# Sustainable Fisheries: Models and Management

## Introduction: Population Dynamics and Fisheries

Population dynamics models are a key component of fisheries sciences to describe the changes in populations over time and their responses to fishing. The need for modeling approaches originates from the difficulties to observe fish directly. Together with the large socioeconomic relevance of fisheries, this has put fisheries models at the forefront of modeling biological systems. The focus has traditionally been on describing changes in population biomass through the growth and decay of a population’s biomass. Historically, models have been divided into biomass models that lump entire populations into one biomass and models that are structured by age or size, allowing for more specific dynamics such as growth of body size, maturation, reproduction, recruitment and mortality. In this section, we will contrast biomass models with their structured counterparts, present models of growth, stock-recruitment and mortality, and outline current applications in a fisheries context.

**Biomass, Age and Size Structure**

In the 1950s, progress in industrial fishing triggered an increasing need to analytically describe the dynamics of fish populations in response to fishing, serving as basis to estimate the productivity of populations and maximize the yield of fisheries. Some of the first models to address these questions were the surplus production models, notably the Schaefer model by Milner B. Schaefer. These types of models describe a population as undifferentiated biomass that grows in response to the population size. Specifically, the Schaefer model (Schaefer 1954) assumes that changes in biomass are governed by population growth rate r, the carrying capacity *К* and the fisheries catch C:

C can be defined in different ways, most typically it is introduced as a product of *В* and a fishing rate (which in turn is usually fishing effort times a catchability coefficient). The core of the equation is, however, the logistic growth of the population, which levels out to zero at *В = К* (and at *В* = 0) and reaches a maximum

*гК К К г К*

of — at *В = —.* Consequently, in this model, *В* remains in perpetuity at — if C = —, which therefore

maximizes the sustainable yield of a fishery. This idea builds the foundation of the concept of a maximum sustainable yield (MSY) that to date dominates global fisheries policies (Hey 2012).

There are various modifications of the Schaefer model (Haddon 2010), such as, for instance, the Fox model:

which accounts for the fact that productivity in many fish species is assumed to be not a symmetric parabola but exhibit a maximum at B<—. An attempt to integrate different functional forms into one generalized surplus production model is the Pella and Tomlinson model:

Here the parameter s defines the shape of the relationship between biomass and productivity, corresponding to the Schaefer model s = 1 and displaying a left- or right-skewed parabola when s < 1 or s > 1, respectively.

Surplus production models can be considered as the first analytical models that have been applied systematically in fisheries science, both to assess the state of fish stocks as well as to determine reference points for management or explore conceptual questions. The biggest advantage of this class of models is their simplicity that requires comparatively little information or input data to generate (somewhat) meaningful predictions. It is for this reason why biomass-based models are still widely in use today, mostly in data-poor fisheries where the lack of information prevents the use of more sophisticated models. However, the simplicity is also the biggest caveat since omitting largely the biology and ecology of a population tends to oversimplify true dynamics. The demographic composition of a population and traits linked to demographics such as growth, maturity and fecundity are important factors of population dynamics. Ignoring these will, therefore, often result in biased or even completely wrong predictions. In the following sections, we will therefore focus on structured models of fisheries dynamics and their different components.

**Growth**

Besides the recruitment of new individuals to the population, increases in population biomass are caused by increasing body size of individual fish, making individual growth one of two major factors of a population’s production. Fish typically grow from millimeter-sized eggs to adults that can reach up to several meters in some species while undergoing dramatic changes in their ecology and morphology. By constraining the available food and potential predation, the size of a fish is a major determinant of its ecological niche and habitat choice, while food acquisition and environmental conditions drive changes in size. Growth is a trait that is shaped by the entire life history and ecology of a species (Enberg et al. 2012), such that, typically, short-lived species tend to grow faster than long-lived ones. Furthermore, the size of fish commonly determines their fecundity and reproductive potential, and thus, the size composition of a population may be important for its productivity (Hixon Johnson, and Sogard 2014). Understanding growth is, therefore, essential to model population dynamics.

