ is defined as a fuzzy subset of Z. The extension principle specifies how to compute this image, that is, how to define the corresponding membership function. In this study, we use the classical Beverton-Holt yield-per-recruit ($Y/R$) model (Beverton & Holt 1957) in its standard (crisp) form and evaluate the propagation of uncertainty when computing $Y/R$ values with natural mortality (M) treated as a fuzzy set. The parameter M is typically treated as a deterministic value; thus, the inherently uncertain and variable nature of the natural processes contributing to non-fishing mortality such as predation, temperature variation, environmental changes, and others is ignored. The formal definition is as follows:

Let $f:X\to Z$ be a function and A a fuzzy subset of X. The Zadeh extension of the function f is the function $\hat{f}$ whose value $\hat{f}(A)$ is a fuzzy subset of Z with membership function defined as follows:

$${\phi }_{\hat{f}(A)}(z)=\{\begin{array}{cc}\underset{f^{-1}(z)}{\max }{\phi }_A(x),\hfill & \mathrm{\text{if}}{f}^{-1}(z)\ne \varnothing \hfill \\ 0,\hfill & \mathrm{\text{if}}{f}^{-1}(z)=\varnothing \hfill \end{array}$$

In this expression, $f^{-1}(z)=\{x\in X\mid f(x)=z\}$ denotes the preimage of z.

In conclusion, fuzzy sets permit the representation of phenomena with imprecise boundaries. $\alpha$-levels provide a tool for analyzing and decomposing this imprecision, and Zadeh’s extension principle enables the application of functions and operations in a fuzzy setting. In this way, all uncertainty in M is coherently propagated through the model, ensuring that more plausible values of M yield more plausible values of $Y/R$ while extreme results are appropriately moderated by the degree of membership of the input. With this integration (fuzzy sets, $\alpha$-levels, and Zadeh’s extension principle), a mathematical framework is obtained that is capable of transparently and reproducibly representing the uncertainty inherent in natural mortality. This approach strengthens the scientific credibility of the results and aligns stock assessment with modern practices for modeling under uncertainty, which is particularly relevant in tropical environments and data-limited fisheries.

Fuzzy set for estimating yield per recruit (Y/R)

In the modeling of the fuzzy set associated with the yield per recruit, we used the algebraic resources inherent to the theory of fuzzy sets: sum of fuzzy numbers, Zadeh extension and alpha-level of fuzzy sets (Zadeh 1965, Barros et al. 2017, Campos et al. 2023).

The dynamic system for variation of yield per recruit investigated in this study is described by the differential equation:

$$\{\begin{array}{c}\frac{dY}{dt}=F\cdot N(t)W(t)\hfill \\ Y(t_c)=Y_{t_c}\hfill \end{array}$$

Where $N(t)$ is the number of individuals subject to capture ($t\ge t_c$); $W(t)$ is the average weight of the individual at time t; $Y(t)$ is the yield obtained from fish stock; F is the fishing mortality rate; $Y_{t_c}$ is the yield at time $t=t_c$; $t_c$ is the first capture age of that species; $t_f$ is the age at end of life.

Let $H:[t_c,t_f]\to \mathbb{ℝ}$ be the function defined by $H(t)=F\cdot N(t)\cdot W(t)$. It is possible to guarantee (Barros 1997) that problem (8) has a unique solution if, and only if, H is continuous and satisfies the following integral equation:

$$Y(t)=Y_{t_c}+{\int }_{t_c}^{t}H(t)dt,t\in [t_c,t_f]$$

The expressions for $N(t)$ and $W(t)$, used in the definition of Equation (8), are given by $N(t)=R{e}^{-M(t_c-t_r)}{e}^{-Z(t-t_c)}$ and $W(t)=W_{\infty }(1-e^{-k(t-t_0)}{)}^b$ respectively. The first expression considers the number of survivors at the age of the first catch ($t_c$) and, in the second, the estimate of nonlinear growth of the species (von Bertalanffy 1938).

The parameters R, M, k, $W_{\infty }$ and b denote the initial number of individuals in the cohort, the natural mortality, the growth rate, the asymptotic weight and the allometric parameter of the length/weight ratio, respectively.

With the help of the expressions $N(t)$, $W(t)$ and Equation (9), the solution of the problem of initial value (8) was calculated, obtaining the yield for an instant $t\in [t_c,t_f]$. For a full life cycle of the species, this yield results in the following value:

$$Y_{t_f-t_c}=Y(t_f)-Y_{t_c}=FR{e}^{-M(t_c-t_r)}{W}_{\infty }{\int }_{t_c}^{t_f}{e}^{-Z(t-t_c)}(1-e^{-k(t-t_0)}{)}^{b}dt$$

The estimate for the yield per recruit, for a complete cycle of the species, can be obtained by dividing (10) by R ($R>0$), according to (11):

$$\frac{Y_{t_f-t_c}}{R}=F{e}^{-M(t_c-t_r)}{W}_{\infty }{\int }_{t_c}^{t_f}{e}^{-Z(t-t_c)}(1-e^{-k(t-t_0)}{)}^{b}dt$$

In the following, we will represent $Y_{t_f-t_c}$, $t\in [t_c,t_f]$, simply by $Y_t$.

