Misplaced Pages

#148851

101-645: Where The solution of the differential equations is deterministic and continuous . This, in turn, implies that the generations of both the predator and prey are continually overlapping. The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known Kolmogorov equations ), which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition , disease, and mutualism . The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth

202-441: A deterministic Turing machine , is a model of computation such that the successive states of the machine and the operations to be performed are completely determined by the preceding state. A deterministic algorithm is an algorithm which, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. There may be non-deterministic algorithms that run on

303-466: A 2013 study indicates the cost versus benefits of iron fertilization puts it behind carbon capture and storage and carbon taxes. While ocean iron fertilization could represent a potent means to slow global warming, there is a current debate surrounding the efficacy of this strategy and the potential adverse effects of this. The precautionary principle is a proposed guideline regarding environmental conservation. According to an article published in 2021,

404-1124: A canonical form of the Hamilton's equations featuring the Hamiltonian H ( q , p ) = V ( x ( q , p ) , y ( q , p ) ) = δ e p − γ p + β e q − α q {\displaystyle H(q,p)=V(x(q,p),y(q,p))=\delta e^{p}-\gamma p+\beta e^{q}-\alpha q} : { q ˙ = ∂ H ∂ p = δ e p − γ , p ˙ = − ∂ H ∂ q = α − β e q . {\displaystyle {\begin{cases}{\dot {q}}={\frac {\partial H}{\partial p}}=\delta e^{p}-\gamma ,\\{\dot {p}}=-{\frac {\partial H}{\partial q}}=\alpha -\beta e^{q}.\end{cases}}} The Poisson bracket for

505-516: A deterministic machine, for example, an algorithm that relies on random choices. Generally, for such random choices, one uses a pseudorandom number generator , but one may also use some external physical process, such as the last digits of the time given by the computer clock. A pseudorandom number generator is a deterministic algorithm, that is designed to produce sequences of numbers that behave as random sequences. A hardware random number generator , however, may be non-deterministic. In economics,

606-400: A global scale. Ocean fertilization occurs naturally when upwellings bring nutrient-rich water to the surface, as occurs when ocean currents meet an ocean bank or a sea mount . This form of fertilization produces the world's largest marine habitats . Fertilization can also occur when weather carries wind blown dust long distances over the ocean, or iron-rich minerals are carried into

707-508: A lack of iron. In 1989 he tested this hypothesis (known as the Iron Hypothesis ) by an experiment using samples of clean water from Antarctica . Iron was added to some of these samples. After several days the phytoplankton in the samples with iron fertilization grew much more than in the untreated samples. This led Martin to speculate that increased iron concentrations in the oceans could partly explain past ice ages. This experiment

808-442: A major component of the carbon-rich deep sea precipitation known as marine snow . Marine snow also includes fish fecal pellets and other organic detritus, and steadily falls thousands of meters below active plankton blooms. Of the carbon-rich biomass generated by plankton blooms, half (or more) is generally consumed by grazing organisms ( zooplankton , krill , small fish, etc.) but 20 to 30% sinks below 200 meters (660 ft) into

909-536: A market with several competitors, complementary platforms and products, a sharing economy, and more. There are situations in which one of the competitors drives the other competitors out of the market and other situations in which the market reaches an equilibrium where each firm stabilizes on its market share. It is also possible to describe situations in which there are cyclical changes in the industry or chaotic situations with no equilibrium and changes are frequent and unpredictable. The Lotka–Volterra predator–prey model

1010-508: A mass basis, each kilogram of iron can fix 83,000 kg of carbon dioxide. The 2004 EIFEX experiment reported a carbon dioxide to iron export ratio of nearly 3000 to 1. The atomic ratio would be approximately: "3000 C: 58,000 N: 3,600 P: 1 Fe". Therefore, small amounts of iron (measured by mass parts per trillion) in HNLC zones can trigger large phytoplankton blooms on the order of 100,000 kilograms of plankton per kilogram of iron. The size of