Based on the model design and their application, growth models in fish can be separated into two groups: process-based growth models and statistical growth models (Enberg, Dunlop, and Jorgensen 2008).

The former type of models defines growth through the underlying biology of an organism and the environment it experiences, shaping the processes that govern growth. Typically, process-based growth models use insights from bioenergetics to predict growth as a product of energy acquisition and transformation, accounting for metabolic processes, behavior, food intake, temperature or life-history tradeoffs, as for instance between growth and reproductive investment. Process-based growth models aim at a fundamental understanding of growth, providing an approximation to the mechanisms that determine growth and enabling them to predict more accurately growth and how it may change in response to environmental and anthropogenic disturbances. In contrast, the second group of growth models focuses on a statistical description of observed sizes with little to none of the underlying biology of growth included. These models contain often fewer parameters and assume a direct relationship between age or size of an organism with growth. Examples are the logistic growth model

or the Gompertz growth model

that assume that the growth rate *G,* is a linear function of absolute or log-transformed weight W at time *t* subject to growth parameter *К* and an asymptotic maximum weight *W„.* While ignoring the biology of growth processes, statistical growth models can be applied to size-at-age data that is readily available for many commercially harvested fish populations. Process-based growth models, on the other hand, require more knowledge of the growth processes and data to fully parameterize the model. Particularly the necessary data on bioenergetic processes is often missing, turning the larger flexibility and predictive power of such models into a disadvantage in a practical context. Consequently, statistical growth models tend to be more common in many areas of fisheries science such as stock assessment, as they are sufficient to describe the observed size compositions of a population.

There is, however, an overlap between the two groups of growth models. Statistical growth models can be extended with additional physical or biological parameters, approximating better the actual drivers behind growth. On the other hand, process-based growth models are often used in purely statistical approaches. A good example for such applications is the von Bertalanffy growth model (VBGM), which stems from a model that includes anabolic and catabolic processes but is commonly fitted to size data like a statistical model.

Ludwig von Bertalanffy originally proposed a general growth model describing the change in length as a differential equation of length at time *t,* the maximum length *L„,* and a growth parameter r:

Mechanistically, this is founded on the differences in how anabolic and catabolic processes scale with body weight W:

where *a* and *c* are the proportionality coefficients for anabolism and catabolism, respectively, and *m _{x}* and

*m*the corresponding scaling exponents. Assuming that the acquisition and thus anabolic processes are

_{2}2

proportional to body mass by *m _{x} = —* and catabolic processes (metabolism and maintenance) by m, =1,

this results with increasing body size in a higher proportion of available resources spent on catabolic processes, leaving less for growth.

The VBGM has later been introduced by Beverton and Holt into fisheries, where it has become and remained the dominating model for fish growth, mainly because it provides typically good fits to length or weight data of most fish species for both individuals and population averages. The standard form of the model is used to calculate the length or weight at time *t* and results in an asymptotic shape, i.e., size approaches a maximum while growth increments decrease over time:

Length or weight are, therefore, determined by the growth coefficient *к* and an asymptotic (maximum) length *L,„* or weight The exponent *b* is derived from the age-length relationship L, *=aW _{t}^{b}* (and in most applications simplified to a cubic relationship), and

*t*is an (hypothetical, negative) age when size is zero. The latter is included to avoid that length or weight equal to zero at hatching (f = 0).

_{0}The VBGM assumes discrete time steps that correspond in most applications to years. However, this may be inadequate from a biological perspective since temperate and boreal species typically do not grow equally throughout the year but show distinct intra-annual growth patterns that align with the seasonal food availability. To account for such dynamics, the standard VGBM can be modified by introducing cyclical growth patterns:

with and *s _{2}* determining the shape of the oscillation and s

_{3}the frequency by subdividing the time step, e.g., in the most common case of annual time steps into months (s

_{3}= 12) or weeks (s

_{3}=52).