In the original solution proposed by Beverton & Holt (1957), for the calculation of yield per recruit, $b=3$ was assumed in Equation (11). However, as "b" varies with environmental conditions, this parameter should not always be assumed equal to 3 (Braga 1997). Therefore, we used the values provided in Table I. To model the fuzzy set associated with the yield per recruit, we analyzed the solution of the model (8), when the parameter M present in the value of $N(t)$ is a fuzzy set (Campos et al. 2023). Equation (12) synthesizes the fuzzy differential equation model with the necessary adaptations for incorporation of $\tilde{M}$:

$$\{\begin{array}{c}\frac{dY}{dt}=\hat{H}(t,\tilde{M})\hfill \\ Y(t_c)=Y_0\hfill \end{array}$$

In this equation, $\hat{H}:[t_c,t_f]\times \mathcal{ℱ}(\mathbb{ℝ})\to \mathcal{ℱ}(\mathbb{ℝ})$ denotes the Zadeh extension of the function H and $\mathcal{ℱ}(\mathbb{ℝ})$ the family of fuzzy sets of real numbers.

The fuzzy solution ${\tilde{Y}}_t(Y_0,\tilde{M})$ of equation (9) was obtained with the help of the alpha-levels $[{\tilde{Y}}_t(Y_0,\tilde{M}){]}^{\alpha }$, $\alpha \in (0,1]$, $t\in [t_c,t_f]$, since every fuzzy set is completely defined with the knowledge of its alpha-levels (Barros et al. 2017, Campos et al. 2023). For this, the solution ${\tilde{Y}}_t={\tilde{Y}}_t(Y_0,\tilde{M})$ of (12) combined with the result of Theorem 2.11 found in Barros et al. (2017), resulting in the following identity:

$$[{\tilde{Y}}_t(Y_0,\tilde{M}){]}^{\alpha }=Y_t(Y_0,[\tilde{M}{]}^{\alpha })$$

In equation (13), $Y_t$ denotes the Zadeh extension of the solution $Y_t$. The alpha level for a complete life cycle of the species, relative to the total yield:

$$[{\tilde{Y}}_{t_f}(Y_0,\tilde{M}){]}^{\alpha }=Y_{t_f}(Y_0,[\tilde{M}{]}^{\alpha })$$

Henceforth, we will represent ${\tilde{Y}}_{t_f}(Y_0,\tilde{M})$ by $\tilde{Y}$. Identity (14) suggests a methodology for fuzzy set analysis $\tilde{Y}/R$ for yield per recruit, based on alpha-levels of $\tilde{M}$. In effect:

$${\left[\frac{\tilde{Y}}{R}\right]}^{\alpha }={\left[\frac{{\tilde{Y}}_{t_f}(Y_0,\tilde{M})}{R}\right]}^{\alpha }=\frac{1}{R}[{\tilde{Y}}_{t_f}(Y_0,\tilde{M}){]}^{\alpha }=\frac{Y_{t_f}(Y_0,[\tilde{M}{]}^{\alpha })}{R},\alpha \in (0,1],R>0$$

Finally, the term on the right in identity (15) can be calculated with the aid of Equation (9), calculating scenarios for each alpha-level $M^{\alpha }$, with $\alpha \in (0,1]$. Three alpha-levels were considered for the estimation of maximum $Y/R$. A less conservative alpha-level (0.01), which presents more possibilities for $Y/R$ values, but with increased uncertainties, a moderate one (0.40) and a more conservative one (0.80), with less flexibility with respect to the possible values for $Y/R$.

Because of Equation (4), the total mortality was assumed as a fuzzy number ($\tilde{Z}$), since the natural mortality (M) component was replaced by the fuzzy number ($\tilde{M}$) and the fishing mortality F was maintained in its crisp form ($[F{]}^{\alpha }=F$). The sum operation has been extended to the sum of fuzzy numbers. The alpha-levels of the set $\tilde{Z}$ were calculated based on the result of Theorem 2.41 in Barros et al. (2017), i.e.:

$$[\tilde{Z}{]}^{\alpha }=[\tilde{M}{]}^{\alpha }\oplus [F{]}^{\alpha }=[\tilde{M}{]}^{\alpha }\oplus F$$

Table I
Population parameters of the species selected for the study in order to estimate the fuzzy sets of yield per recruit (Y/R). The values in normal font were available in the literature and the values in bold were estimated.

Nº	Species	${\mathit{𝑻}}_{\mathit{𝒄}}\mathit{/}{\mathit{𝑻}}_{\mathit{𝒓}}$ (years)	${\mathit{𝑾}}_{\mathit{\infty }}$ (gr)	k (${\mathrm{\text{year}}}^{\mathit{-}\mathbf{𝟏}}$)>	b	${\mathit{𝑻}}_{\mathrm{\text{max}}}$ (year)	Strategy	Environment
1	Cichla temensis	1.88	5037.63	0.20	3.232	14.00	Equilibrium	Freshwater
2	Prochilodus nigricans	0.78	967.45	0.44	2.720	6.80	Seasonal
3	Triportheus angulatus	0.39	350.36	0.77	2.859	3.89	Opportunist
4	Cynoscion acoupa	0.73	16,112.66	0.13	3.028	10.00	Equilibrium	Marine
5	Lutjanus synagris	0.91	2,200.00	0.22	2.890	18.00	Seasonal
6	Sardinella brasiliensis	0.56	180.33	0.26	3.130	10.73	Opportunist

• Source: 1 - Campos et al. (2020); 2 - Catarino et al. (2014); 3 - Prestes et al. (2010); 4 - Oliveira et al. (2020); 5 - Aschenbrenner et al. (2017); 6 - Costa et al. (2018).

RESULTS

Considering the IPCC (Intergovernmental Panel on Climate Change) scenarios: RCPs 2.6, 4.5, 6.0 and 8.5 (IPCC 2014), the fuzzy yield per recruit ($Y/R$) models for the chosen species showed the following patterns: (i) gradual reduction of $Y/R$ from the current scenario to the C4 scenario and (ii) increased uncertainties in $Y/R$ estimates, especially to F higher than the F correspondent to the maximum $Y/R$ (Figures 2 and 3). Equilibrium species showed lower uncertainty in $Y/R$ estimates in all the IPCC scenarios. Seasonal species presented $Y/R$ estimates with intermediate uncertainty values, whereas species with opportunistic strategies exhibited the greatest uncertainties in the estimated $Y/R$ values for all the scenarios.