1111-477: A minor effect on mitigating CO2-induced acidification at the surface ocean." Unfortunately, the impact on ocean acidification would likely not change due to the low effects that iron fertilization has on CO 2 levels. Consideration of iron's importance to phytoplankton growth and photosynthesis dates to the 1930s when Dr Thomas John Hart , a British marine biologist based on the RRS ; Discovery II in

SECTION 10

#1732764814149

1212-565: A significant amount of carbon into the deep ocean, where it was expected to remain for centuries to millennia. The eddy was chosen because it offered a largely self-contained test system. As of day 24, nutrients, including nitrogen, phosphorus and silicic acid that diatoms use to construct their shells, declined. Dissolved inorganic carbon concentrations were reduced below equilibrium with atmospheric CO 2 . In surface water, particulate organic matter (algal remains) including silica and  chlorophyll increased. After day 24, however,

1313-441: A significant benefit to the marine food chain in addition to sequestering carbon for long periods of time. A 2009 study tested the potential of iron fertilization to reduce both atmospheric CO 2 and ocean acidity using a global ocean carbon model. The study found that, "Our simulations show that ocean iron fertilization, even in the extreme scenario by depleting global surface macronutrient concentration to zero at all time, has

1414-521: A significant effect on the concentration of atmospheric carbon dioxide by altering rates of carbon sequestration. In fact, fertilization is an important process that occurs naturally in the ocean waters. For instance, upwellings of ocean currents can bring nutrient-rich sediments to the surface. Another example is through transfer of iron-rich minerals, dust, and volcanic ash over long distances by rivers, glaciers, or wind. Moreover, it has been suggested that whales can transfer iron-rich ocean dust to

1515-450: A significant role in supplying the world's oceans with iron. Volcanic ash is composed of glass shards, pyrogenic minerals, lithic particles and other forms of ash that release nutrients at different rates depending on structure and the type of reaction caused by contact with water. Increases of biogenic opal in the sediment record are associated with increased iron accumulation over the last million years. In August 2008, an eruption in

1616-449: A simple expression in terms of the usual trigonometric functions , although they are quite tractable. If none of the non-negative parameters α , β , γ , δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in x , and the second one in y , the parameters β / α and δ / γ are absorbable in the normalizations of y and x respectively, and γ into

1717-399: A small amount of carbon. Ocean iron fertilization is an example of a geoengineering technique that involves intentional introduction of iron-rich deposits into oceans, and is aimed to enhance biological productivity of organisms in ocean waters in order to increase carbon dioxide ( CO 2 ) uptake from the atmosphere, possibly resulting in mitigating its global warming effects . Iron

1818-570: A surface complex with the Fe (III) metal center of an iron-containing mineral (such as hematite or goethite ). On exposure to solar radiation the complex is converted to an excited energy state in which the ligand, acting as bridge and an electron donor , supplies an electron to Fe(III) producing soluble Fe(II). Consistent with this, studies documented a distinct diel variation in the concentrations of Fe (II) and Fe(III) in which daytime Fe(II) concentrations exceed those of Fe(III). Volcanic ash has

1919-495: A turn over in the plankton masses that are produced. This results in no beneficial effects and actually causes an increase in CO 2 . Finally, a 2010 study showed that iron enrichment stimulates toxic diatom production in high-nitrate, low-chlorophyll areas which, the authors argue, raises "serious concerns over the net benefit and sustainability of large-scale iron fertilizations". Nitrogen released by cetaceans and iron chelate are

2020-402: A variety of measurements, combining ship-borne and remote sampling, submarine filtration traps, tracking buoy spectroscopy and satellite telemetry . Unpredictable ocean currents can remove experimental iron patches from the pelagic zone, invalidating the experiment. The potential of fertilization to tackle global warming is illustrated by the following figures. If phytoplankton converted all

2121-455: Is Euler's number . Deterministic system In mathematics , computer science and physics , a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state. Physical laws that are described by differential equations represent deterministic systems, even though

