Surplus Production Models JON T. SCHNUTE AND LAURA J. RICHARDS

6 Surplus Production Models JON T. SCHNUTE AND LAURA J. RICHARDS

6.1 INTRODUCTION

process analogous to the growth of individual animals. Thus, the collective biomass has an

A sustainable fishery requires a productive fish asymptotic capacity similar mathematically to stock. On an evolutionary scale, most stocks have the asymptotic size of a single fish. Our analysis in adapted to a long history of changing circum- this chapter makes this analogy precise through a stances. Theoretically, this implies an ability to differential equation that links historical growth respond to sudden new causes of mortality, such models with the production models of Schaefer as that imposed by a fishery. If the population (1954, 1957), Pella and Tomlinson (1969) and Fox responds with additional productivity to compen- (1970, 1975). Smith (Chapter 4, this volume) pro- sate for fishing mortality, then this increase can be vides an historical perspective on the development called ‘surplus’ production.

of surplus production models.

A finite carrying capacity represents one pos- The modern approach extends these classical sible mechanism for surplus production. Accord- models by including more biological detail, such ing to this scenario, the biomass expands to a level as explicit assumptions about recruitment, mor- that can be sustained by the environment. If the tality and growth (Sparre and Hart, Chapter 13, biomass becomes larger, then competition for this volume). The emphasis has shifted from dif- scarce resources increases overall mortality and ferential to difference equations that relate more causes the biomass to contract. Fishing imposes obviously to the data available from most fish- mortality that reduces the biomass below carrying eries. All models, whether classical or modern, capacity. At this lower level, reduced competition deal with three essential questions: among surviving animals and greater relative

1 What processes govern the population dynamics abundance of resources allows the population to re- of fished stocks? spond with surplus production. Theoretically, the

2 Given these processes, what is a ‘safe’ harvest biomass can be held indefinitely at a level below level that produces an ‘optimal’ catch while main- carrying capacity, and the fishery can continue to taining a ‘healthy’ stock? operate on the corresponding surplus production.

3 What do the available data reveal about the Essentially, every assessment model of a sus- assumptions in question 1 and the answers to tainable fishery involves the concept of surplus question 2? production, although models vary in their detailed

Question 2 requires defining certain key cri- descriptions of population dynamics. Early pro- teria encapsulated by vague adjectives: safe, opti- duction models used differential equations to de- mal and healthy. One classical approach comes scribe a single stock restricted by a finite carrying from the carrying capacity scenario described capacity. These represent population growth by a above. A closed fishery produces no catch, and the

Chapter 6

stock maintains itself at carrying capacity. By con- trast, a very intense fishery drives the population to extinction, leading eventually to zero catch. Theoretically, some intermediate level of fishing mortality should give the maximum sustainable yield (MSY) under equilibrium conditions. In the Schaefer model, for example, the equilibrium rela- tionship between yield and biomass has a para- bolic form, where maximum yield occurs when the biomass is held at 50% of carrying capacity. Thus, the model suggests this level of abundance as a reference point for a healthy stock size associ- ated with an optimal catch. Both classical and modern models can be used to define a variety of similar reference points, depending on goals set for the fishery and the marine ecosystem.

Mathematically, the three questions cited above correspond to three distinct phases of the analysis. First, a model must be articulated that de- scribes the hypothetical stock dynamics. Second, model properties must be investigated to deter- mine the effects of various fishing strategies. Third, quantities within the model must be compared with actual data to estimate reference points needed to implement the chosen strategy. In mod- ern terminology, quantities internal to the model arecalled‘states’andthemodelitselfisa‘state space model’. For example, the model might describe the dynamics of stock biomass, which is not observed directly. This hidden state variable does, however, influence various observed quantities, such as the catch or an abundance index. The available data may or may not be adequate to estimate all the hid- den states, as well as fishery reference points.

Anyone wishing to pursue the history of pro- duction models can expect a fairly intense math- ematical journey. A reader need only glance at the historical papers by Pella and Tomlinson (1969), Fox (1970, 1975) or Schnute (1977) to find equa- tions in abundance. Mathematical models serve as metaphors of reality. The strength of mathematics lies in its ability to explore consequences. A few assumptions can produce an elaborate description of fish population dynamics, where mathematical results guide biological interpretation and under- standing. On the other hand, mathematics is limited by its complete dependence on the

assumptions, which might be wrong. Some of the models presented in this chapter have remarkable elegance, but this does not guarantee that they correctly represent nature. Given the complexity of marine ecosystems, a modeller must treat any stock assessment model with extreme scepticism directed towards assumptions that might fail. No single approach offers a panacea that applies to all situations (Sparre and Hart, Chapter 13, this volume).