Figure 2
Yield per recruit (Y/R), incorporating fuzzy natural mortality (M), for Cichla temensis, Prochilodus nigricans and Triportheus angulatus and their degree of relevance (vertical bar on the right), for each IPCC scenario after the application of the Zadeh extension. (C1 = RCP 2.6; C2 = RCP 4.5; C3 = RCP 6.0; C4 = RCP 8.5; IPCC 2014).

Figure 3
Yield per recruit (Y/R), with the incorporation of fuzzy natural mortality (M), for Cynoscion acoupa, Lutjanus synagris and Sardinella brasiliensis and their degree of relevance (vertical bar on the right), for each IPCC scenario after the application of the Zadeh extension (C1 = RCP 2.6; C2 = RCP 4.5; C3 = RCP 6.0; C4 = RCP 8.5; IPCC 2014).

The maximum $Y/R$ possibilities for each IPCC scenario for both freshwater and marine species are presented in Tables II and III , respectively. For each alpha-level, the tables detail the range of yield per recruit ($Y/R$) possibilities alongside their respective fishing mortalities (F). Specifically, $Y/R_1$ corresponds to the lowest $Y/R$ value (in bold), with $F_1$ representing the fishing mortality that yields this value. Conversely, $Y/R_2$ corresponds to the highest $Y/R$ value (in italic), and $F_2$ represents the fishing mortality that yields $Y/R_2$. The choice of a more or less conservative scenario, i.e., a higher or lower alpha-level, should be considered mainly for seasonal species and those with opportunistic characteristics, since $\mathrm{\Delta }F$ is higher for these species (Tables II and III).

For all the species, regardless of the environment they inhabit, it is observed that, with the decrease in the $Y/R$ maximum from one scenario to another, the variation between the $Y/R$ maximum ($\mathrm{\Delta }Y/R$ maximum) increases, i.e., there is greater uncertainty in the estimation of the $Y/R$ maximum in scenarios with greater uncertainties in natural mortality (M). However, the relative variation of the decrease in $Y/R$ from the current scenario to the C4 scenario is greater for species with opportunistic characteristics, both freshwater and marine, independent of the alpha-level (Tables II and III).

Table II
Yield per Recruit (Y/R) possibilities for species inhabiting freshwater environments, considering different $\mathit{𝜶}$-levels, for each IPCC scenario (${\mathit{𝑪}}_{\mathbf{𝟎}}$ = current; ${\mathit{𝑪}}_{\mathbf{𝟏}}$ = RCP 2.6; ${\mathit{𝑪}}_{\mathbf{𝟐}}$ = RCP 4.5; ${\mathit{𝑪}}_{\mathbf{𝟑}}$ = RCP 6.0; ${\mathit{𝑪}}_{\mathbf{𝟒}}$ = RCP 8.5, IPCC 2014). The lowest Y/R values ($\mathit{𝒀}\mathit{/}{\mathit{𝑹}}_{\mathbf{𝟏}}$) and their corresponding fishing mortalities (${\mathit{𝑭}}_{\mathbf{𝟏}}$) are in bold, while the highest Y/R values ($\mathit{𝒀}\mathit{/}{\mathit{𝑹}}_{\mathbf{𝟐}}$) and their corresponding fishing mortalities (${\mathit{𝑭}}_{\mathbf{𝟐}}$) are in italic.

ENVIRONMENT		FRESHWATER
	Cichla temensis						Prochilodus nigricans						Triportheus angulatus
SCENARIOS	Alpha-level	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$
C0	0.01	198.56	0.571	200.57	0.56	2.01	0.010	53.38	1.310	53.86	1.29	0.48	0.022	20.10	1.29	20.39	1.264	0.29	0.028
	0.40	199.21	0.571	199.86	0.56	0.65	0.004	53.54	1.310	53.70	1.302	0.15	0.008	20.20	1.28	20.29	1.274	0.09	0.008
	0.80	199.44	0.571	199.63	0.57	0.19	0.000	53.60	1.306	53.645	1.304	0.04	0.002	20.24	1.278	20.26	1.276	0.02	0.002
C1	0.01	193.92	0.60	199.68	0.568	5.76	0.03	52.28	1.372	53.66	1.304	1.37	0.068	19.46	1.360	20.27	1.276	0.81	0.084
	0.40	194.52	0.596	198.98	0.572	4.45	0.024	52.43	1.366	53.49	1.312	1.06	0.054	19.55	1.350	20.18	1.284	0.63	0.066
	0.80	194.74	0.594	198.75	0.574	4.00	0.020	52.48	1.362	53.44	1.314	0.96	0.048	19.58	1.348	20.14	1.288	0.56	0.060
C2	0.01	191.67	0.612	197.37	0.580	5.70	0.032	51.74	1.402	53.11	1.330	1.37	0.072	19.15	1.396	19.95	1.308	0.80	0.088
	0.40	192.26	0.608	196.74	0.584	4.48	0.024	51.89	1.394	52.96	1.338	1.07	0.056	19.23	1.386	19.86	1.318	0.63	0.068
	0.80	192.43	0.608	196.52	0.586	4.09	0.022	51.93	1.392	52.90	1.340	0.97	0.052	19.26	1.384	19.83	1.320	0.57	0.064
C3	0.01	190.43	0.620	196.56	0.584	6.13	0.036	51.45	1.418	52.92	1.340	1.46	0.078	18.98	1.418	19.84	1.320	0.85	0.098
	0.40	191.00	0.616	195.94	0.588	4.93	0.028	51.59	1.412	52.76	1.348	1.17	0.064	19.06	1.408	19.74	1.330	0.68	0.078
	0.80	191.21	0.614	195.71	0.590	4.50	0.024	51.64	1.408	52.71	1.350	1.07	0.058	19.09	1.404	19.71	1.332	0.63	0.072
C4	0.01	186.53	0.644	193.40	0.602	6.88	0.042	50.52	1.476	52.16	1.380	1.64	0.096	18.46	1.488	19.39	1.368	0.94	0.120
	0.40	187.07	0.640	192.76	0.606	5.69	0.034	50.65	1.468	52.01	1.388	1.37	0.080	18.52	1.478	19.31	1.378	0.78	0.100
	0.80	187.22	0.640	192.59	0.606	5.37	0.034	50.69	1.464	51.97	1.390	1.28	0.074	18.55	1.474	19.28	1.382	0.73	0.092