SECTION 20

#1732764814149

2222-425: Is 0.29 W/m of globally averaged negative forcing, offsetting 1/6 of current levels of anthropogenic CO 2 emissions. These benefits have been called into question by research suggesting that fertilization with iron may deplete other essential nutrients in the seawater causing reduced phytoplankton growth elsewhere — in other words, that iron concentrations limit growth more locally than they do on

2323-403: Is a limiting nutrient in many ocean waters. They hoped that the iron would fertilize algae, which would bolster the bottom of the marine food chain and sequester carbon as uneaten algae died. The experiment was demolished by a storm, leaving inconclusive results. The maximum possible result from iron fertilization, assuming the most favourable conditions and disregarding practical considerations,

2424-411: Is a trace element in the ocean and its presence is vital for photosynthesis in plants, and in particular phytoplanktons, as it has been shown that iron deficiency can limit ocean productivity and phytoplankton growth. For this reason, the "iron hypothesis" was put forward by Martin in late 1980s where he suggested that changes in iron supply in iron-deficient seawater can bloom plankton growth and have

2525-533: Is an important iron source. Satellite images and data (such as PODLER, MODIS, MSIR) combined with back-trajectory analyses identified natural sources of iron–containing dust. Iron-bearing dusts erode from soil and are transported by wind. Although most dust sources are situated in the Northern Hemisphere, the largest dust sources are located in northern and southern Africa, North America, central Asia and Australia. Heterogeneous chemical reactions in

2626-400: Is at (1, 1/2). In the model system, the predators thrive when prey is plentiful but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a population cycle of growth and decline. Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of

2727-522: Is generally consumed by other organisms (small fish, zooplankton , etc.) and substantial part of rest of the deposits that sink beneath plankton blooms may be re-dissolved in the water and gets transferred to the surface where it eventually returns to the atmosphere, thus, nullifying any possible intended effects regarding carbon sequestration. Nevertheless, supporters of the idea of iron fertilization believe that carbon sequestration should be re-defined over much shorter time frames and claim that since

2828-659: Is known as the community matrix . When evaluated at the steady state of (0, 0) , the Jacobian matrix J becomes J ( 0 , 0 ) = [ α 0 0 − γ ] . {\displaystyle J(0,0)={\begin{bmatrix}\alpha &0\\0&-\gamma \end{bmatrix}}.} The eigenvalues of this matrix are λ 1 = α , λ 2 = − γ . {\displaystyle \lambda _{1}=\alpha ,\quad \lambda _{2}=-\gamma .} In

2929-481: Is one of several in which iron fertilization could be conducted—the Galapagos islands area another potentially suitable location. Some species of plankton produce dimethyl sulfide (DMS), a portion of which enters the atmosphere where it is oxidized by hydroxyl radicals (OH), atomic chlorine (Cl) and bromine monoxide (BrO) to form sulfate particles, and potentially increase cloud cover. This may increase

3030-580: Is relatively inexpensive compared to scrubbing , direct injection and other industrial approaches, and can theoretically sequester for less than €5/ton CO 2 , creating a substantial return. In August, 2010, Russia established a minimum price of €10/ton for offsets to reduce uncertainty for offset providers. Scientists have reported a 6–12% decline in global plankton production since 1980. A full-scale plankton restoration program could regenerate approximately 3–5 billion tons of sequestration capacity worth €50-100 billion in carbon offset value. However,

3131-432: Is represented in the equation above by the term αx . The rate of predation on the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy . If either x or y is zero, then there can be no predation. With these two terms the prey equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus

Lotka–Volterra equations - Misplaced Pages Continue

3232-607: Is thus attained at the stationary (fixed) point ( γ δ , α β ) {\displaystyle \left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)} and amounts to K ∗ = ( α β e ) α ( γ δ e ) γ , {\displaystyle K^{*}=\left({\frac {\alpha }{\beta e}}\right)^{\alpha }\left({\frac {\gamma }{\delta e}}\right)^{\gamma },} where e

