How Else Can a Population Continue to Maintain High Amounts of Species

Carrying Capacity

M.A. Hixon , in Encyclopedia of Ecology, 2008

Basic Ecology

Carrying capacity is most often presented in ecology textbooks as the constant K in the logistic population growth equation, derived and named by Pierre Verhulst in 1838, and rediscovered and published independently by Raymond Pearl and Lowell Reed in 1920:

$N_t=\frac{K}{1+{\text{e}}^{a-rt}}\left(\text{integral form}\right)$

$\frac{\text{d}N}{\text{d}t}=rN\left(\frac{K-N}{K}\right)\left(\text{differential form}\right)$

where N is the population size or density, r is the intrinsic rate of natural increase (i.e., the maximum per capita growth rate in the absence of competition), t is time, and a is a constant of integration defining the position of the curve relative to the origin. The expression in brackets in the differential form is the density-dependent unused growth potential, which approaches 1 at low values of N, where logistic growth approaches exponential growth, and equals 0 when N  = K, where population growth ceases. That is, the unused growth potential lowers the effective value of r (i.e., the per capita birth rate minus the per capita death rate) until the per capita growth rate equals zero (i.e., births   =   deaths) at K. The result is a sigmoid population growth curve ( Figure 1 ). Despite its use in ecological models, including basic fisheries and wildlife yield models, the logistic equation is highly simplistic and much more of heuristic than practical value; very few populations undergo logistic growth. Nonetheless, ecological models often include K to impose an upper limit on the size of hypothetical populations, thereby enhancing mathematical stability.

Figure 1. The definition of carrying capacity most frequently used in basic ecology textbooks. (a) Logistic population growth model, showing how population size (N) eventually levels off at a fixed carrying capacity (K) through time (t). (b) Logistic population growth rate (dN/dt) as a function of population size. Note that the growth rate peaks at 0.5 K and equals zero at K.

Of historical interest is that neither Verhulst nor Pearl and Reed used 'carrying capacity' to describe what they called the maximum population, upper limit, or asymptote of the logistic curve. In reality, the term 'carrying capacity' first appeared in range management literature of the late 1890s, quite independent of the development of theoretical ecology (see below). Carrying capacity was not explicitly associated with K of the logistic model until Eugene Odum published his classic textbook Fundamentals of Ecology in 1953.

The second use in basic ecology is broader than the logistic model and simply defines carrying capacity as the equilibrial population size or density where the birth rate equals the death rate due to directly density-dependent processes.

The third and even more general definition is that of a long-term average population size that is stable through time. In this case, the birth and death rates are not always equal, and there may be both immigration and emigration (unlike the logistic equation), yet despite population fluctuations, the long-term population trajectory through time has a slope of zero.

The fourth use is to define carrying capacity in terms of Justus Liebig's 1855 law of the minimum that population size is constrained by whatever resource is in the shortest supply. This concept is particularly difficult to apply to natural populations due to its simplifying assumptions of independent limiting factors and population size being directly proportional to whatever factor is most limiting. Moreover, unlike the other three definitions, the law of the minimum does not necessarily imply population regulation.

Note that none of these definitions from basic ecology explicitly acknowledges the fact that the population size of any species is affected by interactions with other species, including predators, parasites, diseases, competitors, mutualists, etc. Given that the biotic environment afforded by all other species in the ecosystem typically varies, as does the abiotic environment, the notion of carrying capacity as a fixed population size or density is highly unrealistic. Additionally, these definitions of carrying capacity ignore evolutionary change in species that may also affect population size within any particular environment.

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Human–Environment Relationship: Carrying Capacity

M.E. Geores , in International Encyclopedia of the Social & Behavioral Sciences, 2001

Carrying capacity is the margin of the habitat's or environment's ability to provide the resources necessary to sustain human life. The earth is the habitat for human life. Estimates of the number of people who can be supported by the earth have ranged widely, with some scholars maintaining that the carrying capacity has been reached, and others certain that the earth can support more people. Human appropriation of the earth's resources can both expand the carrying capacity and diminish it, depending on how the resources are used. Some scholars believe that human innovation will continue to expand the carrying capacity, while others believe that the carrying capacity is finite. These two views fuel the debate about the need for population control.

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Carrying Capacity, Concept of☆

G. Hartvigsen , in Reference Module in Life Sciences, 2017

Abstract

Carrying capacity is the maximum number, density, or biomass of a population that a specific area can support sustainably. This likely varies over time and depends on environmental factors, resources, and the presence of predators, disease agents, and competitors over time. The concept of carrying capacity has been explicitly recognized for more than 175 years, and its use has waxed and waned during this time. Currently, the use of carrying capacity to describe any particular population requires caution, although the concept remains intuitive and invokes questions that challenge our fundamental understanding of factors that regulate populations over time and space.

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Foodshed analysis and carrying capacity estimation

Christian J. Peters , in Food Systems Modelling, 2022

6.4.2 The meaning of carrying capacity estimates

Carrying capacity studies typically estimate the maximum population that a geographic area can feed given some set of starting conditions. The meaning is clear, but the practical implications are not. As with SSR estimates, how low is too low? Does a geographic area need to stay below its carrying capacity, or can it rely on supplemental capacity in the form of food imports? Moreover, maximum population may not be the most meaningful indicator. Rather, countries might be constrained by the carrying capacity in the poorest year.

To have meaning, carrying capacity estimates, like all results from models, must be put into context. What are the starting assumptions and input data? The amounts and types of food needed depend on the quantity of energy needed to support physical activity, the quality of the diet relative to nutritional recommendations, the cultural preferences of the diet (particularly for animal products), and the losses and waste that occurs in the supply chain. Capacity to produce food depends on crop yields, livestock feed needs, availability of irrigation water, and the areas of arable and grazing lands. These factors vary not only from place to place but also over time. They vary across different views of what constitutes a sufficient diet or a sustainable food system. Rather than a source of weakness or uncertainty, differences in approach invite conversations about the feasibility of starting assumptions or the reliability of input data and how the results would change given different assumptions.