Table III
Yield per Recruit (Y/R) possibilities for species inhabiting marine environments, considering different $\mathit{𝜶}$-levels, for each IPCC scenario (${\mathit{𝑪}}_{\mathbf{𝟎}}$ = current; ${\mathit{𝑪}}_{\mathbf{𝟏}}$ = RCP 2.6; ${\mathit{𝑪}}_{\mathbf{𝟐}}$ = RCP 4.5; ${\mathit{𝑪}}_{\mathbf{𝟑}}$ = RCP 6.0; ${\mathit{𝑪}}_{\mathbf{𝟒}}$ = RCP 8.5, IPCC 2014). The lowest Y/R values ($\mathit{𝒀}\mathit{/}{\mathit{𝑹}}_{\mathbf{𝟏}}$) and their corresponding fishing mortalities (${\mathit{𝑭}}_{\mathbf{𝟏}}$) are in bold, while the highest Y/R values ($\mathit{𝒀}\mathit{/}{\mathit{𝑹}}_{\mathbf{𝟐}}$) and their corresponding fishing mortalities (${\mathit{𝑭}}_{\mathbf{𝟐}}$) are in italic.

ENVIRONMENT		MARINE
	Cynoscion acoupa						Lutjanus synagris						Sardinella brasiliensis
SCENARIOS	Alpha-level	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$	$Y/R_1$	$F_1$	$Y/R_2$	$F_2$	$\mathrm{\Delta }Y/R$	$\mathrm{\Delta }F$
C0	0.01	241.48	0.226	244.27	0.226	2.79	0.000	58.50	0.434	60.25	0.424	1.75	0.010	2.43	0.490	2.53	0.480	0.09	0.010
	0.40	242.48	0.226	243.34	0.226	0.86	0.000	59.08	0.430	59.63	0.428	0.55	0.002	2.46	0.488	2.49	0.484	0.03	0.004
	0.80	242.77	0.226	242.98	0.226	0.21	0.000	59.28	0.430	59.43	0.428	0.15	0.002	2.47	0.486	2.48	0.484	0.01	0.002
C1	0.01	232.89	0.230	242.69	0.226	9.80	0.004	56.39	0.450	59.817	0.426	3.42	0.024	2.32	0.506	2.50	0.482	0.18	0.024
	0.40	233.77	0.230	241.70	0.226	7.92	0.004	56.92	0.446	59.22	0.430	2.30	0.016	2.35	0.502	2.47	0.486	0.12	0.016
	0.80	234.11	0.230	241.41	0.228	7.30	0.002	57.11	0.444	59.02	0.432	1.91	0.012	2.36	0.500	2.46	0.488	0.10	0.012
C2	0.01	228.61	0.230	238.53	0.228	9.92	0.002	55.38	0.456	58.75	0.434	3.37	0.022	2.27	0.514	2.45	0.490	0.18	0.024
	0.40	229.48	0.230	237.63	0.228	8.15	0.002	55.88	0.452	58.18	0.436	2.29	0.016	2.30	0.510	2.42	0.494	0.12	0.016
	0.80	229.81	0.230	237.28	0.228	7.47	0.002	56.05	0.452	57.98	0.438	1.93	0.014	2.30	0.508	2.41	0.494	0.10	0.014
C3	0.01	226.32	0.232	237.00	0.228	10.68	0.004	54.83	0.460	58.35	0.436	3.52	0.024	2.24	0.518	2.43	0.492	0.18	0.026
	0.40	227.17	0.232	236.10	0.228	8.93	0.004	55.31	0.456	57.80	0.440	2.49	0.016	2.30	0.514	2.40	0.496	0.13	0.018
	0.80	227.43	0.232	235.76	0.228	8.33	0.004	55.49	0.456	57.61	0.440	2.12	0.016	2.28	0.512	2.40	0.496	0.11	0.016
C4	0.01	218.86	0.234	231.14	0.230	12.28	0.004	53.09	0.474	56.87	0.446	3.78	0.028	2.15	0.532	2.35	0.502	0.20	0.030
	0.40	219.61	0.234	230.27	0.230	10.66	0.004	53.53	0.470	56.35	0.450	2.82	0.020	2.18	0.528	2.32	0.506	0.15	0.022
	0.80	219.86	0.234	229.94	0.230	10.08	0.004	53.68	0.470	56.18	0.450	2.50	0.020	2.18	0.526	2.31	0.508	0.13	0.018

DISCUSSION

Our results show the effects of global climate change on population parameters, in the case of natural mortality (M) of fish and, consequently, in the estimates of yields of catches from fish stocks. These effects evidenced the decrease in the yield per recruit levels in all the IPCC scenarios, as previously observed in several studies (Cheung et al. 2010, Serpetti et al. 2017, Gaines et al. 2018, Cisneros-Mata et al. 2019, Free et al. 2019, Plagányi 2019). Estimating the impact of global warming on fishery production and maximum yield per recruit for a given fishery is an essential approach for subsidizing fishery management proposals in a global environment with high dynamics (Free et al. 2019).