Such modifications also reveal the main limitation of the standard VBGM: lacking functional flexibility and very simplified or erroneous biology. Specifically, the underlying scaling of anabolic and catabolic processes with size has been empirically shown as very similar, with values ranging between 0.7 and 0.8 for both exponents in most fish species. Furthermore, the VBGM neglects crucial insights from life-history theory and, therefore, captures juvenile growth less accurately than adult growth. The reason for this is that the model does not account for maturation and reproduction. Reproductive investment is very energy-demanding and, thus, requires organisms to allocate a major share of the acquired energy to it. Since resources for basic maintenance processes can only be reduced to a very limited extent, reproductive investment mainly comes at the expense of growth. Consequently, there is a trade-off between growth and reproduction, resulting in different growth trajectories before and after maturation. The VBGM does not incorporate these dynamics and, therefore, tends to underestimate the growth rates of juvenile fish, which typically approximates linear growth. The following two models attempt to address these limitations by allowing for more functional flexibility or specifically incorporating life-history considerations.

Generalized models serve the purpose of aggregating different models into one equation that can take different functional forms depending on the parameter values. The model developed by Schnute and Richards generalizes several of the growth models used in fisheries, including the VBGM and logistic model as well as models previously proposed by Gompertz, Chapman, Richards, or Schnute:

Setting parameters *c* and *d* equal to one, for instance, reduces the model to the VBGM. A generalized model such as the one from Schnute and Richards allows therefore for a better representation of observed data and biology of a population. Nevertheless, this presents a statistical approach rather than a processed-based one, possibly explaining data well but without biological understanding. Additionally, the large number of parameters may make the fitting process challenging, particularly if no prior (mechanistic) knowledge is included.

A different approach has been taken by biphasic growth models that build explicitly on life-history theory and incorporate the trade-off between growth and reproduction. These build on bioenergetics and the assumption that the change in somatic weight *W* depends on the acquired energy *E,* and the energy invested in gonadal weight *G _{ni}:*

Provided that weight is a cubic function of length, W = L^{3}, this can be used to model the growth in length. The model by Roff (1983) is one application of this idea, assuming a linear growth for juvenile fish, i.e., when age *a* is lower than the maturation age *a,„ _{al>}* whereas growth will depend on the gonado- somatic index (GSI) from the onset of maturation:

GSI is gonad mass divided by somatic mass, which means the larger the investment in *R, _{+]}>* the more will growth in length be reduced.

A very similar approach has been taken by the modified model based on Quince et al. (2008) in which the growth is also shaped by the reproductive investment *R, _{H}* from of the GSI (Boukal et al 2014):

For fish below*a, _{mt},* R

_{l+1}equals zero, whereas from

*a*onward

_{m}„*R*> 0. The modified Quince et al. model resolves some limitations of Roff’s model and incorporates a larger functional flexibility. This includes a less constrained length-weight relationship than Roff’s model by assuming

_{t+l}*W, = aLf,*a conversion factor between somatic and gonadic investment

*q,*and by not enforcing strictly linear growth for juvenile individuals. Both models, however, incorporate the same key feature that results in biphasic growth trajectories, separating quasi-linear growth prior to maturation from a decreased growth after the onset of maturation that depends on the reproductive investment. These models illustrate how our understanding of life-history processes can be included in growth models to represent better how growth can vary between different life stages and achieve better fits to empirical data.

Besides the trade-off between growth and reproduction, there are other trade-offs that may directly or indirectly affect growth. A major driver is survival and, therefore, everything that affects mortality, most notably predation and fishing. For instance, very size-selective mortality can increase survival for fish that invest more into growth (instead into basic maintenance processes such as the immune system) to grow faster through the size range of increased mortality. Similarly, behavioral adaptations may result in decreased or increased growth. For instance, passive behavior such as hiding can be used to reduce predation risks at the expense of reduced foraging, reducing the resources available for growth and reproduction. In contrast, growth may be increased through more active foraging and a bolder behavior; however, this may also expose fish to higher predation and reduce the probability of survival (Claireaux, Jorgensen, and Enberg 2018).

