Fish Stock Assessment Models for Developing Nations with Emphasis on the Use of the Classic Gordon–Schaefer Model: A Review

Abstract

The paper discusses fish stock assessment methods, emphasising methods for assessing stocks in developing nations. We present the advantages and disadvantages of each method discussed. Approaches to fish stock assessment include single-species, multi-species, multi-gear and ecosystem approaches. We discuss the Gordon–Schaefer (GS) model, a single-species surplus production model, as an alternative method for assessing fishery stocks in developing nations, with Malawi as an example of a developing nation. Although the GS model is not a contemporary method, it is still suitable for the situation in Malawi. We review how the GS model has been applied globally, in general, and in Malawi, in particular. The review shows that most studies have concentrated on the calculation of maximum sustainable yield or maximum economic yield, leaving out open access yield and optimum sustainable yield which is the dynamic reference point. Using all reference points is crucial in making correct management decisions. Bifurcation analysis, calculation of annual sustainable production, and calculation of depletion are missing in most studies. Future research should focus on integrating the use of all four reference points, bifurcation analysis, and calculation of depletion as well as annual sustainable production.

1. Introduction

A fish stock is defined as a subpopulation of a species, considered as the basic taxonomic unit with common parameters, such as growth, location and mortality [1]. Stock assessment is the study of the abundance of a fish stock and the possible consequences of different management systems. Stock assessment helps understand whether the stock amount is within, below or above a target point, determining if there is overexploitation. Stock assessment can also indicate if a catch level will change or maintain the stock amount. A wide range of model-based stock assessment methods exist with varying complexity, suited to different situations of data availability [2]. While some methods are data-limited, other methods offer data-rich assessments.

Approaches to fish stock assessment include single-species, multi-species, multi-gear and ecosystem approaches. Another approach, called integrated analysis, is also used. Integrated analysis combines several data sources and information into one analysis, by integrating traditionally independent analyses into a single analysis [3]. However, it is not feasible to collect different types of data required for most of these methods in a developing country like Malawi [4]. According to [5], traditional fisheries in Malawi appear to be stable when analysed using multi-species methods yet are noted to be declining when analysed using single species methods. This suggests that multi-species methods may not be suitable for use in Malawi. Additionally, the availability and accuracy of collected data in Malawi remain a challenge [6]. Therefore, the use of data-rich methods may not be feasible. However, M’balaka et al. argued against using lack of quality and sufficient data as an excuse for failure to manage fishery resources [7].

Malawi is a land-locked country which has a total surface area of 118,484 km². About 20% of the area (24,405 km²) is water, supporting over 1000 fish species, which accounts for about 15% of global fish species. The fisheries sector in Malawi is categorised into three main areas: capture fisheries, aquaculture and aquarium [8,9]. The main sector is the capture fisheries since it contributes over 90% of total fish catch [10]. Five major water bodies in Malawi are lakes Malawi, Chilwa, Malombe, and Chiuta, the upper Shire river, and the lower Shire river [11]. Notably Lake Malawi has more fish species than any lake in the world [12]. Two divisions for capture fisheries are small-scale (also called traditional or artisanal) and commercial (or semicommercial). Artisanal fisheries, which are mostly open access, contribute about 90% of fish landing in Malawi and the target species are chambo (Oreochromis species), Utaka (Copadichromis species), Kambuzi (Haplochromis species), Kampango (Bargrus meridionalis), Usipa (Engraulicypris sardella), and Mlamba (Clariid gariepinus). Small-scale fishers in Malawi use vessels such as peddle-powered dugout canoes and plunk boats, while commercial fishers use engine boats, trawlers or pair trawlers [12]. The fishing gears used include chambo seine nets, gillnets, nkacha seine nets, kambuzi seine nets, chilimira seine nets, handlines, longlines and fish traps [13].

Stock assessment in Malawi began in 1976 by the department of fisheries following advice from the FAO. Data on commercial fisheries rely on trawler owners submitting monthly reports on catch and effort records to the fisheries department, a requirement for their licenses. Data on small-scale fisheries is obtained through the catch assessment system (CAS) and Malawi Traditional Fisheries (MTF) introduced in 1976 and 1990, respectively [5]. The available data primarily consist of time series records of catch and effort. Other forms of data are difficult to find. For example, age-structure and length-frequency data do not exist [14].

For most developing countries, the objective of fisheries management is to ensure that fish stocks are sustainable by maintaining the Maximum Economic Yield (MEY) or Maximum Sustainable Yield (MSY) of fish. Most developing countries have failed to meet this objective as witnessed by the observed declining trends of some stocks [4]. Fish management in Malawi is now focused on the objective of achieving MEY where the management strategy for the fishery includes restrictions on the fishing areas, fishing gears and fishing times [5,15].

In this paper, we discuss various stock assessment methods, presenting the advantages and disadvantages of each method discussed. We also explore application of the Gordon–Schaefer (GS) model as an alternative method for assessing sustainability of the fishery in developing countries including Malawi. Furthermore, we review global and national application of the GS model with a focus on its use in Malawi.

