3. Age Structured Models

Question 1

What is the Beverton-Holt Model?

Answer 1

1. Widely used
   * Widely used in evaluation of fish populations
   * Helps to predict the number of new individuals (recruits) that will join the population, based on the number of parent fish (spawners)
   * Particulary useful in regions such as North Atlantic, where many important fish species spawn once a year and then die, making the model a good fit for these life cycles
2. Follow each year class from recruitment to virtual disappearance
   * “year class” is a group of fish spawned in the same year.
   * The model allos us to track number of surviving individuals in each year class
   * Provides valuable insight into the survival and mortality rates at different ages, who are key factors in managing the sustainability of the fish population.
3. Annual catch = catch from all year classes in the population
   * The number of fish caught per year includes fish caught from every year classes in the population.
   * Reflect that commercial fishing target a range of ages (both young and old)
4. Discrete time (year, month, quarter), but continuous time mortality, originally continuous-time growth (now usually age-specificc)
   a) Discrete time:
   Time is divided into distinct, separate intervals or units. This might be a year, month or quarter. For instance, certain type of fish spawns once a year during the spring
   b) Continuous time:
   Time is not divided into separate intervals, but flows continuously. For example, growth and mortality (death), which can occur at any time and aren’t necessarily time specific.

Question 2

Give an intuitive explanation of the Beverton-Holt equation

Answer 2

1. No*e^-(F+M): Number of fish that survive at years end
2. No-N0*e^-(F+M): Fish that died during that year
3. F/(F+M): Fraction that died because of fishing

Question 3

Beverton-Holt Model: Can fishing mortality be > 1?

Answer 3

Yes

No*e^-1 = 0.367879

With instantaneous mortality = 1 over one year there would still be 37% of the fish left!

Question 4

Beverton-Holt Model: How high F would it take to catch all the fish before the year is over?

Answer 4

Would need a very big F.

See Document!

Question 5

Beverton-Holt Model: How do you get catch in weight?

Answer 5

Multiply by weight at age

Not quite correct, because fish grow over time, and the higher F the earlier they are fished and the lower is the average weight

As fish grow over time, the size that fish reach before being caught will depend on the intensity of fishing. If the fishing mortality rate (F) is high, fish are more likely to be caught at a younger age, when they are smaller. This leads to a lower average weight of fish in the catch compared to a situation where fishing mortality is low and fish have time to grow to a larger size before they are caught.

So, while you can estimate the total catch in weight by multiplying the number of fish caught at each age by their average weight at that age, this estimate won’t be accurate if it doesn’t account for the effect of fishing mortality on the average weight of the fish. This effect is due to the fact that high fishing pressure tends to skew the age structure of the population towards younger, smaller fish.

Question 6

Beverton-Holt Model: Annual Catch

Answer 6

Sum of catch from all year classes in a
population

[SEE DOCUMENT FOR FORMULA]

h* = age of recruitment to the population
h** = oldest surviving year class
s = selection parameter
N0,h: determined by previous life history
F = fising mortality, assumed constant over
the period
M = natural mortality, assumed constant (mostly)

Question 7

Basic idea of Beverton-Holt

Answer 7

The basic idea is that there’s a relationship between the number of adult fish (the “stock”) and the number of baby fish, or “recruits,” that are added to the population each year. This relationship is influenced by the environment, predation, disease, and other factors.

Question 8

Beverton: Density Dependence:

Answer 8

The model assumes that when the adult population is low, there is less competition for resources, so a higher proportion of the baby fish survive. However, as the adult population gets larger, competition increases and a smaller proportion of the baby fish survive. This is a concept known as “density dependence.”

Question 9

Beverton: Assumptions

Answer 9

The Beverton-Holt model assumes that the environment is stable and that the rate of survival of the baby fish is solely determined by the number of adult fish.

Question 10

Q: What is age-structured models?

Answer 10

These models are used to study the dynamics of fish populations considering the age structure of the population. These models are crucial in understanding fish population dynamics and informing sustainable fishing practices. This is because fish at different ages or stages have different survival rates, reproductive rates, and are differently affected by fishing.

Question 11

Q: What is the Beverton-Holt model?

Answer 11

The Beverton-Holt model is a type of recruitment model that describes the relationship between the spawning stock biomass (adult population size) and the recruitment (new additions to the population). The equation is as follows:

R=aS/(1+S/K)

• R represents recruitment, the number of new individuals added to the population.
• S represents the spawning stock biomass, the total weight of sexually mature fish in the population.
• a, K are parameters of the model, usually determined by fitting the model to empirical data.