3333-623: The CO 2 uptake and that due to the ocean's albedo increase, however the amount of cooling by this particular effect is very uncertain. Beginning with the Kyoto Protocol , several countries and the European Union established carbon offset markets which trade certified emission reduction credits (CERs) and other types of carbon credit instruments. In 2007 CERs sold for approximately €15–20/ton CO 2 . Iron fertilization

3434-630: The Hudson's Bay Company and the moose and wolf populations in Isle Royale National Park . Secondly, the population equilibrium of this model has the property that the prey equilibrium density (given by x = γ / δ {\displaystyle x=\gamma /\delta } ) depends on the predator's parameters, and the predator equilibrium density (given by y = α / β {\displaystyle y=\alpha /\beta } ) on

3535-515: The Ramsey–Cass–Koopmans model is deterministic. The stochastic equivalent is known as real business-cycle theory . Iron fertilization Iron fertilization is the intentional introduction of iron -containing compounds (like iron sulfate ) to iron-poor areas of the ocean surface to stimulate phytoplankton production. This is intended to enhance biological productivity and/or accelerate carbon dioxide (CO 2 ) sequestration from

3636-607: The Sahara desert fertilizes the Atlantic Ocean and the Amazon rainforest . The naturally occurring iron oxide in atmospheric dust reacts with hydrogen chloride from sea spray to produce iron chloride, which degrades methane and other greenhouse gases, brightens clouds and eventually falls with the rain in low concentration across a wide area of the globe. Unlike ship based deployment, no trials have been performed of increasing

3737-521: The South Atlantic . India was also involved. As part of the experiment, the German research vessel Polarstern deposited 6 tons of ferrous sulfate in an area of 300 square kilometers. It was expected that the material would distribute through the upper 15 metres (49 ft) of water and trigger an algal bloom. A significant part of the carbon dioxide dissolved in sea water would then be bound by

3838-608: The Southern Ocean speculated - in "On the phytoplankton of the South-West Atlantic and Bellingshausen Sea, 1929-31" - that great "desolate zones" (areas apparently rich in nutrients, but lacking in phytoplankton activity or other sea life) might be iron-deficient. Hart returned to this issue in a 1942 paper entitled "Phytoplankton periodicity in Antarctic surface waters", but little other scientific discussion

3939-538: The albedo of the planet and so cause cooling—this proposed mechanism is central to the CLAW hypothesis . This is one of the examples used by James Lovelock to illustrate his Gaia hypothesis . During SOFeX, DMS concentrations increased by a factor of four inside the fertilized patch. Widescale iron fertilization of the Southern Ocean could lead to significant sulfur-triggered cooling in addition to that due to

4040-467: The food chain for other marine organisms . There are two ways of performing artificial iron fertilization: ship based direct into the ocean and atmospheric deployment. Trials of ocean fertilization using iron sulphate added directly to the surface water from ships are described in detail in the experiment section below. Iron-rich dust rising into the atmosphere is a primary source of ocean iron fertilization. For example, wind blown dust from

4141-458: The nitrate and phosphate present in the surface mixed layer across the entire Antarctic circumpolar current into organic carbon , the resulting carbon dioxide deficit could be compensated by uptake from the atmosphere amounting to about 0.8 to 1.4 gigatonnes of carbon per year. This quantity is comparable in magnitude to annual anthropogenic fossil fuels combustion of approximately 6 gigatonnes. The Antarctic circumpolar current region

Lotka–Volterra equations - Misplaced Pages Continue

4242-415: The paradox of enrichment ). A demonstration of this phenomenon is provided by the increased percentage of predatory fish caught had increased during the years of World War I (1914–18), when prey growth rate was increased due to a reduced fishing effort. A further example is provided by the experimental iron fertilization of the ocean. In several experiments large amounts of iron salts were dissolved in

4343-450: The precautionary principle (PP) is a concept that states, "The PP means that when it is scientifically plausible that human activities may lead to morally unacceptable harm, actions shall be taken to avoid or diminish that harm: uncertainty should not be an excuse to delay action." Based on this principle, and because there is little data quantifying the effects of iron fertilization, it is the responsibility of leaders in this field to avoid