2. Population Model

The general population model of the fish stock takes into account what would induce an increase in the population and what would lead to a decrease [16]. Two factors which can lead to population increase are recruitment and population growth. On the other hand, natural mortality and fishing mortality are two factors that lead to population decrease. However, most researchers have only considered fishing mortality as a factor that leads to population decrease. The general model of the fish stock takes the form

$$X_{t+1}=X_t+G\left(X_t\right)-H\left(t\right)$$

(1)

where $X_t$ is the stock size at time t, $G\left(X_t\right)$ is the growth generated by the stock and $H\left(t\right)$ is the harvest [17]. According to this model, the size of the stock in any given year is given by the sum of stock size and growth for the previous year minus harvest for the previous year. By implication, if no catches occured in the previous year, the stock size would simply be the sum of the stock size and growth. If $G\left(X_t\right)=H\left(t\right)$ then the population stops growing and we say that the stock has reached an equilibrium point.

3. Stock Assessment

The ability to conduct fish stock assessment is crucial in informing management decisions in fisheries science [2]. The main purpose of conducting stock assessment is to estimate the status of fish stocks so that, in the long run, there is self-sustainability of the fish stocks. Stock assessment gives information on the current status of the fish stock, what has been happening to the stock in the past, what is expected to happen to the stock in future if there is no change in fishing levels as well, as what will happen if alternative management choices are adopted [1]. However, not all stock assessment methods give all this information.

Stock assessment methods can be described and classified in several ways. The classification of stock assessment methods depends on whether the methods are qualitative or quantitative, deterministic or stochastic, equilibrium or dynamic, age-based, or length-based, biomass dynamic or analytical models and whether they include stock-recruitment relationships or just make assessments per recruit [18]. According to [1], the three commonly used stock assessment models are surplus production models, empirical models and analytical models (yield models). Other stock assessment methods exist, such as delay difference [19] and virtual population analysis [20].

Stock assessment methods have undergone significant development over time. Such improvements are mainly attributed to development of new computational techniques, advances in data modelling methodologies and the need to provide best estimates and assess the precision and reliability of such estimates [21]. Contemporary approaches to stock assessment methods include the state-space models, such as AMSY, catch-MSY, CMSY [22,23,24], and Bayesian methods like JABBA (Just Another Bayesian Biomass Assessment method) [25]. The choice of which stock assessment method to use is influenced by the kind of data available for assessment [2].

Table 1. Benefits and limitations of common stock assessment methods for data-limited fisheries.

Method	Benefits	Limitations	References
Yield Models	They can be used to forecast the effects of effort, fishing gears, or mesh sizes on yield or biomass	They are difficult to implement in cases where data availability is a problem	[1,26]
Empirical Models	They are simple, quick and cost effective since they use readily available data	They are dependent on previous studies whose results may change; they do not estimate fishing effort	[27]
Surplus Production Models	They are simple and they consider stock as homogeneous biomass; they require simple data such as catch and effort data	They assume stock has stabilised at the current rate of fishing; they ignore complexities of age and spatial structure	[1,28]
Delay Difference Models	They are simple, just like surplus production models, but they have the additional advantage that they account for both recruit and spawner effects	Where data is scarce, delay difference models offer no advantage to surplus production models	[26]
Length-based and Age-based Models	They make use of age-specific and length-specific information	They require many observations and include many parameters	[19]

shows the advantages and disadvantages of common stock assessment methods.

4. Empirical Models

Empirical models are used as estimators to predict yield or production in lakes and reservoirs. The idea behind these models is that production of fish is mainly determined by the level of primary production in an aquatic system [27]. So, the prediction is made based on observed production from water bodies in the same aquatic system or water bodies of similar characteristics. According to [1], empirical models consider a variety of factors including the morphometry of the water body (for example, lake depth, surface area and temperature), nutrient status (such as conductivity), fish age, water chemistry, and biological structures and functions. In other words, these are models which incorporate environmental factors.

An example of an empirical model is the Morpho-edaphic Index (MEI), originally developed by [29]. The following equation

$$logY=1.4071+0.3697log\left(MEI\right)-0.00005465A,$$

(2)

where Y is the potential sustainable yield, MEI is the morpho-edaphic index or conductivity (μmho cm⁻¹ at 20 °C)/mean depth in meters and A is the surface area, is the recently modified morpho-edaphic index model.

The advantages of empirical models are that they are simple, quick and cost effective since they use data readily available from previous studies. These models offer an easy way to obtain parameter estimates when the parameter cannot be measured directly since it is either difficult or costly to measure [1]. They provide a preliminary estimate of potential fish production where there is no actual production data.

The main disadvantage of empirical models is their dependence on previous studies whose results may not withstand the passage of time. For example, the previously derived relationships may not continue to hold, especially when conditions change [27]. These models are also too restrictive since they require that the water body being studied should be very similar to the one in the original database. Another disadvantage is that the models do not give any idea of what is happening in the fishery in terms of stock changes, and they do not give an estimate of maximum or sustainable fishing effort, so that with these models, it is not possible to understand the effects of changes in effort levels [1].

5. Yield Models

Yield models are used to estimate fish yield using catch and effort data structured with length, age, growth and mortality parameters of the population. The two types of yield models are the Beverton and Holt yield per recruit model and the Thompson and Bell model. The Beverton and Holt model deals with variables obtained from samples drawn from the commercial fishery while the Thompson and Bell model requires data obtained from total records of catch and total population size by age group [1].