The key feature of the Beverton-Holt model is that it assumes density-dependent survival, which means that as the number of adult fish increases, each individual offspring has a lower probability of survival. This model results in an asymptotic relationship between spawning stock and recruitment, where increasing the spawning stock beyond a certain point does not result in a substantial increase in recruitment.

Question 12

Q: What is the Ricker Model?

Answer 12

The Ricker model is another widely used model in fishery science. It includes a term for density-dependent mortality, which can cause populations to exhibit fluctuations or even chaotic dynamics under certain conditions. The equation is as follows:

R=aSexp(-bS)

• R is the recruitment,
• S is the spawning stock,
• a, b are parameters of the model, usually determined by fitting the model to empirical data.

In the Ricker model, recruitment initially increases with spawning stock size but eventually decreases when the spawning stock size gets too large. This is because the model assumes that as the population grows, the resources per individual decrease leading to a decline in survival and hence recruitment. This leads to an inverted-U shaped curve, unlike the Beverton-Holt model.

Question 13

Q: Importance of age-structured models in Fisheries Management?

Answer 13

Important as they allow us to understand how different levels of fishing pressure might affect the population in the long term. They can be used to estimate the maximum sustainable yield (MSY), the largest yield (or catch) that can be taken from a species’ stock over an indefinite period under constant environmental conditions.

Fishery scientists and managers can use these models to predict how changes in fishing mortality, fishing gear, or fishing methods might affect the yield and the age structure of the population. They can also be used to understand how measures such as fish quotas or limiting the number of fishing days could affect the population.

Remember that these models are simplifications of complex biological systems, and the real-world behavior of fish populations may be influenced by many other factors not included in these models (e.g., environmental variability, interactions with other species, changes in growth rates or natural mortality rates, etc.)

Question 14

Q: What is the difference between the Beverton-Holt model and the Ricker model?

Answer 14

1. Both describe the relationship between the spawning stock biomass and the recruitment (i.e., the addition of new young fish to the population).
2. However, they make different assumptions about how population dynamics work, leading to different predicted relationships between spawning stock size and recruitment.

Beverton-Holt Model:
This model assumes density-dependent survival. As the number of adult fish increases, each individual offspring has a lower probability of survival. This is often due to increased competition for resources. As a result, the model predicts that recruitment will increase with spawning stock size, but at a decreasing rate. Beyond a certain point, increasing the spawning stock size doesn’t lead to much increase in recruitment. The relationship between spawning stock size and recruitment is asymptotic (i.e., it approaches but never reaches a limit).

The equation for the Beverton-Holt model is: R = aS / (1 + S/K)

Here, R is recruitment, S is the spawning stock, and a and K are parameters.

Ricker Model:

This model includes density-dependent mortality. This model assumes that at low population sizes, recruitment will increase with spawning stock size. However, when the population gets too large, resources per individual decrease and survival rates decline, causing recruitment to decrease. This leads to an inverted-U shaped relationship between spawning stock size and recruitment, which can even lead to population fluctuations or chaotic dynamics under certain conditions.
The equation for the Ricker model is: R = aS * exp(-bS)

Here, R is recruitment, S is the spawning stock, and a and b are parameters.

In summary, the main difference between the two models lies in their assumptions about how spawning stock size affects recruitment. The Beverton-Holt model predicts an asymptotic relationship, where recruitment plateaus at high spawning stock sizes, while the Ricker model predicts an inverted-U shaped relationship, where recruitment can decline at high spawning stock sizes due to density-dependent mortality.

Question 15

Explain how age-structured models are used in fisheries management and discuss their limitations.

Answer 15

Age-structured models are used in fisheries management to predict the behavior of fish populations over time, considering both biological factors (e.g., growth, mortality, and reproduction) and human factors (e.g., fishing pressure). These models account for differences in fish survival, growth, and reproduction at different ages, providing a nuanced understanding of population dynamics. However, they have limitations, including reliance on several simplifying assumptions (e.g., constant natural mortality rate, no age errors), dependency on accurate data which is often hard to get, and inability to fully account for environmental variability or human behavior changes. Age-structured models also typically ignore spatial structure, potentially leading to misleading conclusions about stock status and optimal harvest levels.

