Population growth

The key assumption behind all sustainable harvesting models such as MSY is that populations of organisms grow and replace themselves – that is, they are renewable resources. Additionally it is assumed that because the growth rates, survival rates, and reproductive rates increase when harvesting reduces population density,^[4] they produce a surplus of biomass that can be harvested. Otherwise, sustainable harvest would not be possible.

Another assumption of renewable resource harvesting is that populations of organisms do not continue to grow indefinitely; they reach an equilibrium population size, which occurs when the number of individuals matches the resources available to the population (i.e., assume classic logistic growth). At this equilibrium population size, called the carrying capacity, the population remains at a stable size.^[13]

Figure 1

The logistic model (or logistic function) is a function that is used to describe bounded population growth under the previous two assumptions. The logistic function is bounded at both extremes: when there are not individuals to reproduce, and when there is an equilibrium number of individuals (i.e., at carrying capacity). Under the logistic model, population growth rate between these two limits is most often assumed to be sigmoidal (Figure 1). There is scientific evidence that some populations do grow in a logistic fashion towards a stable equilibrium – a commonly cited example is the logistic growth of yeast.

The equation describing logistic growth is:^[13]

${\displaystyle N_{t}={\frac {K}{1+{\frac {K-N_{0}}{N_{0}}}e^{-rt}}}}$ (equation 1.1)

The parameter values are:

${\displaystyle N_{t}}$=The population size at time t

${\displaystyle K}$=The carrying capacity of the population

${\displaystyle N_{0}}$= The population size at time zero

${\displaystyle r}$= the intrinsic rate of population increase (the rate at which the population grows when it is very small)

From the logistic function, the population size at any point can be calculated as long as ${\displaystyle r}$, ${\displaystyle K}$, and ${\displaystyle N_{0}}$ are known.

Figure 2

Differentiating equation 1.1 give an expression for how the rate of population increases as N increases. At first, the population growth rate is fast, but it begins to slow as the population grows until it levels off to the maximum growth rate, after which it begins to decrease (figure 2).

The equation for figure 2 is the differential of equation 1.1 (Verhulst's 1838 growth model):^[13]

${\displaystyle {\frac {dN}{dt}}=rN\left(1-{\frac {N}{K}}\right)}$ (equation 1.2)

${\displaystyle {\frac {dN}{dt}}}$ can be understood as the change in population (N) with respect to a change in time (t). Equation 1.2 is the usual way in which logistic growth is represented mathematically and has several important features. First, at very low population sizes, the value of ${\displaystyle {\frac {N}{K}}}$ is small, so the population growth rate is approximately equal to ${\displaystyle rN}$, meaning the population is growing exponentially at a rate r (the intrinsic rate of population increase). Despite this, the population growth rate is very low (low values on the y-axis of figure 2) because, even though each individual is reproducing at a high rate, there are few reproducing individuals present. Conversely, when the population is large the value of ${\displaystyle {\frac {N}{K}}}$ approaches 1 effectively reducing the terms inside the brackets of equation 1.2 to zero. The effect is that the population growth rate is again very low, because either each individual is hardly reproducing or mortality rates are high.^[13] As a result of these two extremes, the population growth rate is maximum at an intermediate population or half the carrying capacity (${\displaystyle N={\frac {K}{2}}}$).

MSY model

Figure 3

The simplest way to model harvesting is to modify the logistic equation so that a certain number of individuals is continuously removed:^[13]

${\displaystyle {\frac {dN}{dt}}=rN\left(1-{\frac {N}{K}}\right)-H}$ (equation 1.3)

Where H represents the number of individuals being removed from the population – that is, the harvesting rate. When H is constant, the population will be at equilibrium when the number of individuals being removed is equal to the population growth rate (figure 3). The equilibrium population size under a particular harvesting regime can be found when the population is not growing – that is, when ${\displaystyle {\frac {dN}{dt}}=0}$. This occurs when the population growth rate is the same as the harvest rate:

${\displaystyle rN\left(1-{\frac {N}{K}}\right)=H}$

Figure 3 shows how growth rate varies with population density. For low densities (far from carrying capacity), there is little addition (or "recruitment") to the population, simply because there are few organisms to give birth. At high densities, though, there is intense competition for resources, and growth rate is again low because the death rate is high. In between these two extremes, the population growth rate rises to a maximum value (${\displaystyle N_{MSY}}$). This maximum point represents the maximum number of individuals that can be added to a population by natural processes. If more individuals than this are removed from the population, the population is at risk for decline to extinction.^[14] The maximum number that can be harvested in a sustainable manner, called the maximum sustainable yield, is given by this maximum point.

Figure 3 also shows several possible values for the harvesting rate, H. At ${\displaystyle H_{1}}$, there are two possible population equilibrium points: a low population size (${\displaystyle N_{a}}$) and a high one (${\displaystyle N_{b}}$). At ${\displaystyle H_{2}}$, a slightly higher harvest rate, however there is only one equilibrium point (at ${\displaystyle N_{MSY}}$), which is the population size that produces the maximum growth rate. With logistic growth, this point, called the maximum sustainable yield, is where the population size is half the carrying capacity (or ${\displaystyle N={\frac {K}{2}}}$). The maximum sustainable yield is the largest yield that can be taken from a population at equilibrium. In figure 3, if ${\displaystyle H}$ is higher than ${\displaystyle H_{2}}$, the harvesting would exceed the population's capacity to replace itself at any population size (${\displaystyle H_{3}}$ in figure 3). Because harvesting rate is higher than the population growth rate at all values of ${\displaystyle N}$, this rate of harvesting is not sustainable.