The Beverton and Holt model is used to estimate fish yield per recruitment [26]. Its main assumptions are that individual growth must occur uniformly and that the age at entry into the fishing grounds must be known or obtained in advance [30]. The model considers age of the first capture and the age of recruitment. As expected, the numbers reduce due to natural mortality. The input data for this model include the age at recruitment, the age at first capture, the natural mortality rate, and the fishing mortality rate.

The Thompson and Bell yield model is used to predict yield on an absolute basis. This model has both length-based and age-based versions. The age-based version is mostly applied in areas where it is easy to undertake age determination. These are temperate regions and in the tropics. The input data for this model include both natural and fishing mortality, total number of fish caught per year, structured by length, the population size per length group, and the mean weight of fish for each length group [1].

A major advantage of yield models is that they are used to predict yield at different fishing effort levels. Therefore, these models can be used to forecast the effects of effort, fishing gears or mesh sizes on the yield or biomass [31]. The main limitation for using these models is data availability. This is so because many fish stocks lack adequate data [1]. For example, the Beverton and Holt model was applied in Malawi by Thompson and Allison to estimate the biomass of offshore pelagic fish of Lake Malawi but most parameters were estimated from an equation due to unavailability of data [32].

6. Surplus Production Models

The essence of stock assessment through surplus production model is that each fish species has the ability to produce an additional amount of fish, over its production capacity, which is called the surplus, and if only this additional amount of fish is harvested, then the fish stock will be able to live sustainably [33]. Three surplus production models which are common are the Gordon–Schaefer (GS), Fox and Pella–Tomlinson models. The GS model is based on a logistic growth equation while the Fox model and the Pella–Tomlinson model depend on the equation of growth proposed by Gompertz and the equation of generalized production, respectively [34]. The GS model is the one that is very popular among all the models.

The advantages of surplus production models include that they are simple and they take the stock as a uniform biomass [1], they are cost-effective in terms of data collection and analysis since they require simple data such as catch and effort, which are easily available [28], and they ignore the complexities of age and spatial structure and assume self-regulating population growth [1]. The limitations of surplus production models are that they include only a single species, they ignore the age structure of the fish population [11], they require long time series catch and effort data, and the ability of the models to accurately describe fish populations depends mainly on the nature of the available data [1]. Despite these disadvantages, the models provide useful information for fisheries management. Surplus production models are usually considered the most complete data-limited assessment methods since they are the only methods that provide a full stock assessment [35].

6.1. The Gordon–Schaefer Model

The bio-economic model, also called the Gordon–Schaefer (GS) model, was developed by Gordon and Schaefer [11]. It is formulated as follows [36]:

$${\displaystyle \frac{d}{dt}}\left[x\left(t\right)\right]=rx\left(t\right)\left(1-{\displaystyle \frac{x\left(t\right)}{K}}\right)-qE\left(t\right)x\left(t\right),x\left(0\right)=x_0$$

(3)

where $x\left(t\right)$ is the fish stock size or biomass of the fish population at time t, $x_0$ is the initial biomass level, r is the intrinsic growth rate of fish biomass, q is the catchability coefficient, $E\left(t\right)$ is the rate of fishing effort at time t, and K is the carrying capacity for the given population. The carrying capacity K is the maximum population size which can be attained and the growth rate r captures mortality, age-structure, reproduction and tissue growth [13].

By looking at Equation (1), we see that the harvest or yield in Equation (3) is given by

$$H\left(t\right)=qE\left(t\right)x\left(t\right).$$

(4)

An equilibrium (or equilibrium point) of a dynamical system generated by an autonomous ordinary differential equation is a solution that does not change with time [37]. The two equilibrium points which are associated with Equation (3) are 0 and

$$x_{eqm}=K\left(1-{\displaystyle \frac{qE}{r}}\right)$$

(5)

which is positive provided that $E<{\displaystyle \frac{r}{q}}$. When $E\ge {\displaystyle \frac{r}{q}}$, $x_{eqm}$ is negative, the population will be reduced up until it goes into extinction.

The GS model gives the relationship between fishing activities and fishery stock biological growth. It is also used to measure the potential economic value of fisheries resources in conservation areas [38]. Its assumptions are that the growth rate is independent of the age composition and is highest when the fishery population is small, the costs and prices of fish remain stable over time [39], the efficiency of the vessels and gears remains constant, there exists no emigration or immigration in the fish population [34], there are no species interactions, no environmental factors affect the population, q is constant, and fishing and natural mortality take place simultaneously [13]. Although many of these assumptions may not be met in practice, if used critically, the GS model is a powerful tool for stock assessment [13].

Substituting Equation (5) into Equation (4) gives the following catch equation, which is also called the yield–effort function

$$H\left(E\right)=qEK\left(1-{\displaystyle \frac{qE}{r}}\right)=qEK-{\displaystyle \frac{q^{2}K{E}^2}{r}}.$$

(6)

Dividing both sides of Equation (6) by effort (E) gives the following equation, which is the relationship between catch per unit effort (CPUE) and effort

$$CPUE={\displaystyle \frac{H}{E}}=qK-{\displaystyle \frac{q^{2}KE}{r}}.$$

(7)

The catch and effort data for a fishery in a given lake or reservoir and for a period of time (usually in years) are usually collected from a database and studied. CPUE is calculated for each year (for all the years under the study) and checked if it is increasing or decreasing. Decreasing CPUE over the years is one of the indicators of overexploitation of the resource [11].