Essential life-history traits such as growth, reproductive investment, survival, and behavior are to a large part determined by an individual’s inherited genotype. This means that traits are shaped by evolution and subject to evolutionary change that depends on the reproductive success of a specific life history within a given environment. Consequently, how resources are acquired and allocated into growth or reproduction is less of an individual decision than the result of an inherited life-history strategy. The trade-offs between growth, reproductive investment, and survival are key to this process and determine the success of a specific strategy under the current environmental conditions in reproducing and thus inheriting the same strategy to the next generation. Because mortality is a major driver of natural selection, changes in the degree or selectivity of mortality may affect the selection landscape and result in evolutionary adaptations in growth or traits that influence growth. This means growth trajectories within a population are not stationary over time but may change in response to environmental change and anthropogenic perturbation such as fishing or increasing sea water temperatures.

Although the growth of an organism is fundamentally shaped by life-history evolution, most observed changes in growth and thus size-at-age occur in the short term as a result of phenotypic plasticity. The most important driver is environmental variability, specifically physical and ecological conditions that influence metabolism and food availability. The latter is particularly relevant, since it determines directly the available energy that can be acquired by an organism and invested into processes such as growth. Food availability per capita is the combined result of food production through the food web and the competition for the available food sources. On a seasonal or annual time scale, growth of an organism can therefore be determined by bottom-up effects through variation in the ecosystem productivity, e.g., through varying nutrient inflow or temperatures, as well as through the abundance of its own population and other competing species. Feedbacks occur between the environmental variability and density dependence, for instance, when high food availability leads to increasing population abundance(s) and thus to competition in the future. Although the causes of environmental variability are often difficult to determine and parametrize in models, density-dependent growth has been empirically established (Zimmermann, Ricard, and Fleino 2018) and may affect the sustainability of fisheries (van Gemert and Andersen 2018). A simple approach to implement density dependence in growth models is the use of an asymptotic length *,* that decreases as a function of a density dependence coefficient *d* and the population biomass B, in each year: *L„j = L„-dB,* (Lorenzen and Enberg 2002). This example illustrates that not only insights from bioenergetics, physiology, and life-history theory are important to modeling growth but also insights from population ecology.

**Recruitment**

Recruitment is a key component of population dynamics and contributes together with body growth to the increase in biomass within a population. Because of the enormous reproductive potential of most fish species, recruitment tends to be the most important factor for the overall productivity of a population and the major driver of fluctuation in population size. Recruitment as such is the combined result of the total number of eggs produced by the mature part of a population and the survival throughout the early life stages from egg to juveniles, which explains the large variation in number of recruits observed in most fish. Typically, fish produce very large numbers of eggs per individuals, reaching millions per spawning event. At the same time, the early life stages are very vulnerable to unsuitable physical conditions, predation or insufficient food, causing large inter-annual variation in survival. Consequently, recruitment can result in favorable years in very large cohorts that sustain a population for many years during which recruitment may be average or fail completely. This variability in recruitment, however, poses also a challenge for any attempt to model and predict recruitment. Nevertheless, because recruitment is fundamental for population dynamics and thus fisheries, various recruitment models have been established. Most of these models rely on the basic assumption that recruitment must be related to the mature part of the population and is subject to some form of density-dependent reduction. The two models that remain most widely used until today are the stock-recruitment models developed by Ricker (1954) and Beverton and Holt (1957).