Two important questions should be resolved by such conversations. First, how sure are we in our estimates? Statistically speaking, one needs to know the uncertainty in the estimates introduced by factors for which variability can be quantified, such as annual variations in crop yield. Second, what are we aiming for? Assumptions about the need for food, for example, implicitly involve making decisions about what kind of eating pattern is acceptable. These implicit assumptions reveal what we value in the food system.

Once we understand the factors that influence the carrying capacity estimates, we must revisit the question, "What do the estimates mean?" One may be comforted by a carrying capacity greater than the current population, but today's buffer in carrying capacity may be consumed by tomorrow's population growth. On the other hand, one might be alarmed by a carrying capacity estimate smaller than the resident population, yet many countries rely on net imports of food. In other words, what is a desirable number? It depends. At the global scale, a carrying capacity equal to or greater than the current population is certainly desirable. However, more is not necessarily better. For example, a high carrying capacity might be achieved by feeding everyone a subsistence diet with input-intensive agriculture. Like any other number, carrying capacity must be interpreted in context.

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Ecological Footprint, Concept of

William E. Rees , in Encyclopedia of Biodiversity (Second Edition), 2013

Glossary

Carrying capacity

Usually defined as the average maximum number of individuals of a given species that can occupy a particular habitat without permanently impairing the productive capacity of that habitat.

Competitive exclusion

The displacement of one species from its habitat or ecological niche by another. When humans appropriate other species' ecological space, it often leads to the local or even the global extinction of the nonhuman organism.

Ecological deficit

An ecological deficit exists when the load imposed by a given human population on its own territory or habitat (e.g., region, country) exceeds the productive capacity of that habitat. Under these circumstances, if it wishes to avoid permanent damage to its local ecosystems, the population must use some biophysical goods and services imported from elsewhere (or, alternatively, lower its material standards).

Human load

The total human load imposed on the environment by a specified population is the product of population size times average per capita resource consumption and waste production. The concept of load recognizes that human carrying capacity is a function not only of population size but also of aggregate material and energy throughput. Thus, the human carrying capacity of a defined habitat is its maximum sustainability supportable load.

Overshoot

A population is in overshoot when it exceeds available carrying capacity. A population in overshoot may permanently impair the long-term productive potential of its habitat, reducing future carrying capacity. It may survive temporarily but will eventually crash as it depletes vital natural capital (resource) stocks.

Patch disturbance

The measurable habitat and ecosystem modification caused by large animals, including humans, as they forage for food or other resources. Patch disturbance is most pronounced near the den site, temporary camp, or other central place within the overall home range of the individual or group.

Sustainability gap

The global ecological deficit – that is, the difference between any excessive human load on the ecosphere and the long-term carrying (or load-bearing) capacity of the planet.

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Carrying Capacity, Concept of

Gregg Hartvigsen , in Encyclopedia of Biodiversity, 2001

III. Definitions of Carrying Capacity

Carrying capacity is the maximum population that a given area can sustain. The measures commonly used include the number of individuals or the total biomass of a population, which are each highly dependent on differences in physiology and age structure among species and across large taxonomic groups. The use of the term carrying capacity has changed over time, but most models suggest that population growth is rapid when density is low and decreases as populations increase toward some maximum. In addition, any definition of this concept improves as we narrow the time and area for the population that we are studying. Population descriptions, therefore, are often depicted as densities, accounting for the number of individuals per unit area. Population density usually varies over time and from place to place. In practice, we generally use population size or density to describe carrying capacity, which is determined either by resource availability or by the influence of enemies (predators and/or pathogens).

Various definitions of carrying capacity arose in the twentieth century, ranging from the suggestion that carrying capacity is that level below which predators have no effect on a population to the population size which can be maximally supported in a given region (previously referred to as the "saturation level"). There also has been a distinction made between "ecological carrying capacity," which refers to the limitation of a population due to resources, and a management-oriented, maximum sustainable yield for a population, referred to as an "economic carrying capacity," which is usually lower than ecological carrying capacity. These definitions clearly lead to difficulty for wildlife managers who have been preoccupied with attempting to determine whether populations are either too high or too low. These debates continue, as exemplified by range management decisions in Yellowstone National Park and issues regarding the increasing frequency of re-introduction programs of top predators.

Carrying capacity may best be expressed mathematically. One of the simplest forms of population change over time can be represented as the differential equation dN/dt = rN, where dN/dt represents the instantaneous change in a population over a short time period, r is the intrinsic growth rate of the population, and N is the size of the population. This yields what is often referred to as a "J" curve, or exponential growth (Fig. 2). In discrete time this relationship is referred to as geometric growth.

Figure 2. Comparison of unregulated exponential growth (solid line) with regulated logistic growth (dotted line).

In 1838, Verhulst modified the exponential growth equation and derived the logistic equation that depicted population growth rate as being inversely related to population size. To slow population growth he added an additional term yielding dN/dt = rN(1−N/K), where K is the population carrying capacity. The term "1−N/K" slows growth rate linearly toward zero as the population (N) approaches the carrying capacity (K). This results in a sigmoidal S-shaped curve for an increasing population over time (Fig. 2). If the population exceeds K (N > K), then 1−N/K is negative, causing growth rate dN/dt to be negative and the population to decline monotonically toward K.

An important attribute to bear in mind is that the logistic equation is deterministic, meaning that if we use the equation to predict population size at the end of a fixed amount of time we will derive the same population each time we start the population over. This assumption is usually violated in field conditions in which random effects, such as accidental deaths, failure to find mates, or fluctuations in environmental conditions, are common. Therefore, it has been argued that we should not expect real populations to behave according to the logistic equation.