6.1.1. Parameter Estimation

Various parameter estimation methods are used when implementing the GS. The first method is by regressing CPUE [40]. The multiple linear regression analysis process is performed following [41] or by using computer software such as SPSS. Other parameter estimation methods include the CYP method used in [42] and the use of computer software programs such as CEDA (Catch and Effort Data Analysis) [28,43] and ASPIC (A Stock Production Model Incorporating Covariates) developed by Prager [44]. However, the regression method is recommended for illustrative analysis [45].

6.1.2. Annual Sustainable Production

The GS model enables researchers to make a comparison between the actual catch and the sustainable catch each year. The estimation of a sustainable production function is performed by using Equation (6) [42]. The result of sustainable production each year is compared with actual production in that year. If the actual catch is greater than the sustainable production, then this is a sign of overexploitation, especially if this happens over several years. Using the sustainable production function, two curves are drawn on the same axes, one for actual production and the other for sustainable production, to display the differences between the two in each year [33].

6.1.3. Depletion

Depletion is important because it provides a more accurate reflection of how much fishery resources have been lost. Depletion is expressed in monetary terms by calculating the corresponding depreciation. The calculation of depletion helps in tracking the catch rate and fishing effort applied to the fishery. Management decisions can then be applied by, among other things, restricting fishing days and gears [42].

6.1.4. Reference Points

In fisheries management, reference points, also called exploitation levels or management indicators, are used depending on the aim of management. The four reference points for the GS model are the maximum sustainable yield (MSY), the maximum economic yield (MEY), the open access yield (OAY) and the optimum sustainable yield (OSY). MSY is a biological exploitation level while the rest of the reference points represent economic situations.

The maximum sustainable yield (MSY) is the greatest amount of catch or yield that can be removed while keeping the stock sustainable. MSY is generally used to biologically conserve fish stocks [39]. Long-term fishing at levels beyond MSY leads to stock depletion.

Another reference point that is considered is the maximum economic yield (MEY). Unlike at MSY where the profit margin decreases, at MEY, the maximum profit is made through fishing [39]. Economists prefer MEY to MSY because MEY is used to increase profit as well as to biologically conserve fish stocks. However, ref. [39] suggested that despite the claims, the advantage of using MEY over MSY is not clear. Some studies have also indicated that using MSY can actually provide more economic output than MEY since MEY only considers individual fishing fleets while MSY is for the whole fishery [40].

Where there is no fisheries management, another reference point exists. This is called the Open Access Yield (OAY) where there are few or no restrictions on fishing. Under this reference point, there are many negative externalities, which means uncontrolled fishing leads to what is called “the tragedy of the commons” [40,46]. Unlike MSY and MEY, normal profit is obtained at OAY and as a result, many fishers are kept in the fishing business and others are attracted to join the industry [39]. OAY is associated with high effort and low catch compared to both MSY and MEY [47] giving zero sustainable net revenue. This results in economic overfishing [36].

When more effort is exerted at OAY than that exerted at MSY, the harvest at OAY exceeds the harvest at MSY, resulting in $x_{OAY}<x_{MSY}$ [36]. This situation is known as biological overfishing and, as a result, there is depletion of the resource. Therefore, an OAY fishing regime can lead to both biological and economic overfishing.

According to [36], the relationship between effort and stock size is that when the stock is low, effort must be high. The net revenue is the difference between the total sustainable revenue ($TR=\mathrm{price}\left(p\right)\times \mathrm{catch}\left(h\right)$) and the total cost ($TC=\cos \mathrm{t}\left(c\right)\times \mathrm{effort}\left(E\right)$):

$$\mathrm{sustainable}\mathrm{net}\mathrm{revenue}=ph-cE=pqEK\left(1-{\displaystyle \frac{qE}{r}}\right)-cE.$$

(8)

The first three reference points, MSY, MEY and OAY, are static, while a fourth reference point, the dynamic equilibrium condition, also exists. It is the Optimum Sustainable Yield (OSY). OSY considers the effects of the discount rate on fish stocks, fishing effort, harvest and sustainable net revenue [36].

Most analyses performed on the static model in (3) rely on the notion of equilibrium. However, in a real sense, equilibrium is a situation which is never attained for most systems due to changing environmental factors [36]. This calls for the need to supplement equilibrium-based or static methods with dynamic analysis to take into account the complex nature of the fishery.

Optimal control is a very useful tool in mathematics and has its applications in different fields, such as aerospace, bioengineering, robotics, finance, economics, biology and ecology.

OSY is the level of effort that maximises the net income, rent or profit. This level of effort maximises the economic profit, or rent, of the resource being utilised. Generally, the effort level at OSY is less than the effort level at MSY.

6.1.5. Bifurcation Analysis

The system in Equation (3) is a dynamical system since it describes the evolution of systems in time. Just like all dynamical systems, the system in Equation (3) has a state for every point in time, and this state is subjected to some rule which determines future states from the initial or current state. The dynamics of the system describe what is happening in terms of whether the system becomes fixed, settles down to equilibrium or fluctuates chaotically [48]. Bifurcation is an occurrence whereby only a small change is enough to move the system from one state to another. In other words, a bifurcation is a change in the stability properties of a dynamical system, including change in the number of equilibrium points, if a parameter is changed. A bifurcation point is the value of the parameter where there is change in the stability dynamics [36].