The Beverton-Holt model assumes a stock-recruitment relationship that increases with increasing biomass of mature fish *В* at time *t,* however, with decreasing number of recruits per spawning individual and thus approaching an asymptotic maximum of recruitment:

with *a* representing the asymptotic maximum for a given *В*, and *p* the population biomass where *all* is reached, defining the steepness of the curve. Biologically, *a* stands for the maximum spawning output that linearly increases with population biomass, whereas *p* defines the density-dependent regulation in recruitment and therefore the productivity of a population at specific biomass levels. The underlying mechanism is the density-dependent survival of early life stages, which is assumed to decrease with increasing amount of eggs spawned by a larger population biomass due to intra-cohort competition for resources, particularly food (Van Poorten, Korman, and Walters 2018), and stronger predation pressure.

The Ricker model takes a similar approach as the Beverton-Holt model, except that it assumes an overcompensatory effect of increasing population biomass on recruitment. This implies that the Ricker stock-recruitment curve reaches a peak recruitment after which the realized number of recruits decreases again, instead of simply approaching an asymptotic maximum. The typical equation to describe this relationship is denoted as:

Here, *a* defines the recruitment at a low biomass of the spawning population and scales the total number of recruits, whereas /? determines the density-dependent decrease in recruits per spawning biomass. As in the Beverton-Holt model, *a* represents the reproductive output of the mature population and *p* the density-dependent mortality experienced by early life stages after spawning. The key difference between the two models is that the recruits per spawning biomass in the Ricker model do not remain at a value larger than zero but approach zero, suggesting that a population biomass above a certain level has such detrimental effects on recruitment that it overcompensates the marginal increase in spawning output. Such effects can occur through substantial negative inter-cohort interactions through cannibalism or competition (Ricard, Zimmermann, and Heino 2016), i.e., older cohorts that deplete the resources of following cohorts or prey directly on them, or other negative feedbacks, for instance when growth at early life stages is density-dependent while predation is size-dependent. This may add up to a substantially increased mortality when cohort density decreases growth rates.

It is noteworthy that the Ricker curve can take an almost identical shape as the Beverton-Holt curve for an observed range of population biomass and number of recruits, making the Ricker model more flexible in representing populations with different recruitment patterns. A step further in this direction is taken by generalized recruitment models that allow for a large functional flexibility with other models as special cases. One example for such an approach is a model suggested by Deriso and later modified by Schnute:

Here *a* and *p* take equivalent roles as in the Beverton-Holt or Ricker models, while parameter *у *determines the form of the recruitment curve. For instance, when *у* goes to 0, the Deriso model corresponds to the Ricker model, and у = -1 transforms it into a Beverton-Holt-type model. A generalized model of this kind avoids the need for prior assumptions on the type of relationship, enabling better fits to data or to test effects of gradual changes in the functional form. The downside of such an approach is, however, that it takes mainly a statistical and not a process-orientated approach to incorporate biological knowledge. Furthermore, recruitment data typically turns out to be very noisy, which may make it very difficult to find reasonable parameter estimates for a model with a high degree of functional freedom.

Empirically, the biomass of the spawning population is in most cases an insufficient predictor of recruitment. As a consequence, models such as Beverton-Holt or Ricker typically fit poorly to data.

Recruitment data is in general very noisy and shows a lot of variation. Because survival at early life stages tends to be much more important than the total reproductive output of the mature individuals, other factors such as environmental conditions, food availability and predation that directly or indirectly affect mortality of eggs, larvae and juveniles are key drivers of recruitment (Zimmermann, Claireaux, and Enberg 2019). Especially large-scale atmospheric and oceanographic processes with cyclical patterns have been identified as important forcing factors. A simple approach to model such cyclical patterns in recruitment is to include an autoregressive term in a stock-recruitment model:

representing a Ricker model that includes a *AR(*l) process defined as *u _{t} =* <

*pu*,

_{4}with

*(p*as autocorrelation coefficient. This allows for capturing temporal autocorrelation in recruitment time series and simulating inter-annual cyclical patterns.