This simple equation has been challenged repeatedly by critics without apparent damage. This resilience of a theory is rather rare in science, which is a discipline that prides itself on being able to quickly dispel hypotheses (or equations) given even a small amount of contradictory data. However, the intuitive nature of the idea that populations are regulated by factors such as food supply helps the logistic equation to remain a staple in ecological texts and classrooms. The reason this equation and carrying capacity (K) endure is that the equation's shortcomings help us better understand the dynamics of real populations, ensuring its utility for many years to come.

The discrete, or difference, form of the logistic equation yields a different prediction of population behavior compared to the previously described continuous version. In particular, the discrete form was the equation used by Sir Robert May to first describe how a simple, deterministic equation could produce chaotic population dynamics, a pattern that emerges when intrinsic growth is relatively high. This chaotic behavior appears to mimic realistic changes in populations over time. Several long-term data records conform to chaotic dynamics, including the change in the number of lynx captured over time in Canada (Fig. 3).

Figure 3. The number of lynx trapped in the Mackenzie River district from 1821 to 1934 (after Elton and Nicholson, 1942).

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Competition, Interspecific

Bryan Shorrocks , in Encyclopedia of Biodiversity (Second Edition), 2001

Glossary

Carrying capacity

Maximum number of organisms that can be supported by a given habitat, based on the amount of resources available (such as food, nutrients, shelter, and space).

Exploitation competition

Interaction among two or more species that use a common resource that is limited, in which one species benefits more than the other.

Interference competition

Interaction among two or more species that use a common resource that is not limiting, in which one species is harmed by having its access to the resource restricted.

Resource partitioning

Ecological arrangement in which two or more species use different, nonoverlapping resources in a given habitat, such as warblers foraging for insects in different locations within a tree or canopy.

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Design and Operating Guide for Aquaculture Seawater Systems - Second Edition

In Developments in Aquaculture and Fisheries Science, 2002

2.8 Carrying capacity guidelines

While the carrying capacity of flow through systems are dependent on species biology and specific conditions, there are some generally valid 'rules of thumb' that can be used for planning purposes, assuming good quality input water. The intensity or carrying capacity of an aquaculture system can be described by a number of parameters. The most common parameters are:

(2.10) $\text{Volumetric}\text{density}(\text{lb}/{\text{ft}}^3)=\frac{\text{mass}\text{of}\text{animals}(\text{lb})}{\text{volume}\text{of}\text{rearing}\text{unit}({\text{ft}}^3)}$

(2.11) $\text{Areal}\text{density}(\text{lb}/{\text{ft}}^2)=\frac{\text{mass}\text{of}\text{animals}(\text{lb})}{\text{area}\text{of}\text{rearing}\text{unit}({\text{ft}}^2)}$

(2.12) $\text{Loading}(\text{lb}/\text{gpm})=\frac{\text{mass}\text{of}\text{animals}(\text{lb})}{\text{flow}\text{to}\text{rearing}\text{unit}(\text{gpm})}$

(2.13) $\text{Exchange}\text{rate}(\text{exchanges}/\mathrm{h})=\frac{(60)(\text{flow}\text{to}\text{unit},\text{gpm})}{\text{volume}\text{of}\text{unit}(\text{gal})}$

Loading, exchange rate and volumetric density are related by:

(2.14) $\text{Loading}(\text{lb}.\text{gpm})=\frac{8.02\times \text{density}}{\text{exchange}\text{rate}}$

Another important measure of rearing intensity is cumulative oxygen consumption (COC). For a single rearing unit, COC is equal to DO_in – DO_out. For n rearing units in series, the COC is equal to:

(2.15) $\text{COC}(\text{mg}/1)=\underset{n}{\overset{i=1}{\Sigma }}({\text{DO}}_{\text{in}}-{\text{DO}}_{\text{out}})$

The COC depends strongly on animal size and temperature and therefore integrates both animal size and metabolic activity.

If loading rates are maintained at low values (high exchange rate), densities in small-scale experimental systems have been as high as 34 lb/ft³ (545 kg/m³). These numbers are well above any practical values. The maximum practical density will depend on water quality considerations, management skills and the ability of the particular species to tolerate crowding. Maximum loading rates in production systems typically range from 4 to 10 lb/gpm (0.5–1.2 kg/1pm). For research purposes maximum loading rates typically range around 1 1b/gpm (0.125 kg/1pm) or less.

Carrying capacity may not be limited by water flow but by volumetric or area density. Surface area limitations often apply for plants due to their need for sunlight and to organisms that require a substrate, such as some shellfish. A substrate requirement may also be combined with quantifiable territorial needs of the organisms. As an example, juvenile or adult lobsters and some crabs are cannibalistic bottom dwellers and generally must be individually isolated to prevent unacceptable mortality. In surface-limited systems, the capacity can sometimes be increased by stacking substrate layers or clever packaging. For organisms that tolerate communal crowding, volumetric density is an important economic parameter, having a direct impact on the required rearing volume and a major impact on capital costs. The maximum density depends on both the loading and behavior characteristics of the animals. Much of the available volumetric data are of questionable use because of the interactions between biomass, density and loading. As an example, if 1000 lb of fish is held in 1000 ft³ of rearing volume with 1000 gpm of flow, this results in a loading of 1 1b/gpm and a density of 1 lb/ft³. If the biomass of fish is doubled both the loading and the density are also doubled. If the flow is cut by half, the loading doubles but the density remains unchanged. If the loading is kept low (high exchange rate), the maximum density is only limited by the ability of the organisms to tolerate crowding. As a general rule, assuming good water quality and amenability to crowding, the following are suggested as maximum densities for organisms that are spread throughout the water column, such as most fish:

Research:	0.01–0.1 lb/ft3;
Production:	1–2 lb/ft3;
Holding:	2–5 lb/ft3.