We focus our discussion on two surplus produc- tion models that encapsulate classical and modern points of view. The first uses a differential equation proposed by Pella and Tomlinson (1969). We compare this framework with difference equations obtained from a more extensive age- structured analysis (Schnute and Richards 1998). These two approaches have features in common, but also significant differences. Each example gives mathematically tractable results that shed intuitive light on the inner workings of fishery models. Our comparison leads to a few results never published previously, although other as- pects of the material appear in greater detail else- where (e.g. Quinn and Deriso 1999).

Consistent with the history of this subject, our presentation has a high technical content. For per- spective, we organize most of the mathematics into tables, and focus primarily on the underlying concepts. We often omit proofs, but emphasize logical relationships between assumptions and conclusions. We regard models as tools for thought, rather than ultimate descriptions of reality. These tools can be used to the greatest advantage by understanding them in detail. Fea- tures of simple models can guide the development of more complex models. We ask readers to ap- proach our material with a sense of adventure. Scaling a mountain to gain perspective requires a passage through numerous difficulties and lesser vistas along the route.

Equations in our summary tables follow a sim- ple numbering system; for example, (T6.2.3) refers to equation 3 in Table 6.2. Subscripts usually indi- cate time or age variables, which may be discrete or continuous, depending on the context. For in-

stance, B t ,w a , and w • denote biomass at time t,

Surplus Production Models

60 ) B Fig. 6.1 Key concepts in surplus

production models, where the equilibrium catch rate C (solid

40 Biomass ( line) and biomass B (broken line)

Catch rate ( 2

depend on the fishing mortality

rate F. Symbols indicate the 20 following conditions: pristine

stock (䊊); fishery at MSY (䉫); 0 0 stock extinction (䊐); and a

precautionary fishery with catch 0.0 0.02 0.04 0.06 0.08 0.10 0.12 rate lower than MSY (+).

Fishing mortality rate (F)

(6.1) An asterisk indicates a value associated with sta-

weight at age a, and asymptotic weight as a Æ •. E t 哫B t ,C t .

ble MSY conditions. Thus, the maximum sustain- In this scenario, the biomass B t usually acts as a able catch rate C* occurs when the biomass B* hidden state that cannot be observed directly. experiences an optimal fishing mortality rate F*. A Thus, in practical terms, (6.1) reduces to prime symbol denotes a value associated with pris-

(6.2) (Remember that ‘prime’ means ‘pristine’.) Thus, B¢

tine conditions, that is, a stable unfished stock. E t 哫C t ,

is the pristine stock biomass, sometimes desig- which translates to the more romantic statement nated B 0 in other fishery literature. We retain the ‘Boats go out to sea and come back with fish.’

temporal meaning of the subscript 0, so that B 0 de-

notes the initial biomass at time t = 0. This may or may not be the pristine biomass; thus, the condi-

6.2 GRAPHICAL MODEL

tion B 0 = B¢ indicates that the model begins with an unfished biomass. Models describe the evolution Before developing formal mathematics, we use of time-dependent states, given a vector Q of model

a simple graph to illustrate the key features of a parameters that typically remain constant in time. surplus production model. Figure 6.1 portrays the Sometimes, however, the distinction between long-term impact of a fishing mortality rate F on a parameters and states weakens, and it becomes stock biomass B. In the absence of fishing (F = 0), simpler to think of every quantity in the model as a the stock maintains itself at the pristine level state.

B ¢ = 100 biomass units (e.g. tonnes), which is the We give models two levels of interpretation. natural carrying capacity. As F increases, B drops First, we emphasize the biological meaning of as- below B¢, and this reduced biomass leaves room for sumptions and corresponding results. Second, we growth. The fishery harvests the resulting surplus examine each model from a systems perspective, production, generates a catch rate C, and main- in which some states act as controls that deter- tains the biomass at level B. All quantities in Fig. mine other states. For example, an effort rate E t

6.1 represent long-term conditions. Thus, the mor- might alter the population biomass B t and produce tality rate F maintains the catch rate C by remov-

a catch rate C t , as indicated by ing exactly the surplus production available from