An important feature of the MSY model is how harvested populations respond to environmental fluctuations or illegal offtake. Consider a population at ${\displaystyle N_{b}}$ harvested at a constant harvest level ${\displaystyle H_{1}}$. If the population falls (due to a bad winter or illegal harvest) this will ease density-dependent population regulation and increase yield, moving the population back to ${\displaystyle N_{b}}$, a stable equilibrium. In this case, a negative feedback loop creates stability. The lower equilibrium point for the constant harvest level ${\displaystyle H_{1}}$ is not stable however; a population crash or illegal harvesting will decrease population yield farther below the current harvest level, creating a positive feedback loop leading to extinction. Harvesting at ${\displaystyle N_{MSY}}$ is also potentially unstable. A small decrease in the population can lead to a positive feedback loop and extinction if the harvesting regime (${\displaystyle H_{2}}$) is not reduced. Thus, some consider harvesting at MSY to be unsafe on ecological and economic grounds.^[14][15] The MSY model itself can be modified to harvest a certain percentage of the population or with constant effort constraints rather than an actual number, thereby avoiding some of its instabilities.^[14]

The MSY equilibrium point is semi-stable – a small increase in population size is compensated for, a small decrease to extinction if H is not decreased. Harvesting at MSY is therefore dangerous because it is on a knife-edge – any small population decline leads to a positive feedback, with the population declining rapidly to extinction if the number of harvested stays the same.^[14][15]

The formula for maximum sustained harvest (${\displaystyle H}$) is one-fourth the maximum population or carrying capacity (${\displaystyle K}$) times the intrinsic rate of growth (${\displaystyle r}$).^[16]

${\displaystyle H={\frac {Kr}{4}}}$

For demographically structured populations

The principle of MSY often holds for age-structured populations as well.^[17] The calculations can be more complicated, and the results often depend on whether density dependence occurs in the larval stage (often modeled as density dependent reproduction) and/or other life stages.^[18] It has been shown that if density dependence only acts on larva, then there is an optimal life stage (size or age class) to harvest, with no harvest of all other life stages.^[17] Hence the optimal strategy is to harvest this most valuable life-stage at MSY.^[19] However, in age and stage-structured models, a constant MSY does not always exist. In such cases, cyclic harvest is optimal where the yield and resource fluctuate in size, through time.^[20] In addition, environmental stochasticity interacts with demographically structured populations in fundamentally different ways than for unstructured populations when determining optimal harvest. In fact, the optimal biomass to be left in the ocean, when fished at MSY, can be either higher or lower than in analogous deterministic models, depending on the details of the density dependent recruitment function, if stage-structure is also included in the model.^[21]

Implications of MSY model

Starting to harvest a previously unharvested population will always lead to a decrease in the population size. That is, it is impossible for a harvested population to remain at its original carrying capacity. Instead, the population will either stabilize at a new lower equilibrium size or, if the harvesting rate is too high, decline to zero.

The reason why populations can be sustainably harvested is that they exhibit a density-dependent response.^[14][15] This means that at any population size below K, the population is producing a surplus yield that is available for harvesting without reducing population size. Density dependence is the regulator process that allows the population to return to equilibrium after a perturbation. The logistic equation assumes that density dependence takes the form of negative feedback.^[15]

If a constant number of individuals is harvested from a population at a level greater than the MSY, the population will decline to extinction. Harvesting below the MSY level leads to a stable equilibrium population if the starting population is above the unstable equilibrium population size.

Uses of MSY

MSY has been especially influential in the management of renewable biological resources such as commercially important fish and wildlife. In fisheries terms, maximum sustainable yield (MSY) is the largest average catch that can be captured from a stock under existing environmental conditions.^[22] MSY aims at a balance between too much and too little harvest to keep the population at some intermediate abundance with a maximum replacement rate.

Relating to MSY, the maximum economic yield (MEY) is the level of catch that provides the maximum net economic benefits or profits to society.^[23][24] Like optimum sustainable yield, MEY is usually less than MSY.

Limitations of MSY approach

Although it is widely practiced by state and federal government agencies regulating wildlife, forests, and fishing, MSY has come under heavy criticism by ecologists and others from both theoretical and practical reasons.^[15] The concept of maximum sustainable yield is not always easy to apply in practice. Estimation problems arise due to poor assumptions in some models and lack of reliability of the data.^[8][25] Biologists, for example, do not always have enough data to make a clear determination of the population's size and growth rate. Calculating the point at which a population begins to slow from competition is also very difficult. The concept of MSY also tends to treat all individuals in the population as identical, thereby ignoring all aspects of population structure such as size or age classes and their differential rates of growth, survival, and reproduction.^[25]

As a management goal, the static interpretation of MSY (i.e., MSY as a fixed catch that can be taken year after year) is generally not appropriate because it ignores the fact that fish populations undergo natural fluctuations (i.e., MSY treats the environment as unvarying) in abundance and will usually ultimately become severely depleted under a constant-catch strategy.^[25] Thus, most fisheries scientists now interpret MSY in a more dynamic sense as the maximum average yield (MAY) obtained by applying a specific harvesting strategy to a fluctuating resource.^[8] Or as an optimal "escapement strategy", where escapement means the amount of fish that must remain in the ocean [rather than the amount of fish that can be harvested]. An escapement strategy is often the optimal strategy for maximizing expected yield of a harvested, stochastically fluctuating population.^[26]

However, the limitations of MSY, does not mean it performs worse than humans using their best intuitive judgment. Experiments using students in natural resource management classes suggest that people using their past experience, intuition, and best judgement to manage a fishery generate far less long term yield compared to a computer using an MSY calculation, even when that calculation comes from incorrect population dynamic models.^[27]

For a more contemporary description of MSY and its calculation see ^[28]