The consumption of oxygen by aquatic animals has been studied extensively by physiologists because it can be used to estimate energy expenditure. The respiration or oxygen consumption of aquatic animals is composed of three components

(2.16) $T=T_{\text{standard}}+T_{\text{activity}}+T_{\text{sda}}$

where T is total respiration (mg/h per individual), T _standard is oxygen consumption of an unfed and resting animal (mg/h per individual), T _activity is additional oxygen consumption due to swimming or movement (mg/h per individual), T _sda is additional oxygen consumption required for digestion, assimilation, and storage of material (mg/h per individual).

The standard metabolic rate can be determined by extrapolation to zero activity from determination of oxygen consumption at various levels of forced activity. The sum of T _standard + T _activity is the routine metabolic rate and is a measure of the random activity of the animal. The routine metabolism of an individual animal may vary more widely than its standard metabolic rate. The active metabolic rate is the maximum sustained metabolic rate of an animal swimming or moving steadily. Most oxygen consumption studies are conducted on unfed animals and measure either the standard or routine metabolic rate.

The effect of weight on oxygen consumption of aquatic animals has been studied extensively. At a given temperature and feeding level (typically a zero feeding rate), the oxygen consumption of a single animal is equal to:

(2.17) $T^{\prime }=a^{\prime }{W}^{b^{\prime }}$

where T' is oxygen consumption in mg/h per individual, a′ and b′ are constants, W is weight of animal in grams.

The oxygen consumption may also be expressed in mg/h per kg biomass basis. Letting a = 1000a' and b = b' – 1, Eq. 2.17 can be rewritten as

(2.18) $T=a{W}^b$

where T is the oxygen consumption in mg/h per kg.

Typical values of a and b are presented in Table 2.8 for aquatic species. The value of b ranges typically from −0.100 to −0.300. For many species, the value of b is independent of temperature and feeding level.

Table 2.8. Constants for the computation of oxygen consumption rates of aquatic animals

Species	Form a	a		b	Variable range		Reference
		α	β		temp. (°C)	wt. (g)
Freshwater fish
Carp (Cyprinus carpio)	S	0.123	2.157	−0.106	10–35	30–400	Beamish, 1964
Channel catfish (Ictalurus punctatus)	R	0.051	2.685	−0.200	24–30	2–1000	Andrews and Matsuda, 1975
	F	3.06	1.540	−0.200	24–30	2–1000
Mozambique tilapia (Sarotherodon mossambicus)	R	0.629	2.078	−0.348	16–37	10–150	Caulton, 1978
Rainbow trout (Oncorhynchus mykiss)	F	36.9	0.866	−0.196	4–10	12–900	Muller-Feuga et al., 1978
	F	50.4	0.903	−0.142	12–22	12–900
Sockeye salmon (Oncorhynchus nerka)	S	11.7	0.944	−0.118	5–20	2–2000	Brett and Glass, 1973
Striped bass (Morone saxatilis)	S	1.87	1.514	−0.250	8–24	0.3–10	Klyashtorin and Yarzhombek, 1975 Kruger and Brocksen, 1978
Marine fish
Aholehole (Kuhlia sandvicensis)	S	[140 mg/h per kg]		−0.213	23	10–80	Muir and Niimi, 1972
Albacore tuna (Thunnus alalunga)	A	[57.6 mg/h per kg]		−0.180	15–19	6000–13,000	Graham and Laurs, 1982
Cod (Gadus morhua)	R	85.8	0.372	−0.159	3–15	90–3200	Saunders, 1963
	F	87.5	0.665	−0.207	3–15	90–3200
Plaice (Pleuronectes platessa)	R	39.1	0.908	−0.374	10–20	4–50	Jobling, 1982
	R	[288 mg/h per kg]		−0.376	10	1–320
Skipjack tuna (Katsuwonus pelamis)	A	[118 mg/h per kg]		−0.190	23–25	400–6000	Graham and Laurs, 1982
Striped mullet (Mugil cephalus)	R	1.14	1.759	−0.145	13–33	3–90	Marais, 1978
Turbot (Scophthalmus maximus)	F	1.87	2.15	−0.252	7–16	4–1000	Brown et al., 1984
Crustaceans
American lobster (Homarus americanus)	R	[114.2 mg/h per kg]		−0.390	22	0.004–0.05	Logan and Epifanio, 1978
	F	[210.7 mg/h per kg]		−0.350	22	0.004–0.05	Logan and Epifanio, 1978
	R	5.52	0.999	−0.120	12–25	0.6–12,300	McLeese, 1964
Blue crab (Callinectes sapidus)	R	10.9	0.785	−0.289	10–25	20–200	Laird and Haefner, 1976
Brine shrimp (Artemia salina)	R	17.4	1.11	−0.194	5–35	0.0077	Decleir et al., 1980
Norway lobster (Nephrops norvegicus)	R	[52.9 mg/h per kg]		−0.139	10	40–210	Bridges and Brand, 1980
Shrimp (Penaeus japonicus)	R	[275 mg/h per kg]		−0.293	22–23	3–18	Egusa, 1961
Spiny lobster (Panulirus japonicus)	R	2.23	1.213	−0.163	17–26	26–350	Nimura and Inoue, 1969
Molluscs
American oyster b (Crassostrea virginica)							Shumway and Koehn, 1982
salinity = 32 g/kg	S	15.3	0.908	−0.490	10–30	0.03–0.7
salinity = 14 g/kg	S	2.77	1.781	−0.553	10–30	0.03–0.7
salinity = 7 g/kg	S	11.9	1.291	−0.615	10–30	0.03–0.7
Clam b (Arctica islandica)	S	[374 mg/h per kg]		−0.399	10	0.03–1.0	Taylor and Brand, 1975
	S	[317 mg/h per kg]		−0.578	10	2.9–16
Cuttlefish (Sepia officinalis)	S	[196 mg/h per kg]		−0.090	17	0.1–1500	Johansen et al., 1982
Dogwhelk b (Thais lapillus)	S	19.6	0.596	−0.400	5–20	?	Stickle and Bayne, 1982
Mussel b (Mytilus californianus)	S	69.1	0.583	−0.352	13–26	0.5–5.0	Bayne et al., 1976
Pacific oyster b (Crassostrea gigas)	S	[922 mg/h per kg]		−0.230	20	0.0–1.7	Gerdes, 1983

a

R = routine; S = standard; F = fed.

b

Based on dry weight excluding shell.