Several studies used as indicator the maximum sustainable yield, estimated using catch and effort data to evaluate the effects of climate change on fisheries. Global analyses of 915 fish stocks showed changes in the global MSY (weighted average) by 1.0%, -1.5%, -5.0% and -25.0% until 2100, under the RCPs 2.6, 4.5, 6.0 and 8.5, respectively (Gaines et al. 2018). However, the authors reported that these modest changes in the overall MSY under the three lowest RCPs mask a huge variation in changes between stocks since, while some stocks become extinct (MSY decreases by 100%), others increase by more than 35% under RCP 6.0. Overall, approximately 41%, 53%, 66%, and 91% of global inventories may experience declines in total MSY by 2100, under RCPs 2.6, 4.5, 6.0, and 8.5, respectively. Serpetti et al. (2017) evaluated the impact of climate change on commercially important fish stocks, considering depleted and growing stocks, and compared the status quo of the scenarios of maximum sustainable yield (MSY) to the estimated values for the IPCC scenarios. These authors noted that capture showed an overall decrease in all the IPCC scenarios compared to the current scenario. This suggests that stocks will be affected by the increase in temperature due to global climate change, regardless of the stage of exploitation of the stock.

The intensity of the effects on the fishing yield are also related to the different life strategies of exploitable stocks. Cisneros-Mata et al. (2019) analyzed the impacts due to climate change that are expected on the productivity of 25 fish stocks, which correspond to more than 70% of Mexico’s total landings. The authors found that approximately 84% stocks are expected to show declines in maximum capture potential compared to the initial MSY over a 30-year time horizon. Declines ranged from -1% to -44%. Among the species studied are Cynoscion othonopterus, with an expected decrease of -2.5%, which belongs to the same genus as Cynoscion acoupa; Lutjanus peru, with an expected decrease of -21%, which belongs to the same genus as Lutjanus synagris; and Sardinops sagax, with an estimated decrease of -24%, which belongs to the same family as Sardinella brasiliensis. Free et al. (2019) measured the effects of increased temperature on the productivity of 235 global fish and invertebrate populations, corresponding to approximately 33% of the reported global catch. The authors used the generated model to convey the temperature-induced changes in MSY and estimated a 4.1% decrease in the combined MSY of all the populations. In addition, they related the influence of temperature with the characteristics of the species’ life history and observed that, despite being species that respond faster to changes, temperature affects more species with opportunistic characteristics, with faster growth, early age of sexual maturity and shorter life expectancy, than strategists in equilibrium.

Similarly to MSY, the maximum yield per recruit ($Y/R$) values are targets of fishery management because they represent the highest fishing yield associated with the number of annual recruits in the stock. Although $Y/R$ and MSY are distinct metrics, with $Y/R$ evaluating the productivity of a cohort on a per-individual basis and MSY estimating the absolute maximum catch for the entire population, both are highly sensitive to variations in natural mortality and serve as critical proxies for stock robustness under climate stress. While direct quantitative comparisons between these measures should be interpreted with caution, studies using MSY qualitatively corroborate our results regarding the distinct responses of species according to their life strategy. Specifically, both metrics project a greater relative variation in the decrease in $Y/R$ and overall productivity for opportunistic species. In addition, the greatest uncertainties in the estimates of $Y/R$ occurred for this group of species. This could be explained by higher plasticity (Jeppesen et al. 2014) and high variability in the population patterns of these species, since they occupy habitats with strong dynamics that demand a high amount of energy throughout the life cycle and, consequently, their population responses to variations in the environment tend to be large in amplitude (King & McFarlane 2003).

Natural mortality (M) is a parameter of yield per recruit models that operates simultaneously and continuously with fishing mortality (F) to produce total mortality (Z), which determines reductions in population abundance over time. However, natural mortality is a difficult parameter to estimate and the yield per recruit models, used in inventory evaluation, are sensitive to the chosen value of M (Pascual & Iribarne 1993, Quinn & Deriso 1999, Quiroz et al. 2010). The effect of incorrect estimation of M in stock valuation models has already been investigated by several authors (Ulltang 1977, Sims 1984, Lapointe et al. 1992, Aanes et al. 2007, Chen et al. 2018, Punt et al. 2021). Erroneous estimates of M will influence the calculations of fish stock productivity (Quiroz et al. 2010). The inclusion of uncertainties of M estimates in fishery stock assessments, based on methodologies that enable mathematical modeling of uncertainty, such as fuzzy set theory (Barros & Bassanezi 2010), has proven to be a promising strategy.

Other studies have incorporated the uncertainties of M in stock valuation: Quiroz et al. (2010) examined prediction errors inherent in empirical methods used to assess M. Gaertner (2015) utilized a Monte Carlo resampling procedure to explore the effects of variability in life-history parameters used to estimate M. Legault & Palmer (2016) assessed the relationship between fishing mortality (F) and natural mortality (M). Chen et al. (2018) employed four empirical methods to estimate M to evaluate the variability of different management procedures. However, the current study offers a unique contribution to fisheries science. It is the first to quantitatively incorporate the uncertainties of M related to global climate change into yield-per-recruit ($Y/R$) models using fuzzy set theory. The choice of fuzzy sets as an alternative to standard probabilistic methods (e.g., Monte Carlo simulations) is justified by the epistemic nature of climate-induced uncertainties. Since future climate scenarios (RCPs) represent unprecedented ecological conditions, fitting empirical probability distributions to M is mathematically challenging. Uncertainty propagation via fuzzy arithmetic avoids the need for arbitrary assumptions about probability distributions (such as uniform or normal priors), mapping the range of $Y/R$ values directly from the physiological thresholds of the target species. As they quantify different types of uncertainty, these are complementary approaches.