4444-516: The 2012 iron fertilization; many factors contribute to predictive models, and most data from the experiment are considered to be of questionable scientific value. On 15 July 2014, the data gathered during the project were made publicly available under the ODbL license. In 2022, a UK/India research team plans to place iron-coated rice husks in the Arabian Sea , to test whether increasing time at

4545-716: The Aleutian Islands deposited ash in the nutrient-limited Northeast Pacific. This ash and iron deposition resulted in one of the largest phytoplankton blooms observed in the subarctic. Previous instances of biological carbon sequestration triggered major climatic changes, lowering the temperature of the planet, such as the Azolla event . Plankton that generate calcium or silicon carbonate skeletons, such as diatoms , coccolithophores and foraminifera , account for most direct sequestration. When these organisms die their carbonate skeletons sink relatively quickly and form

4646-699: The Australian-based Ocean Nourishment Corporation, planned to engage in fertilization projects. These companies invited green co-sponsors to finance their activities in return for provision of carbon credits to offset investors' CO 2 emissions. LOHAFEX was an experiment initiated by the German Federal Ministry of Research and carried out by the German Alfred Wegener Institute (AWI) in 2009 to study fertilization in

4747-463: The Lotka–Volterra model shows two important properties of predator and prey populations and these properties often extend to variants of the model in which these assumptions are relaxed: Firstly, the dynamics of predator and prey populations have a tendency to oscillate. Fluctuating numbers of predators and prey have been observed in natural populations, such as the lynx and snowshoe hare data of

4848-416: The absence of prey. Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate. The Lotka–Volterra predator-prey model makes a number of assumptions about the environment and biology of the predator and prey populations: None of the assumptions above are likely to hold for natural populations. Nevertheless,

4949-658: The atmosphere for at least a period of time. This technique is controversial because there is limited understanding of its complete effects on the marine ecosystem , including side effects and possibly large deviations from expected behavior. Such effects potentially include release of nitrogen oxides , and disruption of the ocean's nutrient balance. Controversy remains over the effectiveness of atmospheric CO 2 sequestration and ecological effects. Since 1990, 13 major large scale experiments have been carried out to evaluate efficiency and possible consequences of iron fertilization in ocean waters. A study in 2017 considered that

5050-425: The atmosphere modify the speciation of iron in dust and may affect the bioavailability of deposited iron. The soluble form of iron is much higher in aerosols than in soil (~0.5%). Several photo-chemical interactions with dissolved organic acids increase iron solubility in aerosols. Among these, photochemical reduction of oxalate -bound Fe(III) from iron-containing minerals is important. The organic ligand forms

5151-501: The atmosphere. Iron is a trace element necessary for photosynthesis in plants. It is highly insoluble in sea water and in a variety of locations is the limiting nutrient for phytoplankton growth. Large algal blooms can be created by supplying iron to iron-deficient ocean waters. These blooms can nourish other organisms. Ocean iron fertilization is an example of a geoengineering technique. Iron fertilization attempts to encourage phytoplankton growth , which removes carbon from

SECTION 50

#1732764814149

5252-878: The canonical variables ( q , p ) {\displaystyle (q,p)} now takes the standard form { F ( q , p ) , G ( q , p ) } = ( ∂ F ∂ q ∂ G ∂ p − ∂ F ∂ p ∂ G ∂ q ) {\displaystyle \{F(q,p),G(q,p)\}=\left({\frac {\partial F}{\partial q}}{\frac {\partial G}{\partial p}}-{\frac {\partial F}{\partial p}}{\frac {\partial G}{\partial q}}\right)} . A less extreme example covers: α = 2/3 , β = 4/3 , γ = 1 = δ . Assume x , y quantify thousands each. Circles represent prey and predator initial conditions from x = y = 0.9 to 1.8, in steps of 0.1. The fixed point