The value of a depends primarily on temperature, but feeding levels and activity can also have significant effects. The impact of temperature on the a value can be modeled as an exponential:

(2.19) $a=\alpha t^{\beta }\text{or}T=\alpha t^{\beta }{W}^b$

where α and β are constants for a specific species and activity level, T is temperature (°C).

When available, values of α and β are presented in Table 2.8. When oxygen consumption data were collected at a single temperature, only the a value has been listed.

The impact of activity level and feeding is much more difficult to estimate. Commonly, the maximum oxygen consumption rate of fed fish is twice the standard rate. The average oxygen consumption rate of fed fish is about 1.4 the unfed routine rate. The impact of activity depends very strongly on species and culture system. For very active fish such as tuna or salmon, the active rate can be as high as 10 times the standard rate.

The metabolic activity of animals depends strongly on size, temperature, and activity level. For example, the oxygen consumption of the American lobster (McLeese, 1964) is equal to:

(2.20) $\text{Oxygen}\text{consumption}(\text{mg}/\mathrm{h}\text{per}\text{kg})=5.52T^{0.999}{W}^{-0.120}$

where T is temperature (°C) and W is wet weight (g). At 12°C, the oxygen consumption of 10 kg of 1, 50, and 1000 g lobsters is equal to:

Size (g)	Oxygen consumption (mg/h per kg)	Oxygen consumption (mg/h)
1	66	661
50	41	413
1000	19	192

Therefore, a given flow of water that would support the oxygen requirements of 10 kg of 50 g lobsters would also support 22 kg of 1000 g lobsters.

An alternative approach to the computation of oxygen demand is based on the ingested ration. For trout, the average daily oxygen demand (Haskell, 1955; Willoughby, 1968) is proportional to the total ration:

(2.21) $\text{Average}\text{daily}\text{oxygen}\text{demand}(\text{lb}/\text{d})=\text{OFR}\times R$

where OFR is oxygen/feed ratio (lb/lb), R is total ration (lb/d).

The oxygen requirement to process a given mass of feed depends on animal size, feeding rate, composition of the ration, digestibility of the feed components, and moisture content and can be described by the oxygen/feed ratio (OFR).

In salmon and trout production systems, OFRs ranging from 0.20 to 0.22 kg oxygen/kg wet feed have been reported (Willoughby, 1968; Westers, 1981). In commercial high density warm-water fish culture, a value of OFR = 1.00 lb oxygen/lb wet feed is commonly used. Limited data are available for OFRs in recycle systems. The oxygen demand from bacterial oxidation of organic compounds, ammonia, and solids strongly depends on the unit processes and their operation. The upper bound for OFR equals the ultimate biochemical oxygen demand (BOD) of the feed, which for channel catfish feed is equal to 1.1 lb O₂/lb dry feed (Harris, 1971). Careful feeding and rapid removal of solids from the system can significantly reduce the OFR.

The feeding rate is commonly reported in terms the amount of actual feed fed (wet feed or on a 'as fed' basis). The feeding rate can also be reported in terms of amount of dry feed. Many commercial dry diets contain only 5–8% moisture, so the difference between the wet and dry values are small. This is not the case for 'semi-moist' diets (i.e., Oregon moist pellets) or diets that are made from unprocessed fish products. In these cases, the moisture content can vary from 30 to 90%. It is likely that the feeding rate used in Eq. 2.21 should be based on a dry fed basis for semi-moist and moist diets, although specific data are lacking.

On a daily basis, the maximum oxygen consumption occurs 4 to 6 h following feeding in a flow-through system. The peaking factor can be reduced by increasing the number of feedings per day. Westers (1981) suggested a peaking factor of 1.44 to account for the maximum daily oxygen consumption rate:

(2.22) $\text{maximum}\text{daliy}\text{oxygen}\text{demand}(\text{kg}/\text{day})=1.44\times \text{OFR}\times R$

Working with freshwater fish, Piper et al. (1982) popularized an approach to estimating stocking density as a function of animal size. This approach is based on a density index (DI) which is equal to:

(2.23) $\text{DI}=\frac{\text{Biomass}(\text{lb})}{\text{Rearing}\text{volume}({\text{ft}}^3)\times \text{Total}\text{length}(\text{inch})}$

The units of the DI are lb/(ft³ × inch). Eq. 2.23 can be rearranged to:

(2.24) $\text{DI}=\frac{\text{Density}(\text{lb}/{\text{ft}}^3)}{\text{Total}\text{length}(\text{inch})}$

or

(2.25) $\text{Density}=\text{DI}\times \text{Total}\text{length}$

For domestic rainbow trout, a DI = 0.50 is commonly used; for anadromous salmon, DI values in the range of 0.08 to 0.15 are used. For a DI = 0.50, 2-inch fish could be held at 1 lb/cf while 8-inch fish could be held at 4 lb/cf.