Chapter 6

the biomass B. Conceptually, F represents the frac- tion of the biomass: F = C/B. By comparison with tion of B removed per unit time to produce the equation (6.3), the additional parameter catch rate C = FB, measured as the biomass cap- tured per unit time.*

C * 5 ty - 1

= 5 % y - A high mortality rate can potentially drive the 1 (6.4)

y=

100 t

population to extinction. In Fig. 6.1, for example,

B = 0 when F = 0.13; thus the stock cannot main- gives an overall measure of stock productivity by tain a removal rate greater than 13% per unit time. relating the maximum sustainable catch rate to Following Schnute and Richards (1998), we denote the pristine biomass. this critical point as F † . (Remember that ‘dagger’

Figure 6.1 makes a useful topic of conversation means ‘death’ to the population, i.e. extinction.) with fishermen embarking on a new fishery. Ac- Thus, two conditions imply a catch rate C = 0: no cording to this scenario, the biomass can initially fishing (F = 0) or extinction fishing (F ≥ F † ). A mor-

be reduced from 100 t to the optimal B* = 57.8 t, tality rate F between these extremes (0 < F < F † ) pro- producing a windfall catch of 42.2 t. This might duces a positive catch rate C > 0. In particular, the happen over two years in a fishery that takes maximum sustainable catch rate C* corresponds more than 21 t per year. Unfortunately, such a to an appropriate mortality rate F*. The following high mortality rate, much greater than F † , would short table summarizes the three stock conditions rapidly drive the stock to extinction. After identified so far, as designated by accents in the the initial windfall, the catch must be curtailed to text and symbols in Fig. 6.1:

a much slower rate near 5 t per year. Fishermen who have become dependent on four times this rate may suffer economic ruin. Furthermore,

Condition Accent

the figure represents a population that declines Pristine

Symbol

more rapidly as the mortality F approaches the critical F MSY † * 䉫 . In the uncertain world of fishery man- Extinction

agement, it might be dangerous to adopt even the theoretically optimal catch rate C* = 5ty -1 .

A more precautionary approach (marked ‘+’ biomass in tonnes (t). Then Fig. 6.1 implies the in Fig. 6.1) could be achieved with B = 78t and

Suppose that time is measured in years (y) and

F = 5.1% y following reference points: B¢ = 100t, F* = -1 , producing an approximate catch rate 8.7% y -1 , B* = 57.8t, C* = 5ty -1

-1 . C =4ty , and F -1 † = 13% y . Such a policy would reduce the risk of In particular,

extinction by maintaining the biomass at a higher, more robust level.

C * 5 ty - 1 The graphical model in Fig. 6.1 has two compo-

B * nents, a dome-shaped catch curve and a descend- . 57 8 t ing biomass curve. A mathematical model that

= 87 .% y - 1 .

More generally in Fig. 6.1, the fishing mortality represents these components must include at least rate F expresses the annual catch rate C as a frac- three parameters: a maximum level for the bio-

mass (B¢), a scale parameter for the catch relative to the biomass (y = C*/B¢), and a location parameter that determines the position of the dome (F*). For example, a relatively high F* in Fig. 6.1 gives a

* In some non-equilibrium contexts, this relationship between the instantaneous rates C and F implies an accu-

dome shifted towards the right end of the catch mulated yield Y = (1 - e -Ft )B during the time interval t.

curve. Any equivalent set of three parameters can Models presented later in the paper clarify such alterna-

be used as surrogates for (B¢, y, F*). For example, tive formulations. For now, experienced readers can take

our first model includes B¢ explicitly, but captures note of the limit C = lim tÆ0 Y /t = FB.

y and F* through two other parameters labelled a

109 and g. Mathematical analysis of the model gives Table 6.1 Production model 1, based on a differential

Surplus Production Models

formulas for (y, F*) in terms of (a, g). Keen readers equation proposed by Pella and Tomlinson (1969). can sneak a glance ahead to equations (T6.3.2) and (T6.3.7) in Table 6.3. Furthermore, for those who Model Q = (B¢, a, g, q

want an even better preview of things to come, (T6.1.1)

1 ,q 2 )

(T6.1.2) and B(F) portrayed in Fig. 6.1. How do the simple g dB

equations (6.12) and (T6.1.6) define the curves C(F) F t =q 1 E t

ideas in the graphical model suggest such prolific

mathematics? Read on.