5353-499: The carbon is suspended in the deep ocean it is effectively isolated from the atmosphere for hundreds of years, and thus, carbon can be effectively sequestered. Assuming the ideal conditions, the upper estimates for possible effects of iron fertilization in slowing down global warming is about 0.3W/m of averaged negative forcing which can offset roughly 15–20% of the current anthropogenic CO 2 emissions. However, although this approach could be looked upon as an easy option to lower

5454-399: The carbon sequestration. This is as predicted by the equilibrium population densities of the Lotka–Volterra predator-prey model, and is a feature that carries over to more elaborate models in which the restrictive assumptions of the simple model are relaxed. The Lotka–Volterra model has additional applications to areas such as economics and marketing. It can be used to describe the dynamics in

5555-705: The circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency ω = α γ {\displaystyle \omega ={\sqrt {\alpha \gamma }}} . The value of the constant of motion V , or, equivalently, K = exp(− V ) , K = y α e − β y x γ e − δ x {\displaystyle K=y^{\alpha }e^{-\beta y}x^{\gamma }e^{-\delta x}} , can be found for

5656-717: The closed orbits near the fixed point. Increasing K moves a closed orbit closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem y α e − β y x γ e − δ x = y α x γ e δ x + β y ⟶ max x , y > 0 . {\displaystyle y^{\alpha }e^{-\beta y}x^{\gamma }e^{-\delta x}={\frac {y^{\alpha }x^{\gamma }}{e^{\delta x+\beta y}}}\longrightarrow \max _{x,y>0}.} The maximal value of K

5757-428: The colder water strata below the thermocline . Much of this fixed carbon continues into the abyss, but a substantial percentage is redissolved and remineralized. At this depth, however, this carbon is now suspended in deep currents and effectively isolated from the atmosphere for centuries. Evaluation of the biological effects and verification of the amount of carbon actually sequestered by any particular bloom involves

5858-407: The concentration of CO 2 in the atmosphere, ocean iron fertilization is still quite controversial and highly debated due to possible negative consequences on marine ecosystems . Research on this area has suggested that fertilization through deposition of large quantities of iron-rich dust into the ocean floor can significantly disrupt the ocean's nutrient balance and cause major complications in

5959-411: The densities of predators for all times. This corresponds to eliminating time from the two differential equations above to produce a single differential equation relating the variables x (predator) and y (prey). The solutions of this equation are closed curves. It is amenable to separation of variables : integrating yields the implicit relationship where V is a constant quantity depending on

6060-817: The derivatives are equal to 0: x ( α − β y ) = 0 , {\displaystyle x(\alpha -\beta y)=0,} − y ( γ − δ x ) = 0. {\displaystyle -y(\gamma -\delta x)=0.} The above system of equations yields two solutions: { y = 0 ,     x = 0 } {\displaystyle \{y=0,\ \ x=0\}} and { y = α β ,     x = γ δ } . {\displaystyle \left\{y={\frac {\alpha }{\beta }},\ \ x={\frac {\gamma }{\delta }}\right\}.} Hence, there are two equilibria. The first solution effectively represents

6161-415: The discrete numbers of individuals might cause the rabbits to actually go extinct, and, by consequence, the foxes as well. This modelling problem has been called the "atto-fox problem", an atto- fox being a notional 10 of a fox. A density of 10 foxes per square kilometre equates to an average of approximately 5×10 foxes on the surface of the earth, which in practical terms means that foxes are extinct. Since

SECTION 60

#1732764814149

6262-675: The eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be V = δ x − γ ln ⁡ ( x ) + β y − α ln ⁡ ( y ) {\displaystyle V=\delta x-\gamma \ln(x)+\beta y-\alpha \ln(y)} on orbits. Thus orbits about

6363-451: The emerging bloom and sink to the ocean floor. The Federal Environment Ministry called for the experiment to halt, partly because environmentalists predicted damage to marine plants. Others predicted long-term effects that would not be detectable during short-term observation or that this would encourage large-scale ecosystem manipulation. A 2012 study deposited iron fertilizer in an eddy near Antarctica. The resulting algal bloom sent