The state dynamics of the model in Equation (3) has two equilibrium points, namely, 0 and $x_{eqm}=K\left(1-{\displaystyle \frac{qE}{r}}\right)$. The bifurcation point is given by $E={\displaystyle \frac{r}{q}}$. This gives the number of trips per year for the fishery.

Following [36], to conduct bifurcation analysis, we compute both the equilibrium point $x_{eqm}$ and the bifurcation point $E={\displaystyle \frac{r}{q}}$. Then, we sketch three solution curves—the first one corresponding to the bifurcation point, the second one corresponding to a value below the bifurcation point (look at $x_{MSY}$ and $x_{MEY}$), and the last one corresponding to a value above the bifurcation point (consider $x_{MSY}$ + ${\displaystyle \frac{r}{q}}$). For any biomass level $x_0>0$, each solution curve shows whether the population approaches the equilibrium point 0 or $x_{eqm}$. The population that approaches $x_{eqm}$ (or any positive value) is the one which can persist while the one approaching 0 will go into extinction. Therefore, we are able to tell whether fish stocks exploited at the bifurcation point, below the bifurcation point or above the bifurcation point will persist or go into extinction. The solution curves also show stability of the system. So, from the solution curves, we are able to tell whether the dynamical system is structurally stable or unstable.

6.1.6. Tax Policy

Implementation of tax policy is one of the ways that are used to regulate the fishing industry. The types of tax policies include landing tax, effort tax and entry tax. The following are the formulas, respectively, as derived in [49]

$$T=p-{\displaystyle \frac{c{E}_{MSY}}{h_{MSY}}},$$

(9)

$$T={\displaystyle \frac{p{h}_{MSY}}{E_{MSY}}}-c$$

(10)

and

$$T=p{h}_{MSY}-c{E}_{MSY}$$

(11)

where T represents the tax, p is price, c is cost, E is effort and h is harvest.

6.1.7. Management Advice

The GS model is a bioeconomic model since it describes both the biological and economic nature of the fishery. It is used to determine both the biological and the economic parameters of the given fishery. By calculating the maximum sustainable yield, the model shows whether catch levels can sustain the fish population. If it is determined that the catch levels can actually lead to extinction of the species under study, the management advice given is to set catch limits. Calculation of maximum effort under MSY and MEY shows the effort levels which can sustain the population and the effort levels which are profitable to the fishers, providing the greatest net revenue [36]. Restricting fishing areas, fishing gears and fishing times can help reduce both catch and effort in the fishery [5]. Implementation of closed seasons ensures stock recovery in the long run. When the fish population increases, fishers incur less costs since they can easily catch fish with less effort [15]. As a result, the objective of maximising economic benefits is met while ensuring sustainable fish population. Bifurcation analysis shows whether it is sustainable to harvest the resource with effort at, below or above the bifurcation point. Tax and license regimes are some of the management actions which can be implemented to ensure long term sustainability of the resource.

6.2. The Fox and Pella–Tomlinson Models

The Fox model is based on the Gompertz growth model and its equation is [13]:

$$B_{t+1}=B_t+r{B}_t\left(1-{\displaystyle \frac{ln{B}_t}{lnK}}\right)-C_t$$

(12)

where B is the biomass, r is the intrinsic rate of population increase, K is the carrying capacity and C is the catch. The difference between the Schaefer and Fox models is that the former assumes that as a result of a high effort level, the stock can become totally exhausted and the yield becomes zero, while the latter assumes that an extreme effort level cannot result in stock becoming totally exhausted up to extinction [1]. However, the two models have no major differences between them.

The following equation is the Pella and Tomlinson model [13]:

$${\displaystyle \frac{dB}{dt}}=rB-{\displaystyle \frac{r{B}^m}{K}}$$

(13)

where B is the biomass, K is the carrying capacity, r is the intrinsic growth rate and m is called the shape parameter [26]. This is a generalised production model [28] which takes any form, including that of the GS when $m=2$ and Fox when $m=1$. One limitation of the Pella and Tomlinson model is that one must estimate an additional parameter m. As a result, this model is not more useful than GS and Fox since the relationship between the performance of the models and the number of parameters to be estimated is often inverse [13]. The preceding discussion perhaps is the reason why the GS model is the most used model among all surplus production models.

7. Other Stock Assessment Methods

7.1. Abundance Maximum Sustainable Yield (AMSY), Catch-MSY and CMSY Methods

The abundance maximum sustainable yield (AMSY) is a fish stock assessment method which uses a time series catch per unit of effort (CPUE), prior information (priors) of growth rate r and relative biomass level $B_t/k$ to estimate the relative population size [50]. Priors are obtained from online databases, expert knowledge or from independent data like length frequencies. This method is suitable for stocks in which catch data are not available. The AMSY method applies a state-space modelling framework. It employs a surplus production model to select values for r and relative carrying capacity $k_q$ to give predictions of catches compatible with CPUE [22].

Catch-MSY is a fish stock assessment method which is used to estimate MSY from catch data, the resilience of fish species and assumptions about stock sizes for two years—the first and last year of catch data. The minimum data requirement for this method include time series catch data, prior ranges of carrying capacity K and rate of population increase r, and biomass in the first and final years. Then, the Schaefer production model is used to predict yearly stock sizes using a given set of r and K values [24].

CMSY is a Monte Carlo fish stock assessment method which uses catch data and resilience of fish species to estimate biomass, MSY, exploitation rate and other reference points. CMSY was developed as an improvement to Catch-MSY. The Monte Carlo method is used to filter possible prior ranges of the maximum rate of population increase r and the carrying capacity K [23].