A different approach to extend a Ricker model is to include explicitly an additional variable that is related with recruitment:

Here *X* is the second variable besides mature biomass *В* and *у* is the corresponding coefficient. Examples for *X* could include any environmental or ecological factor that is expected to affect the recruitment of a specific population, such as annual sea surface temperature, zooplankton indices or the biomass of a predator. This approach can be further extended with additional, equally defined terms.

Classic stock-recruitment models assume that the relationship between mature population biomass and recruitment remains stationary over time. However, this assumption may often not hold because the reproductive potential of a population as well as the mechanisms of density regulation can change over time due to external factors, both anthropogenic and natural. Regime shifts can, for instance, occur when changes in the environment or anthropogenic impacts alter the ecosystem productivity with implications on the reproductive success of a population and thus the relationship between the mature population and recruitment (Vert-pre et al. 2013). To incorporate two different regimes, a Ricker model can be modified:

*a* and *p* depend now on *i* = 1,2 which can be defined as two different periods in a time series.

The number of regimes can be extended further if required. This comes, however, with an equally increasing number of parameters as caveat. Considering that time series of recruitment data usually only cover a few decades, the number of parameters can become easily disproportional compared to the number of data points. A solution to this problem is to introduce a time-invariant parameter that captures gradual changes in the relationship between population and recruitment over time (Perala and Kuparinen 2015). For instance, if we assume that the productivity of the mature biomass varies with time, time-variant *a,* can be introduced:

where parameter *a* follows a random walk process *a, = a,-i +* cr„ with *a* assumed to be normally distributed. This enables the productivity term in the stock-recruitment relationship to vary gradually over time, representing changes in the population’s reproductive output for instance through changes in the population’s demographic structure due to fishing or because of increasing temperatures, as well as subsequent evolutionary adaptation (Enberg et al 2010).

**Mortality and Population Dynamics**

Whereas individual growth and recruitment represent the increase in biomass of a population, mortality constitutes the loss term in the equation. Survival is therefore a crucial component in fisheries models, making them very sensitive to the underlying assumptions and specifications of mortality. Mortality is typically given in rates that define the survival of individual fish over time. Two main sources of mortality are distinguished: mortality from natural causes, most notably predation and diseases, commonly denominated as natural mortality *M,* and mortality from anthropogenic harvesting, commonly termed fishing mortality *F.* These instantaneous rates of mortality can be summed to total mortality Z = *F+M, *which translates into proportional survival after a given time step as *e~ ^{z}* and, reciprocally, relative mortality as 1 -

*e*~

^{z}. In an age-structured population, the abundance reduces in one time step

*t*by:

or alternatively an abundance at a given age *a* of:

with *N _{0}* as initial abundance of a cohort, typically corresponding to recruitment

*R,.*

Mortality is, however, in most fish species dependent on size and, thus, age of an individual. This applies both for *F* and *M.* While other sources of mortality such as diseases, parasite or starvation can be relevant as well, *M* is to a large degree shaped by predation. Vulnerability to predation is influenced by various traits, including the individual behavior, but most importantly body size because most predators have a size window for prey based on limitations in perception and handling. For a given stock, smaller and younger fishes tend to be much more vulnerable to predation than bigger and older ones, which means that mortality decreases substantially with increasing size and age.

*F* is subject to similar size selectivity, indirectly because parts of populations such as pre-recruits often do not share the same habitat as the ones targeted by fisheries and directly through size-selective fishing gear. In contrast to *M, F* increases in most cases with age and size: while small, young fishes are usually excluded from fisheries and experience very low fishing, the *F* for the targeted age and size classes can be substantial and outweigh *M.* These dynamics can be captured by using age-specific mortality rates *F„* and M„, leading to

This model can be further extended by allowing *F* and *M* also to vary in time:

The two-dimensional mortality matrices here are age- and time-specific, which is common for stock assessment models where annual *F _{a}’s* are estimated.