Question 16

Compare and contrast the Beverton-Holt and Ricker recruitment models, providing the key assumptions and equations of each.

Answer 16

Both the Beverton-Holt and Ricker models are fundamental models in fisheries science describing the relationship between spawning stock biomass and recruitment. The Beverton-Holt model assumes density-dependent survival with recruitment increasing with spawning stock size, but at a decreasing rate. The relationship is asymptotic (R = aS / (1 + S/K)). The Ricker model assumes density-dependent mortality, predicting an inverted-U shaped relationship between spawning stock size and recruitment (R = aS * exp(-bS)). Here, high spawning stock sizes can lead to lower recruitment due to overpopulation and resource scarcity.

Question 17

Discuss the relationship between fishing mortality, sustainable yield, and economic efficiency, referring to gear selectivity as a fisheries management tool.

Answer 17

Fishing mortality rate, sustainable yield, and economic efficiency are closely intertwined. The sustainable yield represents the catch that can be taken from a stock over an indefinite period without depleting it. However, increasing fishing mortality rates without controlling access to fisheries can lead to overfishing and economic inefficiency, with too much effort spent to achieve the same yield. Gear selectivity, which targets specific age classes, can potentially increase yield and efficiency by harvesting fish at the optimal age. However, in an open-access fishery, economic equilibrium may still be inefficient if fishing mortality is not adequately controlled, leading to high fishing effort and low profitability.

Question 18

Explain the difficulties associated with predicting recruitment in fisheries populations.

Answer 18

Predicting recruitment is challenging due to the numerous variables and uncertainties involved. There is often a high degree of variability in recruitment, making it difficult to identify clear relationships between spawning stock size and the number of recruits. Multiple factors can influence recruitment, including environmental conditions, availability of food, predation rates, disease, and genetics, which can vary unpredictably. Despite the larger spawning stock producing more roe and milt, recruitment is not guaranteed due to the complexity of survival factors influencing early life stages of fish.

Question 19

Discuss the use and implications of the Ricker recruitment model in the context of Pacific salmon fisheries.

Answer 19

The Ricker recruitment model was originally developed for Pacific salmon, which are semelparous, spawning only once and then dying. The model posits an inverted-U relationship between the spawning stock and the recruits, indicating that there’s an optimal population size that maximizes recruitment. Too high or too low spawning stock sizes lead to fewer recruits due to increased competition for resources or too little breeding potential, respectively. For fisheries management, this model highlights the importance of maintaining the spawning stock at an optimal level to ensure the maximum sustainable yield. However, in real-world application, many factors can complicate the situation. Environmental conditions, genetic variability, disease, and predation can all impact the expected outcomes predicted by the Ricker model. There are also potential issues related to the spatial distribution of the fish and fishing pressure. In this regard, it’s important for fisheries managers to not rely solely on models, but use them as part of a broader toolkit of assessment methods and strategies, while keeping a keen eye on actual fishery data and ecological conditions.

Question 20

Define “selective fishing” and “non-selective fishing” and discuss how each relates to the age-structured models.

Answer 20

Selective fishing refers to fishing practices that selectively target specific age groups, species, or sizes of fish. This could be achieved through the use of specific gear, fishing techniques, or fishing in particular areas or times of year. Non-selective fishing, in contrast, does not discriminate between different groups of fish. The concept of selectivity is crucial in age-structured models as it influences the age distribution of the remaining stock and potentially affects the future stock dynamics. Selective fishing can be employed to optimise the yield and economic return from the fishery by targeting the most productive age groups, although it also poses risks such as altering the genetic structure of the population.

Question 21

Discuss the principle of “recruitment overfishing” and “growth overfishing” and how they are accounted for in age-structured models.

Answer 21

“Recruitment overfishing” occurs when the parent population size is reduced to a point where it no longer has the reproductive capacity to replenish the stock.

“Growth overfishing” occurs when fish are caught at an age that is too young, i.e., before they have reached their maximum growth potential.

Age-structured models can help identify and prevent these types of overfishing. The models can estimate the impact of fishing on different age classes and can be used to set harvest policies that protect young and spawning-age fish. However, the effectiveness of such measures depends on the accuracy of the data and assumptions, as well as compliance with the regulations.