For many hatchery fish, the behavioral impacts of density are not a significant problem. The density computed from Eq. 2.25 accounts for the impact of size on metabolic activities that would occur when the water flow over the production cycle is relatively constant. The intrinsic impact of density is much more important for crustaceans and many marine fish, especially for small juveniles. Regardless, of these impacts, Eq. 2.25 offers a simple way to evaluate the impact of animal size on metabolic carrying capacity. Water reconditioning and reuse internal to a flow-through system will raise the carrying capacity. This assumes that the water parameters are the limiting factor, which is often but not always the case. It is also critical to know the specific limiting water quality parameter, because improving a nonlimiting parameter will not have much effect. Predicting the carrying capacity of a system with considerable water reuse approaching a closed system is complex and beyond the scope of this book (see Appendix C).

In a flow-through system, flow is needed to supply oxygen and remove ammonia, carbon dioxide, soluble organic compounds, uneaten feed, and fecal matter. Typically, the most limiting water quality parameters are dissolved oxygen, un-ionized ammonia, and carbon dioxide.

The flow needed to maintain a given water quality criterion can be developed from mass-balance considerations. For a given control volume (Fig. 2.4), the mass-balance on a single parameter under steady-state conditions (concentration is not changing) is simply:

Fig. 2.4. Mass balance relationship for flow-through systems.

(2.26) $\text{in}+\text{generation}-\text{decay}=\text{out}$

The mass of compound 'x' leaving the control volume is equal to the mass entering, plus any generation within the control volume, minus any decay within the control volume.

The ability to estimate mass (or concentrations) using the mass balance approach depends strongly on the complexity of the system and how well the biological and chemical processes are understood. Fig. 2.4 has generation of ammonia within the control volume but no decay. This is appropriate for a flow-through system where detention times are in the range of 20 to 60 min, but would be totally invalid in pond systems.

The following equations are based on typical flow-through systems where gas transfer across the air-water or water-bottom surfaces is commonly small and may be neglected and the only metabolic loads are from the culture animals. These equations are not valid for pond systems because a significant part of the pond's metabolic activity may be from algae or bacteria and the gas transfer across the air-water and water-soil interface can not be neglected. Modeling of these types of systems is much more complex.

Application of the mass-balance equation to the system presented in Fig. 2.4 and neglecting mass transfer across the air-water and water-substrate interfaces results in the following two equations:

(2.27) $Q_{\text{oxygen}}=\frac{K_{\text{oxygen}}\times R}{\text{DO}(\text{in})-\text{DO}(\text{out})}$

(2.28) $Q_{\text{ammonia}}=\frac{-\alpha K_{\text{ammonia}}\times R}{{\text{NH}}_3-\text{N}(\text{in})-{\text{NH}}_3-\text{N}(\text{out})}$

where Q is flow required to maintain a given oxygen (Q _oxygen) and un-ionized ammonia criterion (Q _ammonia), K is production rate per unit of feed for oxygen (K _oxygen) and un-ionized ammonia (K _ammonia), R is total ration, DO is influent (DO(in)) and effluent dissolved oxygen concentrations (DO(out)), NH₃-N is influent (NH₃-N(in)) and effluent un-ionized ammonia concentrations (NH₃-N(out)), α is mole fraction of un-ionized ammonia.

The values of DO(out) and NH₃-N(out) are set to the water quality criteria for the specific species under consideration. Values of K _oxygen and K _ammonia are typically in the range of 200 and 30 g/kg of feed, respectively. Typically, the influent concentrations of ammonia in nearshore waters are small and can be neglected. Based on the above assumptions and unit selections of lpm for Q, kg/day for R, and mg/1 for all concentrations, Eqs. 2.27 and 2.28 can be written as:

(2.29) $Q_{\text{oxygen}}=\frac{0.694\times S{F}_{\text{oxygen}}\times K_{\text{oxygen}}\times R}{\text{DO}(\text{in})-\text{DO}(\text{out})}$

(2.30) $Q_{\text{ammonia}}=\frac{0.694\times {\text{SF}}_{\text{ammonia}}\times \alpha K_{\text{ammonia}}\times R}{{\text{NH}}_3-\text{N}(\text{out})}$

A safety factor (SF) term has been added to each equation. The K values are based on daily averages. This safety factor is used to adjust the flow requirement for periods of higher than average metabolic activity. The instantaneous K values can range from 0.5 to 3.0 (or larger) depending on feeding or other activities. The maximum K values for active fish such as tuna or striped bass may range as high as 10 times the average value. For typical fish, a SF of 2 is probably adequate. For less active crustaceans or mollusks, a value of 1.25 is suggested. For large systems, this safety factor can represent a significant cost and pilot-scale determination may be prudent.

A similar equation could be written for carbon dioxide. When oxygen is not limiting, the consumption of 1 mole of oxygen produces approximately 1 mole of carbon dioxide (1.375 mg carbon dioxide per mg oxygen). Under normal surface water carbon dioxide concentrations (<1 mg/1) and common loadings, carbon dioxide is rarely limiting. This may not be true if deep ground waters and/or pure oxygen aeration are used. Carbon dioxide does have a significant impact on flow requirements due to its impact on pH and un-ionized ammonia concentrations.

Eqs. 2.29 and 2.30 can be used to estimate water requirements to maintain oxygen and un-ionized ammonia concentrations for a specific species under conditions typical of common flow-through marine culture systems. The water flow estimates will depend strongly on the water criteria values selected. The actual design requirement will be the largest of the two flows.

For general design considerations, it is also useful to consider the reuse ratio:

(2.31) ${\text{RR}}_{\text{ammonia}/\text{oxygen}}=\frac{Q_{\text{oxygen}}}{Q_{\text{ammonia}}}$

If the reuse ratio is below one (Eq. 2.31), the maximum allowable ammonia concentration has already been reached and aeration will not help. The reuse ratio for typically marine conditions (salinity of 35 g/kg) is presented in Fig. 2.5 as a function of pH and temperature. The reuse ratio decreases as the pH increases due to the increase in the mole fraction of un-ionized ammonia. At a given pH, the reuse ratio first decreases as temperature is increased, then increases as the temperature increases above 20°C. Over typical pHs and temperatures, the reuse ratio varies from 2 to 3, so aeration can often increase the carrying capacity. Above 25°C the reuse ratio rapidly increases, but the actual carrying capacity is decreasing because the available DO is rapidly decreasing. Above 32°C, the reuse ratio is zero because influent DO is less than the DO criteria.