The main advantage of the AMSY, Catch-MSY and CMSY methods is that they require simple data for catch statistics or CPUE. The disadvantage of the AMSY method is that it is not suitable for use in making management decisions. Due to a lack of statistical catch data as input data, there may be wide margins of uncertainty for estimates of exploitation. This makes the method less appropriate for management. AMSY, Catch-MSY and CMSY share a common disadvantage that, if priors are uncertain or if they are not available, there may be bias in the outputs of the models, resulting in incorrect management recommendations [22,23,24].

7.2. Just Another Bayesian Biomass Assessment Method

Just another Bayesian biomass assessment (JABBA) is an open-source software used for fish population modelling. JABBA, which runs on the R working environment, is a Bayesian state-space model. It is used to estimate fish biomass and other fish population parameters [25]. The minimum data required for JABBA are two comma-separated value files, which are the catch by weight and abundance indices (CPUE) and the priors for the intrinsic growth rate r and the carrying capacity K. Missing catch input values are not allowed but missing abundance index values are allowed [51].

The advantages of the JABBA software include that it is flexible, can be run on a personal computer, is user-friendly, runs quickly and its estimates can be reproduced [52]. The limitations for using JABBA relate to the quality and availability of data. The model requires detailed data on catch, effort and abundance. Missing values or poor-quality data have the potential to affect the outcomes of this model [51].

7.3. Delay Difference Models

Delay difference models are an extension to surplus production models by including some biological parameters such as the age and size of the population [19]. They differ from surplus production models because they also model age-structured dynamics and the lag between spawning and recruitment, but they are not formal age-structured or size-structured models since they use simple assumptions of growth and survival. The minimum data requirements for delay difference models include time series of catch and effort data. Other biological information may be included as data or may be modelled. According to [26], delay difference models can be expressed using the following two equations:

$$B_t=s_{t-1}\alpha N_{t-1}+s_{t-1}\rho B_{t-1}+W{t}_{\mathrm{Recruit}\mathrm{age}}{R}_t$$

(14)

and

$$N_t=s_{t-1}{N}_{t-1}+R_t$$

(15)

where B is the biomass, R is the recruitment, N is the population number, $W{t}_{\mathrm{Recruit}\mathrm{age}}$ is the weight at age of recruitment and t is an index of time. $\alpha$ and $\rho$ are parameters of the growth equation:

$$W{t}_{\mathrm{age}}=\alpha +\rho W{t}_{\mathrm{age}-1}.$$

(16)

Delay difference models share advantages of surplus production models including that they are easy to implement [26]. They have an additional advantage that they account for both a per recruit and spawner effect [19]. However, in data-poor assessments, delay difference models offer little or no advantage over surplus production models since they require extra information [26].

7.4. Virtual Population Analysis

Virtual Population Analysis (VPA) is a stock assessment method which calculates past stock abundances by considering stocks in the past without any assumptions. VPA is also called cohort analysis. The basic model for VPA, according to [20], is

$$X_{t+1}=X_t-C_t-N_t$$

(17)

where $X_{t+1}$ is the fish population next year, $C_t$ is the catch this year, $X_t$ is the population this year and $N_t$ is the natural mortality this year. The data requirements for VPA include effort and weight-at-age data, and catch-at-age and age-specific abundance indices [53]. The assumptions of VPA include that there is no error in collecting and measuring catch data [26]. The main limitation for using VPA methods is that they are data intense [18] since they require high-quality and complete data which are not available for most stocks.

7.5. Length- and Age-Based Data-Limited Models

Length-based models are used to estimate abundance and fishing mortality by length given growth parameters and assumptions on natural mortality and the catch length-frequency distribution from a population which is assumed to be at equilibrium. The Beverton and Holt model and the Thompson and Bell model discussed above are examples of equilibrium length-based methods [26]. Another example is the length-based spawning potential ratio (LBSPR), which uses the spawning potential ratio (SPR) reference point to inform management decisions [54].

An example of an age-based method is catch-curve analysis. This is a fish stock assessment method used to estimate fish mortality rates and the age structure of fish populations. The input data are age-frequency data, which are also known as catch-curve data [55].

According to [26], age-structured methods have been applied to crustacean stocks in place of catch at age and virtual population analysis (VPA)-based methods in which there are problems with ageing. There are also models which integrate both age and length data, but such models are more complicated than age-based or length-based models [19].

The advantage of length-based or age-based models is that they make use of age and length-specific information [26]. Their disadvantage is that they require many observations and include many parameters [19].

7.6. Time Series and Forecasting

A forecast is a prediction of some future event or events. Most forecasting problems involve the use of time series data. A time series is a sequence of observations on a variable under consideration. There are three ways which are most widely used to conduct time series analysis. These are smoothing models, regression models and general time series models [56]. Regression models explain changes in fishery variables, such as catch and recruitment, in terms of changes in various biotic variables, for example, spawning stock, or abiotic variables, such as fishing effort and climate change [48]. The formal basis of most regression models is the method of least squares. Smoothing models use a simple function of previous observations to provide a forecast of the variable under consideration. General time series models use the properties of the historical data to derive a formal model and then estimate parameters which are unknown (mostly) by least squares. General time series models include autoregressive (AR), moving average (MA) and autoregressive integrated moving average (ARIMA) models [56].