Question 22

Explain the concept of the “biomass dynamic model” and compare it with the age-structured models in terms of their assumptions and applications

Answer 22

Biomass dynamic models, also known as surplus production models, focus on the change in total population biomass over time as a function of biological processes such as growth and mortality, as well as fishing pressure. They are simpler than age-structured models as they do not consider the age structure of the population, treating the population as a homogeneous unit. While this can be a disadvantage in terms of detail, it is also an advantage in situations where detailed age data are not available. Biomass dynamic models are often used when a more general view of stock dynamics is sufficient, or when age data is difficult to obtain or not available. However, they lack the detail provided by age-structured models, which can reveal important dynamics linked to age structure, such as recruitment and growth overfishing.

Question 23

What is a cohort and why is it important in age-structured models?

Answer 23

A cohort is a group of individuals of the same age within a population. In the context of age-structured models, cohorts are significant because they allow us to track the survival, growth, and reproduction of a specific group of individuals over time. This tracking aids in the prediction of population dynamics and assists in making informed management and conservation decisions.

Question 24

How does fishing pressure affect the age structure of a fish population, and how is this reflected in the Beverton-Holt and Ricker models?

Answer 24

Fishing pressure, especially when selective towards certain age groups, can significantly affect the age structure of a fish population. Overfishing can lead to a disproportionate number of young fish in the population, reducing the average age and potentially affecting the reproductive capacity of the population. Both the Beverton-Holt and Ricker models can capture these effects. These models reflect changes in population dynamics due to fishing pressure by altering the survival probabilities of different age classes in the population. However, these models assume constant recruitment and survival rates, which may not be accurate in the face of variable fishing pressure or environmental conditions.

Question 25

Explain how the parameters of the Beverton-Holt and Ricker models are typically estimated.

Answer 25

The parameters of the Beverton-Holt and Ricker models are usually estimated using data collected from the field. This includes data on the age structure of the population, growth rates, fecundity rates, natural mortality rates, and fishing mortality rates. Typically, methods such as maximum likelihood estimation or Bayesian estimation are used to fit the model to the data and estimate the parameters. These methods aim to find the parameter values that make the observed data most likely under the model. It’s important to remember that these estimates are only as good as the data they are based on, and they come with some level of uncertainty. Therefore, the models’ predictions should always be interpreted with care and in the light of the data and assumptions used.

Question 26

In the context of the Beverton-Holt model, what is meant by ‘recruitment’?

Answer 26

‘Recruitment’ in the Beverton-Holt model refers to the number of new individuals that survive to enter the fishable population. This concept is central to the Beverton-Holt model which is designed to predict how a harvested population’s age structure will evolve over time, based on the rate of recruitment and the survival probabilities of different age classes.

Question 27

What are the fundamental assumptions made by the Beverton-Holt model regarding survival and fecundity rates?

Answer 27

The Beverton-Holt model assumes that survival and fecundity rates are constant for all age classes. This means that it assumes a constant rate of recruitment, which refers to the number of new individuals entering the fishable population, and also a constant survival probability for each age class, regardless of the population’s overall age structure.

Question 28

How does the Ricker model differ from the Beverton-Holt model in terms of its assumption about the relationship between recruitment and population size?

Answer 28

Unlike the Beverton-Holt model, which assumes a linear relationship between the number of spawners and the number of recruits, the Ricker model assumes a non-linear, specifically a dome-shaped relationship. According to the Ricker model, recruitment initially increases with an increasing number of spawners, but after a certain point, further increases in the number of spawners lead to a decrease in recruitment due to factors such as resource competition or cannibalism.

Question 29

Q: Explain and draw the Beverton-Holt model graph

Answer 29

s. 89
* Y-axis: Million tonnes / Number of recruits
* X-axis: F / Spawners

1. Initially, it rises very steeply.
2. Then, levels off as the number of spawners increases
   This shape reflects the model’s assumption of density-independent survival, meaning that the survival rate of offspring does not decrease as the population size increases. The model assumes that the environment has unlimited resources and thus does not limit the population’s growth.

Question 30

Q: Explain and draw the Ricker-model graph

Answer 30

S. 89
* Y-axis: Million tonnes / Number of recruits
* X-axis: F / Spawners

Ricker model shows an inverted parabola or a “hill”-shaped curve.’

1. Initially rises steeply, reach a peak, then decrease (almost like U, but steeper on the left side)
   This shape reflects the model’s assumption of density-dependent survival, meaning that the survival rate of offspring decreases as the population size increases. This is due to competition for limited resources in the environment. The peak of the curve represents the maximum sustainable yield, the largest number of individuals that can be removed from the population without causing it to decline.