Fig. 2.5. Reuse ratio (RR_{oxygen/ammonia}) as function of pH and temperature. Based on K _oxygen = 200 g/kg, K _ammonia = 30 g/kg, NH₃-N_out = 10 μg/1, DO_in = saturation, DO_out = 6 mg/1 and salinity of 35 g/kg. Note that between 32° and 33°C, the reuse ratio is equal to zero because DO_in &lt; 6.00 mg/1.

In general, Eq. 2.30 will over-estimate the flows needed to maintain the required un-ionized ammonia concentration. This is due to chemical reactions of metabolic carbon dioxide in water and the impacts of metabolic carbon dioxide on pH and un-ionized ammonia. The mole fraction of ammonia (α in Eq. 2.30) depends strongly on pH. If the metabolic carbon dioxide excreted by the animal reduces the pH, Eq. 2.30 will significantly over-estimate the value of Q _ammonia. The impact of these two factors on the required flows depends strongly on the amount of carbon dioxide retained in the water (under normal pHs, negligible ammonia (NH₃) is lost to the atmosphere). Compared to oxygen, carbon dioxide is a very soluble gas and is much more difficult to remove by aeration. To accurately predict the impact of metabolic carbon dioxide on water requirements, it is necessary to estimate carbon dioxide losses to the atmosphere across the water surface (due to aeration or other processes). Limited information is available on the transfer of carbon dioxide to the atmosphere under marine culturing conditions. As a result, this book will neglect all reactions of carbon dioxide in water. This is equivalent to assuming that pH is constant and carbon dioxide gas is an inert gas. For more information on the impact of metabolic carbon on water chemistry and water flow, see Colt and Orwicz (1991a).

While the reuse ratio increases at high temperatures, the actual carrying capacity decreases. This is better demonstrated by looking at the total amount of dissolved oxygen that can be used before the un-ionized ammonia criterion is exceeded. Assuming that the influent water is saturated and the effluent DO concentration is a minimum of 6.00 mg/1, the maximum available oxygen to the animals ranges from 4.107 mg/1 at 5°C to 0.236 mg/1 at 30°C (Fig. 2.6). Even though the reuse ratio is higher at 30°C than at 5°C, the potential available dissolved oxygen is less.

Fig. 2.6. Potential available dissolved oxygen. Based on same conditions and assumptions as in Fig. 2.5.

If the oxygen consumption rate of the culture animal is known, the loading rate can be computed. The loading rate will depend on dissolved oxygen and un-ionized ammonia criteria, temperature, and size of animals. Guidelines for maximum loadings are presented for holding (Fig. 2.7A), production (Fig. 2.7B), and research (Fig. 2.7C) conditions. For holding conditions, aeration can increase the loading rate. For production conditions, it has been assumed that aeration will be used. The maximum loading for unaerated production conditions is typically 50 to 60% of the aerated values but falls off significantly above about 25°C. For research conditions, maximum loading is controlled by the ammonia criteria and aeration will have no effect on loading. Note that the recommended maximum loading is higher for large animals and at low temperature. In the absence of more specific loading criteria, this information can be used in system design. While these recommended loadings should, in most circumstances, be conservative, they should still be used with caution.

The loading rates in Fig. 2.7 are significantly less than under freshwater conditions, primarily due to the high pH of seawater increasing the toxicity of un-ionized ammonia. Depending on the pH, temperature, and water quality criteria, un-ionized ammonia is often the most limiting parameter and when this limit is reached aeration cannot be used to increase carrying capacity. In fact, above 20°C, only 1–2 mg/1 of dissolved oxygen can be used before the un-ionized ammonia criterion is exceeded (Fig. 2.6). Increasing the loading above these somewhat arbitrary numbers results in rapidly increasing risk. The given values assume good conditions but no internal water reconditioning and may represent the carrying capacity of even a reuse system, whose water reconditioning has either mechanically or biologically failed. Earlier life stages are usually much more demanding on water quality and quantity then juveniles and adults and this must be considered when using data such as Fig. 2.7 in hatchery applications (see Example 2.9 for production and Example 2.10 for holding). Except as an emergency measure, it would also be unwise to initially plan to use a seawater system at its maximum capacity until experience is acquired with the system.

A great deal remains to be learned about water quality and water reconditioning. Success with heavily loaded systems is in many ways an art. The best policy for high water reuse or recycling systems is not to exceed very conservative limits, unless one is very experienced, is researching such questions or enjoys high risk.

In those cases where little is known about the effects of ammonia on growth or mortality, chronic bioassays may be needed to develop design loading information (Meade, 1988). These tests are expensive but will allow optimum design and should be cost effective in the long-run, especially for large commercially oriented culture systems.

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RECREATION | Inventory, Monitoring and Management

T. Sievänen , in Encyclopedia of Forest Sciences, 2004

Approaches and Concepts Related to Managing Recreation Resources and Visitors

The carrying capacity concept describes a sustainable level of recreational use. The ecological carrying capacity is defined as the number of visitors or visits an area can sustain without degrading natural resources. The social carrying capacity refers to level of recreational use where the fulfillment expectations of visitor experiences are not threatened because of crowding or misbehavior of other visitors. Most professionals agree that both ecological and social carrying capacity factors must be considered for effective area planning and management. For managerial applications, it is essential to learn about the user attitudes, user preferences, and site use impacts relating to management objectives.