Forecasting is important because the prediction of future events helps in planning and the decision-making process [56]. However, it is difficult to forecast accurately [48] since good predictions are not always easy to achieve [56].

8. Studies on Stock Assessment and Application of the GS Model

Stock assessment has been carried out across the globe in order to determine if yearly catches can sustain the stocks forever. Most studies have employed surplus production models, where catch and effort data analysis (CEDA) was conducted [43,57,58]. The most common surplus production model used is the GS model [1,26,38]. Although most research studies have reported overexploitation of the fishery resource, they fail to explain the depletion in monetary values called depreciation of the resource [33,59,60,61]. However, work by [42], which considered both MSY and MEY, has analysed fish depletion by comparing the potential sustainable yield and actual production. Depreciation was calculated in terms of the monetary loss due to depletion. This research was conducted in Indonesia.

Application of the GS model to determine biomass, yield and effort on all the four reference points, MSY, MEY, OAY and OSY, has not been widely undertaken. Most studies have considered MSY only ([4,33,62,63,64,65,66]), with other studies considering both MSY and MEY ([57,59,61,67,68]). Some studies considered the three reference points of MSY, MEY and OAY ([38,39,40,60,69,70,71]). Research by [36] considered all four of the reference points. OSY was considered by looking at the effects of discounting on biomass, effort and harvest or yield. This study, which was conducted in Ghana, also considered bifurcation analysis. Bifurcation analysis of dynamical systems helps understand how the system behaves and to find out possible causes of the system’s behaviour by reporting on the occurrence and changes in stability of equilibrium points and helping to model these changes and transitions from the stable to unstable case as some parameters change [72]. Bifurcation analysis helped the researchers to determine sustainable effort levels for the fishery. Another study by [49], conducted in the United States of America, also considered all the four reference points. The different tax policies to be imposed on fishers were also calculated. The imposition of a prohibitive tax in the form of license fees has been suggested by [73] as one of the ways which can help to sustain the fishery.

Table 2. Studies on fish stock assessment using the Gordon–Schaefer model.

Country	MSY	MEY	OAY	OSY	Bifurcation	Depletion	ASP	References
Egypt	Yes	No	No	No	No	No	No	[62]
Egypt	Yes	No	No	No	No	No	No	[64]
Egypt	Yes	Yes	No	No	No	No	No	[57]
Egypt	Yes	No	No	No	No	No	No	[74]
Indonesia	Yes	Yes	No	No	No	Yes	Yes	[42]
Indonesia	Yes	Yes	Yes	No	No	No	No	[38]
Indonesia	Yes	Yes	Yes	No	No	No	No	[71]
Indonesia	Yes	Yes	Yes	No	No	No	No	[70]
Indonesia	Yes	No	No	No	No	No	Yes	[33]
Indonesia	Yes	No	No	No	No	No	No	[75]
Oman	Yes	No	No	No	No	No	No	[59]
USA	Yes	Yes	Yes	No	No	No	No	[69]
USA	Yes	Yes	Yes	Yes	No	No	No	[49]
Morocco	Yes	No	No	No	No	No	No	[4]
China	Yes	Yes	Yes	No	No	No	No	[40]
Ghana	Yes	Yes	No	No	No	No	No	[76]
Ghana	Yes	Yes	Yes	Yes	Yes	No	No	[36]
Pakistan	Yes	No	No	No	No	No	No	[43]
Pakistan	Yes	Yes	No	No	No	No	No	[68]
Pakistan	Yes	No	No	No	No	No	No	[34]
Pakistan	Yes	No	No	No	No	No	No	[77]
Pakistan	Yes	No	No	No	No	No	No	[28]
Pakistan	Yes	Yes	Yes	No	No	No	No	[39]
Zanzibar	Yes	Yes	No	No	No	No	No	[65]
India	Yes	No	No	No	No	No	No	[63]
India	Yes	Yes	Yes	No	No	No	No	[47]
Iran	Yes	Yes	No	No	No	No	No	[58]
Kenya	Yes	Yes	No	No	No	No	No	[60]
Malawi	Yes	Yes	No	No	No	No	No	[73]
Malawi	Yes	No	No	No	No	No	No	[7]
Malawi	Yes	Yes	No	No	No	No	No	[73]
Malawi	Yes	Yes	Yes	No	No	No	No	[45]
Malawi	Yes	Yes	No	No	No	No	No	[15]

is a summary of the articles retrieved in this review. It includes the countries where studies were conducted, the reference points considered, whether bifurcation analysis was conducted, whether depletion was calculated and whether annual sustainable production was calculated.

In Malawi, various methods have been used to assess fish stocks of different species in lakes and rivers. For example, ref. [6] analysed catch and effort data between 1976 and 1999 in the Nkhotakota district. This study established that there was an increase in the number of boats, especially those without engines, total catches and effort for some fishing gears and decrease in CPUE for other gears. Similar results were obtained by [8]. However, these studies failed to use the GS or Fox models to perform further analyses due to the large fluctuations exhibited in the catch and effort data [6]. As a result, the studies failed to propose any management strategies for the fishery. However, the studies recommended maintaining effort levels coupled with accurate data collection and reporting.