6464-422: The extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α , β , γ , and δ . The stability of

6565-598: The fixed point are closed and elliptic , so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency ω = λ 1 λ 2 = α γ {\displaystyle \omega ={\sqrt {\lambda _{1}\lambda _{2}}}={\sqrt {\alpha \gamma }}} and period T = 2 π / ( λ 1 λ 2 ) {\displaystyle T=2{\pi }/({\sqrt {\lambda _{1}\lambda _{2}}})} . As illustrated in

6666-525: The fixed point at the origin can be determined by performing a linearization using partial derivatives . The Jacobian matrix of the predator–prey model is J ( x , y ) = [ α − β y − β x δ y δ x − γ ] . {\displaystyle J(x,y)={\begin{bmatrix}\alpha -\beta y&-\beta x\\\delta y&\delta x-\gamma \end{bmatrix}}.} and

6767-487: The fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover. Evaluating J at

6868-566: The harmful effects of this procedure. This school of thought is one argument against using iron fertilization on a wide scale, at least until more data is available to analyze the repercussions of this. Critics are concerned that fertilization will create harmful algal blooms (HAB) as many toxic algae are often favored when iron is deposited into the marine ecosystem. A 2010 study of iron fertilization in an oceanic high-nitrate, low-chlorophyll environment, however, found that fertilized Pseudo-nitzschia diatom spp., which are generally nontoxic in

6969-418: The initial conditions and conserved on each curve. An aside: These graphs illustrate a serious potential limitation in the application as a biological model: for this specific choice of parameters, in each cycle, the rabbit population is reduced to extremely low numbers, yet recovers (while the fox population remains sizeable at the lowest rabbit density). In real-life situations, however, chance fluctuations of

7070-590: The initial state were known exactly, then the future state of such a system could theoretically be predicted. However, in practice, knowledge about the future state is limited by the precision with which the initial state can be measured, and chaotic systems are characterized by a strong dependence on the initial conditions. This sensitivity to initial conditions can be measured with Lyapunov exponents . Markov chains and other random walks are not deterministic systems, because their development depends on random choices. A deterministic model of computation , for example

7171-476: The iron particles is critical. Particles of 0.5–1 micrometer or less seem to be ideal both in terms of sink rate and bioavailability. Particles this small are easier for cyanobacteria and other phytoplankton to incorporate and the churning of surface waters keeps them in the euphotic or sunlit biologically active depths without sinking for long periods. One way to add small amounts of iron to HNLC zones would be Atmospheric Methane Removal . Atmospheric deposition

7272-496: The islands of Haida Gwaii . The Old Massett Village Council financed the action as a salmon enhancement project with $ 2.5 million in village funds. The concept was that the formerly iron -deficient waters would produce more phytoplankton that would in turn serve as a "pasture" to feed salmon . Then-CEO Russ George hoped to sell carbon offsets to recover the costs. The project was accompanied by charges of unscientific procedures and recklessness. George contended that 100 tons

7373-485: The late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or Arditi–Ginzburg model . The validity of prey- or ratio-dependent models has been much debated. The Lotka–Volterra equations have a long history of use in economic theory ; their initial application is commonly credited to Richard Goodwin in 1965 or 1967. The equations have periodic solutions. These solutions do not have

7474-679: The method is unproven; the sequestering efficiency was low and sometimes no effect was seen and the amount of iron deposits needed to make a small cut in the carbon emissions would be in the million tons per year. However since 2021, interest is renewed in the potential of iron fertilization, among other from a white paper study of NOAA, the US National Oceanographic and Atmospheric Administration, which rated iron fertilization as having "moderate potential for cost, scalability and how long carbon might be stored compared to other marine sequestration ideas" Approximately 25 per cent of

7575-449: The model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point . The instability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as