The ROS is a management framework designed to respond to the diversity of experiences desired by recreationists and is used by many recreation resource management agencies all over the world. The original ROS framework describes six levels of recreation opportunities as a spectrum of natural to more developed categories – primitive, semiprimitive, nonmotor, semiprimitive motor, roaded natural, rural, and urban. Recreation opportunities comprise of activity, setting, and recreation experience.

The term limits of acceptable change (LAC) is the management process developed for recreation and wilderness planning and management. The focus is to determine the degree of change caused by recreationists which is acceptable in a specific area. The LAC principles include ecological, economic, and social dimensions of recreation and nature-based tourism. The LAC concept is based on nine steps, where different parameters, such as vegetation and littering, and their indicators (e.g., presence of seedlings and litter) are monitored to detect when the limits are reached. In the LAC process, the general principles of recreation and nature tourism management are divided into more detailed aims and indicators. Furthermore, the management actions will be defined beforehand if the LAC of a certain indicator is being approached or reached. The LAC process can also be applied as a tool for assessing the impacts of recreation and nature tourism on natural areas as well as managing visitor conflicts and other visitor-related problems.

Applying theoretical approaches of carrying capacity and limits of acceptable change into planning and management processes sets a demand of monitoring both of recreational use and its impacts on natural resources. A contemporary framework for managing carrying capacity in the US national parks is visitor experience and resource protection (VERP), which focuses on formulating indicators and standard of quality for desired future conditions of park resources and visitor experiences.

A broad management framework was developed in order to combine both resource and visitor management, paying more attention to the final desired outcomes of resource use. The benefit-based management (BBM) approach focus on optimizing net benefits of use for recreation resources. The BBM requires benefits-oriented management prescriptions, guidelines, and standards to assure provision of optimal recreation opportunities to citizens.

The most advanced visitor management approach is the outcomes approach to leisure (OAL). It focuses on both ecologically and culturally sustainable use of natural resources and the realization of satisfying recreation experiences of recreationists. It stresses applying science-based knowledge in planning and management systems. It also includes the notion of creating and maintaining collaborative partnerships with affected stakeholders. OAL covers all aspects of recreation production, both input and output elements, facilitating outputs as well as final outcomes, i.e., benefits gained on an individual and societal level. Inputs refer to the agency efforts such as time, knowledge, and capital investments used for the production of recreation opportunities as a whole. Facilitating outputs are the results of provider actions, i.e., recreation services such as trails and information. Outcomes can be beneficial or unwanted consequences resulting from the management and use of recreation resources.

Related concepts and frameworks on visitor resources are discussed in the article on VRM (see LANDSCAPE AND PLANNING | Visual Resource Management Approaches).

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Principles of Salmonid Culture

William Pennell , William E. McLean , in Developments in Aquaculture and Fisheries Science, 1996

Uses of Oxygen Supplementation

Increased Carrying Capacity . Oxygen supplementation can be used to increase the carrying capacity if oxygen is the first limiting factor. This is its main purpose in fish farming where biomass harvested is the prime objective. It has proven to be a more economical method of increasing production than developing new water supplies and pumping more water.

Colt and Orwicz (1991b) have shown that the potential for increased production is determined by a complex interaction of site specific factors. These factors include the background water quality, the type of oxygen addition technology used, and the overall configuration of the fish production system. As noted previously (see Table 5), carbon dioxide, ammonia and suspended solids accumulations can also limit production and therefore must be considered.

These complex interactions will be explored later (see page 433) using a computer model. This model shows that increasing carrying capacity by addition of oxygen always results in a trade-off in environmental quality. This trade-off may not be acceptable where smolt quality is of prime importance — e.g., in ocean ranching where adult returns (rather than biomass produced in the pond) is the criteria for success.

Removal of Nitrogen Gas. Systems using pure oxygen strip nitrogen more effectively than simple aeration towers operating under atmospheric conditions. With these systems it is possible to reduce both nitrogen and total gas pressure to below 100% of saturation. This feature was an important reason for installing oxygenation systems in Michigan hatcheries (Westers et al. 1987).

Low persistent levels of gas supersaturation have been suspected of causing chronic stress in hatcheries for some time. Fish may be more susceptible to supersaturation in hatcheries because they are constrained to shallow water. In this case there is no mitigating effect of water depth and the fish experience the full excess gas pressure (see page 415).

Hatcheries using conventional aeration systems often have low chronic levels of gas supersaturation. Mean levels of 103% (AP   =   23   mmHg) are common. Fidler (1983) has pointed out that AP is not stable — in fact it fluctuates in response to changes in temperature and barometric pressure. If AP is 23   mmHg and the barometric pressure suddenly drops from 760 to 730   mmHg, the ∆P experienced by the fish increases to 53   mmHg (TGP%   =   107%). The water supply eventually adjusts to the new barometric pressure but there is a time delay during which the fish experience an elevated ∆P of 53   mmHg. Oxygenation systems can reduce nitrogen below 100% and eliminate concerns over chronic exposure to supersaturation.

Optimize Rearing Environment. Oxygen can be introduced down the length of a raceway to eliminate the dissolved oxygen gradient that normally exists between inflow and outflow. This may improve fish distribution in the pond and help to improve the overall quality of the environment.

Back-up or Emergency Life Support. Oxygenation systems can be used in an emergency to compensate for a reduction in water flows. An oxygenation system was used to oxygenate a remote spawning channel during an emergency that developed in September 1989 (Seppala 1991). A large run of pink salmon returned to the Glendale system during drought conditions. Low flows coupled with high temperatures and large biomass caused the dissolved oxygen to drop to 2   mg/l. This lead to mass suffocation of adult fish. Ceramic stones and liquid oxygen tanks were transported to the site and provided life support for 70,000 salmon in a spawning channel during the most critical period. These systems can also act as a back-up for pumped water. If pumps fail, dissolved oxygen drops to critical levels within minutes. An oxygenation system can extend the survival time to many hours giving time for remedial action (McLean et al. 1993a ).

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