Time series models have been used to forecast future catches for different fish species using catch data from the lakes in Malawi. The results of these studies are mixed, with some studies predicting an increase in catch after some years [78,79,80] and others predicting a decrease in fish catch after several years [81,82]. While these studies provide information on increase or decrease in fish catches, they fail to relate the catches to the amount of effort exerted. As a result, it is difficult to know how much effort is required to be exerted in order to achieve a sustainable catch.

Work on stock assessment using the GS model in Malawi includes [7], in which catch and effort data were used to analyse CPUE and MSY. Some studies considered both MSY and MEY [15,73], while other studies considered the three reference points of MSY, MEY and OAY [11,45]. Chambo (Oreochromis sp.) is the most studied species particularly because of its importance as food and its significance in Malawi’s economy [7,11,15]. Other species are usipa (Engraulicypris sardella) [73] and kambuzi (Haplochromis sp.) [45]. All these studies have reported overexploitation of the species under study but the studies have not clearly given an account of what was lost in terms of depletion and depreciation. Again, these studies did not consider the OSY reference point and bifurcation analysis was not performed. In addition, the cited studies did not discuss the stock assessment of other important species, such as mlamba (Clariid gariepinus), kampango (Bargrus meridionalis) and utaka (Copadichromis sp.).

9. Suitability of the GS Model to Malawian Fish Stocks

The GS model requires time series data of catch and effort. In Malawi, collection of such data started in 1976 following advice from the Food and Agriculture Organisation (FAO). Commercial fishery data are obtained through commercial fishers who submit monthly reports to the department of fisheries in Mangochi. Data on traditional fisheries are obtained through the catch assessment system and the Malawi traditional fisheries programs [5].

Inaccuracy of collected data prevented some researchers from applying any surplus production model, including the GS model. For example, [6] analysed fish data from the Nkhotakota district from 1976 to 1999. The analysis concentrated only on fish species and gears since the accuracy of such data was questionable because there were a lot of fluctuations in both catch and effort values. The fish species considered included chambo (Oreochromis sp.), utaka (Copadichromis sp.), usipa (Engraulicyppris sardella) and catfish (Clariid gariepinus) and the gears discussed included gillnets, chilimira seine, kambuzi seine, longlines, handlines and illegal gears, such as mosquito nets. The researchers failed to implement any surplus production model due to inaccuracy of data. Similar limitations of the data were observed in the data for the southeast arm of Lake Malawi from 1976 to 1999 [8]. The studies recommended improvement in data collection and reporting.

A study by Maguza-Tembo in 2002 used catch and effort data for the chambo (Oreochromis sp.) fishery and the whole fishery from Lake Malombe for 1976 to 1999 (Table S1 and Figure S1). Effort was standardised over three gears—gillnets, chiliira and kambuzi seine—to obtain the relative effort. Although the data values were noted to be inconsistent, they were still considered to be sufficiently reliable for modelling, which then enabled conclusions to be drawn about the stock status of the chambo (Oreochromis sp.) fishery. The data were inconsistent due to underestimation of the catches [11].

Improvements in the data collected were manifested in several studies conducted after a study performed by Maguza-Tembo (Table S1 and Figure S2). For example, a study by Singini successfully applied the GS model to the kambuzi (haplochromine cichlids) in Lake Malombe from 1976 to 2011 [45]. Another study by Gumulira also applied the GS model to fish data of usipa (Engraulicypris sardella) species for the southeast arm of Lake Malawi from 2000 to 2015. This study successfully concluded that usipa was fully exploited [73]. A similar study was conducted using data for the chambo (Oreochromis sp.) fishery from lake Malawi for the years 2000 to 2015 [7].

According to Zwieten, the catch and effort data from Lake Malombe and the southeast arm of Lake Malawi are among the best available in Malawi [83]. The availability of catch and effort data for key species, such as chambo (Oreochromis sp.), kambuzi (Haplochromis sp.) and usipa (Engraulicyppris sardella), and the fact that the GS model has been successfully applied to such data are an indication that the GS model is suitable for the Malawian fish sector.

10. Conclusions and Recommendation for Future Research

This paper has highlighted various fish stock assessment methods. It is difficult to implement data-rich methods, such as yield models and age and length models, in developing countries including Malawi because, in general, achieving accuracy of the collected data in Malawi is a challenge. Delay difference models, with data scarcity, do not offer any advantage over surplus production models. Empirical models and time series models have a common disadvantage that they do not estimate the optimal fishing effort. We therefore recommend surplus production models to be used in Malawi.

There is no major difference between the GS model and the Fox model except that the former assumes that, with high levels of effort, the stocks can become totally exhausted, leading to zero yield, an assumption which is not true with the latter model. One limitation of the Pella and Thomlinson model is that it requires estimation of additional parameters. Therefore, we recommend the GS model as the surplus production model to be used in Malawi. Depletion and depreciation of the resource should be calculated in order to give an account of what is being lost in monetary values. We recommend the use of all the four reference points, MSY, MEY, OAY and OSY, to describe the biomass, yield and effort. Inclusion of the OSY level in analysis ensures use of the dynamic version of the GS model. We also recommend bifurcation analysis to be performed in order to determine the optimal fishing effort. Future research on GS should integrate use of all four reference points, bifurcation analysis and calculation of depletion and annual sustainable yield.

The future direction of stock assessment in Malawi should include considering implementation of contemporary models, such as dynamic and state-space models. Further improvement in data collection is required to allow use of data-rich models.