SURPLUS PRODUCTION REVISITED
Extensions to the Graham-Schaefer Production Model
The standard Graham-Schaefer surplus production model is based on various simplifying
assumptions, including the following:
Fishing mortality is proportional to fishing effort

MODELS FOR VARIABLE RECRUITMENT (continued)
Fitting Real Data to the Spawner-Recruit Models
One strategy for fitting these spawner-recruit models to real data is to linearize the models by
means of some suitable transformation and then apply standard line

THE CATCH PROCESS (continued)
In our previous derivation of the relationship between CPUE and fish abundance we assumed that
all the fishing units and all the fish were spatially homogeneous. Now we explore what happens
when this is not the case.
Spatial

TOTAL EQUILIBRIUM YIELD
Recall that the yield-per-recruit model enabled us to examine the problem of growth overfishing
but, because we assumed that recruitment was constant, the model did not tell us whether a given
level of fishing might result in recru

YIELD-PER-RECRUIT
So far on this course we have been laying the groundwork for constructing the "real" models of
fisheries science. We have a model for survival, a model for growth, and a model for the fishing
process. Now we will put them all together.
T

TOTAL EQUILIBRIUM YIELD (continued)
Harvest Management policies Based on Yield Analysis
We have seen in the yield and yield-per-recruit models that equilibrium yield rates (catch per year)
are controlled by the age-at-entry te and by the rate of fishing m

SURPLUS PRODUCTION (continued)
W e just saw that at equilibrium there is a linear relationship between catch (weight)- per-unit-effort
and effort for a population whose growth follows the logistic model. In practice, fishing effort and
CPUE are almost nev

SURPLUS PRODUCTION (continued)
Transition to a New Equilibrium
The following materials are adapted from Fletcher (1978), on the Recommended Reading list.
Because B(t) approaches the new equilibrium value asymptotically, it takes an infinite amount of
time

BIOECONOMIC MODEL OF A FISHERY
Now we are going to develop and analyze a model for the economics of fishing. Anderson (1986),
on the Recommended Reading list, provides a brief overview of basic economic theory and
terminology and covers much of the materi

BIOECONOMIC MODEL OF A FISHERY (continued)
Dynamic Maximum Economic Yield
In our derivation of maximum economic yield (MEY) we examined a system at equilibrium and our
analysis made no distinction between profits in the present versus profits in the futur

BIOECONOMIC MODELS (continued)
These bioeconomic models that we have developed are all for long-run equilibrium situations. To
examine the short-run dynamics we need a model for changes in fishing effort. Clark (1985)
discusses the following simple model

THE CATCH PROCESS (continued)
In our previous derivation of the relationship between CPUE and fish abundance we assumed that
all the units of fishing were equivalent. Now we explore what happens when they are not.
Fishing Power
Suppose there are two boats

MODELS FOR VARIABLE RECRUITMENT (continued)
The other model commonly used to relate recruitment strength with the size of the parental
spawning population is a model developed by Beverton and Holt (1957, Section 6), which is on the
Recommended Reading lis

THE CATCH PROCESS
Usually we cannot harvest all the fish from a population all at the same time. Instead, we catch fish
over some period of time and gradually diminish the size of the population. Now we will explore a
model for this catch process.
Suppose

SURPLUS PRODUCTION REVISITED (continued)
The Assumption of Constant r and K
Suppose the parameters r and K of the Graham-Schaefer surplus production model are not
constant, but instead are subject to random shocks. One qualitative approach for studying th

FW 431/531 - Dynamics of Marine Biological Resources
One important job of fishery managers is to decide how fish stocks should be harvested. For
example, when the harvests should occur and how many fish should be taken. In this course we
will study some o

MORTALITY: a mathematical model for the death (survival) process
Let N(t) denote the number of animals in some closed population. In this population there is no
immigration, no emigration, and no reproduction. The only thing that can happen to change the

YIELD-PER-RECRUIT (continued)
We will now do a bit more exploring of the yield-per-recruit surface. We will start with the
yield-per-recruit analysis for North Sea plaice, based on Beverton and Holt's original development
and application of the approach.

GROWTH IN LENGTH: a model for the growth of an individual
Consider the following differential equation:
dL
= L' = J K L
dt
What are the dimensions
for parameters J and K?
As was the case with the differential equation relating N' with N, once again there

YIELD-PER-RECRUIT (continued)
Grow overfishing results in sub-optimal stock sizes and wasted economic resources (too much
fishing effort and the fuel and labor required to produce the fishing effort). Allen's Method is a
simple technique for determining w

AGE DISTRIBUTIONS
The Mean Age of an Individual
If we pick an individual at random from a cohort, on average how long will that individual live? The
average age attained by an individual in a cohort is also known as the expectation of life and is
closely

YIELD-PER-RECRUIT (continued)
The yield-per-recruit model applies to a cohort, but we saw in the Age Distributions lecture that the
properties of a cohort do not apply in general to a collection of cohorts, which is what we usually
encounter in practice.

THE "DELTA" METHOD
The equations for N(t), L(t), W(t), and B(t) all assume that one set of parameters applies for all
individuals in the population being modeled. It is more realistic to assume that characteristics vary
among individuals and that the para

MODELS FOR VARIABLE RECRUITMENT
The Beverton and Holt model for yield-per-recruit applies to situations in which the influx of new
recruits does not change from year to year, as in the cumulative yield that results from harvesting a
single cohort during i

MULTISPECIES MODELS
All of the models that we have examined so far have been based on the notion that we have a
fishery harvesting from a single stock of fish. This is a gross oversimplification of the real situation
in most fisheries. Most types of fishi