By demonstrating how these climate-induced uncertainties propagate differently depending on a species’ life history strategy, our model bridges a critical gap between theoretical climate projections and the practical estimation of biological reference points. In addition, since the data presented a pattern, it would also be possible to make inferences in a more holistic approach, extrapolating these BRPs to species groupings according to their life strategies. However, this holistic approach should be employed with caution. According to Chen et al. (2018), the greater the uncertainty of M in stock assessment models, the greater the risks of overfishing, implying that the risk of overfishing should be assumed as a priority with a large uncertainty of M. Species with opportunistic characteristics presented the greatest uncertainties in the variations of $Y/R$ and, therefore, are more prone to risk. In this case, it is necessary to make more conservative decisions when choosing fishing management strategies for species with this life strategy. Consequently, this study provides a practical and innovative framework for adapting fisheries management to the realities of global climate change.

Acknowledgements

We thank the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, grant 1441407).

Data availability

The data that support the findings of this study are included in the article. Further inquiries can be directed to the corresponding author.

References

• AANES S, ENGEN S, SAETHER B & AANES R. 2007. Estimation of the parameters of fish stock dynamics from catch-at-age data and indices of abundance: can natural and fishing mortality be separated? Can J Fish Aquat Sci 64(8): 1130-1142. doi: 10.1139/f07-074.
• ALBOUY C, DELATTRE V, DONATI G, FRÖLICHER T, ALBOUY-BOYER S, RUFINO M, PELLISSIER L, MOUILLOT D & LEPRIEUR F. 2020. Global vulnerability of marine mammals to global warming. Sci Rep 10: 548. doi: 10.1038/s41598-019-57280-3.
• ALFONSO S, GESTO M & SADOUL B. 2020. Temperature increase and its effects on fish stress physiology in the context of global warming. J Fish Biol 98: 1496-1508. doi: 10.1111/jfb.14599.
• ANDRADE B, SINGH C, SANTOS J, GONÇALVES V, SIQUEIRA-SOUZA F & FREITAS C. 2019. Efeitos das mudanças climáticas sobre as comunidades de peixes na Bacia Amazônica. Rev Cien Soc 2(4): 107-124. Available at: http://www.ufopa.edu.br/portaldeperiodicos/index.php/revistacienciasdasociedade.
• ARAÚJO A, LIMA L, NASCIMENTO W, YAMAMOTO M & CHELLAPPA S. 2012. Características morfométricas-merísticas e aspectos reprodutivos da sardinha de água doce, Triportheus angulatus (Osteichthyes: Characiformes) do rio Acauã do bioma Caatinga. Biota Amazonia 2: 59-73. doi: 10.18561/2179-5746/biotaamazonia.v2n1p59-73.
• ASCHENBRENNER A, FREITAS M, ROCHA G, MOURA R, FRANCINI-FILHO R, MINTE-VERA C & FERREIRA B. 2017. Age, growth parameters and fisheries indices for the lane snapper in the Abrolhos Bank, SW Atlantic. Fish Res 194: 155-163. doi: 10.1016/j.fishres.2017.06.004.
• BARROS D & ALBERNAZ A. 2013. Possible impacts of climate change on wetlands and its biota in the Brazilian Amazon. Braz J Biol 74(4): 810-820. doi: 10.1590/1519-6984.04013.
• BARROS LC. 1997. Sobre Sistemas Dinâmicos Fuzzy - Teoria e Aplicações. Tese de doutorado, Universidade Estadual de Campinas, 114 p. Available at: https://www.ime.unicamp.br/ laeciocb/tese_doutorado.pdf.
• BARROS LC & BASSANEZI RC. 2010. Tópicos de Lógica Fuzzy e Biomatemática. IMECC, São Paulo, 3rd ed., 382 p.
• BARROS LC, BASSANEZI RC & LODWICK WA. 2017. A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. Berlin: Springer-Verlag. doi: 10.1007/978-3-031-50492-1.
• BEVERTON R & HOLT S. 1957. On the dynamics of exploited fish populations. Fishery Investigations, Series II, Vol. 19. London: H.M. Stationery Office, 533 p.
• BITAR S, CAMPOS C & FREITAS C. 2016. Applying fuzzy logic to estimate the parameters of the length-weight relationship. Braz J Biol 76: 611-618. doi: 10.1590/1519-6984.20014.
• BLANCK A & LAMOUROUX N. 2007. Large-scale intraspecific variation in life-history traits of European freshwater fish. J Biogeogr 34(5): 862-875. doi: 10.1111/j.1365-2699.2006.01654.x.
• BRAGA F. 1997. Análise da equação alométrica na relação peso e comprimento e o fator de condição em Plagioscion squamosissimus (Teleostei, Sciaenidae). Rev Bras Biol 57(3): 417-425.
• BRODZIAK J, IANELLI J, LORENZEN K & METHOT R. 2011. Estimating natural mortality in stock assessment applications. Technical Memorandum NMFS-F/SPO-119. NOAA. Available at: https://fisheriessolutions.org/wp-content/uploads/2015/02/Brodziak_et_al_2011.pdf
  » https://fisheriessolutions.org/wp-content/uploads/2015/02/Brodziak_et_al_2011.pdf
• CAMPOS C, BITAR S & FREITAS C. 2023. Uncertainties regarding the natural mortality of fish can increase due global climate change. PeerJ 11: e14989. doi: 10.7717/peerj.14989.
• CAMPOS C, CATARINO M & FREITAS C. 2020. Stock assessment of the peacock bass Cichla temensis (Humboldt, 1821), an important fishing resource from the Middle Negro river, Amazonas, Brazil. Braz J Biol 80(3): 506-510. doi: 10.1590/1519-6984.203124.
• CAMPOS D, VAL A & ALMEIDA-VAL V. 2018. The influence of lifestyle and swimming behavior on metabolic rate and thermal tolerance of twelve Amazon forest stream fish species. J Therm Biol 72: 148-154. doi: 10.1016/j.jtherbio.2018.02.002.
• CATARINO M, CAMPOS C, GARCEZ R & FREITAS C. 2014. Population dynamics of Prochilodus nigricans caught in Manacapuru Lake (Amazon basin, Brazil). Bol Inst Pesca 40(4): 589-595. Available at: https://institutodepesca.org/index.php/bip/en/article/view/1064.
• CHEN N, ZHANG C, SUN M, XU B, XUE Y, REN Y & CHEN Y. 2018. The impact of natural mortality variations on the performance of management procedures for Spanish mackerel (Scomberomorus niphonius) in the Yellow Sea, China. Acta Oceanol Sin 37(8): 21-30. doi: 10.1007/s13131-018-1234-0.
• CHEUNG W, LAM V, SARMIENTO J, KEARNEY K, WATSON R, ZELLER D & PAULY D. 2010. Large-scale redistribution of maximum fisheries catch potential in the global ocean under climate change. Glob Change Biol 16(1): 24-35. doi: 10.1111/j.1365-2486.2009.01995.x.
• CISNEROS-MATA M, MANGIN T, BONE J, RODRIGUEZ L, SMITH S & GAINES S. 2019. Fisheries governance in the face of climate change: assessment of policy reform implications for Mexican fisheries. PLoS One 14(10): e0222317. doi: 10.1371/journal.pone.0222317.
• COSTA M, TUBINO R & MONTEIRO-NETO C. 2018. Length-based estimates of growth parameters and mortality rates of fish populations from a coastal zone in the Southeastern Brazil. Zoologia 35: e22235. doi: 10.3897/zoologia.35.e22235.
• DUTIL J & LAMBERT Y. 2000. Natural mortality from poor condition in Atlantic cod (Gadus morhua). Can J Fish Aquat Sci 57(4): 826-836. doi: 10.1139/f00-023.
• FLORIDES G & CHRISTODOULIDES P. 2009. Global warming and carbon dioxide through sciences. Environ Int 35: 390-401. doi: 10.1016/j.envint.2008.07.007.
• FREE C, THORSON J, PINSKY M, OKEN K, WIEDENMANN J & JENSEN O. 2019. Impacts of historical warming on marine fisheries production. Science 363(6430): 979-983. doi: 10.1126/science.aau1758.
• FREITAS C, RIVAS A, CAMPOS C, SANT’ANA I, KAHN J, CORREA M & CATARINO M. 2013. The potential impacts of global climatic changes and dams on Amazonian fish and their fisheries. In: Turker H (Ed), New Advances and Contributions to Fish Biology, p. 175-195. London: IntechOpen. doi: 10.5772/54549.
• GAERTNER D. 2015. Indirect estimates of natural mortality rates for Atlantic skipjack (Katsuwonus pelamis), using life history parameters. Collect Vol Sci Pap ICCAT 71(1): 189-204.
• GAINES S ET AL. 2018. Improved fisheries management could offset many negative effects of climate change. Sci Adv 4(8): eaao1378. doi: 10.1126/sciadv.aao1378.
• GRIFFITHS D. 1997. Local and regional species richness in North American lacustrine fish. J Anim Ecol 66(1): 49-56. doi: 10.2307/5963.
• IPCC - INTERGOVERNMENTAL PANEL ON CLIMATE CHANGE. 2014. Climate Change 2014: Synthesis Report. Contribution of Working Groups I, II and III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. In: Pachauri RK & Meyer LA (Eds), Geneva, Switzerland. Available at: https://www.ipcc.ch/report/ar5/syr/
  » https://www.ipcc.ch/report/ar5/syr/
• IPCC - INTERGOVERNMENTAL PANEL ON CLIMATE CHANGE. 2021. Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report. In: Masson-Delmotte V et al. (Eds), Cambridge University Press, Cambridge. doi: 10.1017/9781009157896.
• JEPPESEN E ET AL. 2010. Impacts of climate warming on lake fish community structure and potential effects on ecosystem function. Hydrobiologia 646(1): 73-90. doi: 10.1007/s10750-010-0171-5.
• JEPPESEN E ET AL. 2014. Climate change impacts on lakes: an integrated ecological perspective based on a multi-faceted approach, with special focus on shallow lakes. J Limnol 73: 84-107. doi: 10.4081/jlimnol.2014.844.
• KENCHINGTON T. 2014. Natural mortality estimators for information-limited fisheries. Fish Fish 15(4): 533-562. doi: 10.1111/faf.12027.
• KING J & MCFARLANE G. 2003. Marine fish life history strategies: applications to fishery management. Fish Manage Ecol 10(4): 249-264. doi: 10.1046/j.1365-2400.2003.00359.x.
• KING M. 1995. Fisheries Biology, Assessment and Management. Oxford University Press, Oxford. doi: 10.1002/9781118688038.
• LAPOINTE M, PETERMAN R & ROTHSCHILD B. 1992. Variable natural mortality rates inflate variance of recruitments estimated from virtual population analysis (VPA). Can J Fish Aquat Sci 49(10): 2020-2027. doi: 10.1139/f92-225.
• LEGAULT C & PALMER M. 2016. In what direction should the fishing mortality target change when natural mortality increases within an assessment? Can J Fish Aquat Sci 73(3): 349-357. doi: 10.1139/cjfas-2015-0232.
• MACEDO P & MADALENO M. 2023. Global temperature and carbon dioxide nexus: evidence from a maximum entropy approach. Energies 16(1): 277. doi: 10.3390/en16010277.
• MALABARBA M. 2004. Revision of the Neotropical genus Triportheus Cope, 1872 (Characiformes: Characidae). Neotrop Ichthyol 2: 167-204. doi: 10.1590/S1679-62252004000400001.
• MAUNDER M, HAMEL O, LEE H, PINER K, COPE J, PUNT A, IANELLI J, CASTILLO-JORDÁN C, KAPUR M & METHOT R. 2023. A review of estimation methods for natural mortality and their performance in the context of fishery stock assessment. Fish Res 257: 106489. doi: 10.1016/j.fishres.2022.106489.
• OJEA E, PEARLMAN I, GAINES S & LESTER S. 2017. Fisheries regulatory regimes and resilience to climate change. Ambio 46(4): 399-412. doi: 10.1007/s13280-016-0850-1.
• OLIVEIRA C, LESSA R, ALMEIDA Z & SANTANA F. 2020. Biology and fishery of Acoupa Weakfish Cynoscion acoupa (Lacepède, 1801): a review. Neotrop Biol Conserv 15(3): 333-349. doi: 10.3897/neotropical.15.e55563.
• OREMUS K, BONE J, COSTELLO C, GARCÍA MOLINOS J, LEE A, MANGIN T & SALZMAN J. 2020. Governance challenges for tropical nations losing fish species due to climate change. Nat Sustain 3(4): 277-280. doi: 10.1038/s41893-020-0476-y.
• OVANDO D, FREE C, JENSEN O & HILBORN R. 2022. A history and evaluation of catch-only stock assessment models. Fish Fish 23(3): 616-630. doi: 10.1111/faf.12637.
• PASCUAL M & IRIBARNE O. 1993. How good are empirical predictions of natural mortality? Fish Res 16(1): 17-24. doi: 10.1016/0165-7836(93)90107-I.
• PEDRYCZ W & GOMIDE F. 1998. An Introduction to Fuzzy Sets: Analysis and Design. Cambridge: MIT Press, 465 p. doi: 10.7551/mitpress/3926.001.0001.
• PLAGÁNYI E. 2019. Climate change impacts on fisheries. Science 363(6430): 930-931. doi: 10.1126/science.aaw5824.
• PRESTES L, SOARES M, SILVA F & BITTENCOURT M. 2010. Dinâmica populacional de Triportheus albus, T. angulatus e T. auritus (Characiformes: Characidae) em lagos da Amazônia Central. Biota Neotrop 10: 177-181. doi: 10.1590/S1676-06032010000300020.
• PUNT A, CASTILLO-JORDÁN C, HAMEL O, COPE J, MAUNDER M & IANELLI J. 2021. Consequences of error in natural mortality and its estimation in stock assessment models. Fish Res 233: 105759. doi: 10.1016/j.fishres.2020.105759.
• QUINN II T & DERISO R. 1999. Quantitative Fish Dynamics. New York: Oxford University Press. doi: 10.1093/oso/9780195076318.001.0001.
• QUIROZ J, WIFF R & CANECO B. 2010. Incorporating uncertainty into estimation of natural mortality for two species of Rajidae fished in Chile. Fish Res 102(3): 297-304. doi: 10.1016/j.fishres.2010.01.001.
• SCHRANK W. 2007. Is there any hope for fisheries management? Mar Policy 31: 299-307. doi: 10.1016/j.marpol.2006.09.001.
• SERPETTI N, BAUDRON A, BURROWS M, PAYNE B, HELAOUET P, FERNANDES P & HEYMANS J. 2017. Impact of ocean warming on sustainable fisheries management informs the Ecosystem Approach to Fisheries. Sci Rep 7(1): 1-15. doi: 10.1038/s41598-017-13220-7.
• SIMS S. 1984. An analysis of the effect of errors in the natural mortality rate on stock-size estimates using Virtual Population Analysis (Cohort Analysis). ICES J Mar Sci 41(2): 149-153. doi: 10.1093/icesjms/41.2.149.
• SPARRE P & VENEMA S. 1997. Introdução a avaliação de mananciais de peixes tropicais. Roma: FAO-Organização das Nações Unidas para a Alimentação e a Agricultura. Available at: https://www.fao.org/4/w5449p/w5449p00.htm
  » https://www.fao.org/4/w5449p/w5449p00.htm
• ULLTANG O. 1977. Sources of errors in and limitations of Virtual Population Analysis (Cohort Analysis). ICES J Mar Sci 37(3): 249-260. doi: 10.1093/icesjms/37.3.249.
• VAL A. 2019. Fishes of the Amazon: diversity and beyond. An Acad Bras Cienc 91: e20190260. doi: 10.1590/0001-3765201920190260.
• VON BERTALANFFY L. 1938. A quantitative theory of organic growth (inquiries on growth laws II). Hum Biol 10(2): 181-213. Available at: https://www.jstor.org/stable/41447359.
• WINEMILLER K. 1989. Patterns of variation in life history among South American fishes in seasonal environments. Oecologia 81: 225-241. doi: 10.1007/BF00379810.
• WINEMILLER K & ROSE K. 1992. Patterns of life-history diversification in North American fishes: implications for population regulation. Can J Fish Aquat Sci 49(10): 2196-2218. doi: 10.1139/f92-242.
• ZADEH L. 1965. Fuzzy sets. Inf Control 8(3): 338-353. doi: 10.1016/S0019-9958(65)90241-X.

Correspondence

Correspondence to: Carlos Edwar de Carvalho Freitas E-mail: [e-mail]

Handling editor

Renan Souza

Publication Dates

• Publication in this collection
  23 Oct 2026
• Date of issue
  2026

History

• Received
  29 Apr 2025
• Accepted
  1 June 2026