7676-473: The model has become known as the "Lotka-Volterra model". The model was later extended to include density-dependent prey growth and a functional response of the form developed by C. S. Holling ; a model that has become known as the Rosenzweig–MacArthur model. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey. In

7777-553: The natural level of atmospheric iron. Expanding this atmospheric source of iron could complement ship-based deployment. One proposal is to boost the atmospheric iron level with iron salt aerosol . Iron(III) chloride added to the troposphere could increase natural cooling effects including methane removal , cloud brightening and ocean fertilization, helping to prevent or reverse global warming. Martin hypothesized that increasing phytoplankton photosynthesis could slow or even reverse global warming by sequestering CO 2 in

7878-471: The normalization of t , so that only α / γ remains arbitrary. It is the only parameter affecting the nature of the solutions. A linearization of the equations yields a solution similar to simple harmonic motion with the population of predators trailing that of prey by 90° in the cycle. Suppose there are two species of animals, a rabbit (prey) and a fox (predator). If the initial densities are 10 rabbits and 10 foxes per square kilometre, one can plot

7979-578: The ocean by glaciers , rivers and icebergs. About 70% of the world's surface is covered in oceans. The part of these where light can penetrate is inhabited by algae (and other marine life). In some oceans, algae growth and reproduction is limited by the amount of iron. Iron is a vital micronutrient for phytoplankton growth and photosynthesis that has historically been delivered to the pelagic sea by dust storms from arid lands. This Aeolian dust contains 3–5% iron and its deposition has fallen nearly 25% in recent decades. The Redfield ratio describes

8080-420: The ocean floor where their carbonate skeletons can form a major component of the carbon-rich deep sea precipitation, thousands of meters below plankton blooms, known as marine snow . Nonetheless, based on the definition, carbon is only considered "sequestered" when it is deposited in the ocean floor where it can be retained for millions of years. However, most of the carbon-rich biomass generated from plankton

8181-426: The ocean surface has ample macronutrients, with little plant biomass (as defined by chlorophyll). The production in these high-nutrient low-chlorophyll (HNLC) waters is primarily limited by micronutrients , especially iron. The cost of distributing iron over large ocean areas is large compared with the expected value of carbon credits . Research in the early 2020s suggested that it could only permanently sequester

8282-438: The ocean. The expectation was that iron, which is a limiting nutrient for phytoplankton, would boost growth of phytoplankton and that it would sequester carbon dioxide from the atmosphere. The addition of iron typically leads to a short bloom in phyoplankton, which is quickly consumed by other organisms (such as small fish or zooplankton ) and limits the effect of enrichment mainly to increased predator density, which in turn limits

8383-592: The open ocean, began producing toxic levels of domoic acid . Even short-lived blooms containing such toxins could have detrimental effects on marine food webs. Most species of phytoplankton are harmless or beneficial, given that they constitute the base of the marine food chain. Fertilization increases phytoplankton only in the open oceans (far from shore) where iron deficiency is substantial. Most coastal waters are replete with iron and adding more has no useful effect. Further, it has been shown that there are often higher mineralization rates with iron fertilization, leading to

8484-554: The particulate matter fell to between 100 metres (330 ft) to the ocean floor. Each iron atom converted at least 13,000 carbon atoms into algae. At least half of the organic matter sank below, 1,000 metres (3,300 ft). In July 2012, the Haida Salmon Restoration Corporation dispersed 100 short tons (91 t) of iron sulphate dust into the Pacific Ocean several hundred miles west of

8585-400: The prey's parameters. This has as a consequence that an increase in, for instance, the prey growth rate, α {\displaystyle \alpha } , leads to an increase in the predator equilibrium density, but not the prey equilibrium density. Making the environment better for the prey benefits the predator, not the prey (this is related to the paradox of the pesticides and to

8686-401: The progression of the two species over time; given the parameters that the growth and death rates of rabbits are 1.1 and 0.4 while that of foxes are 0.1 and 0.4 respectively. The choice of time interval is arbitrary. One may also plot solutions parametrically as orbits in phase space , without representing time, but with one axis representing the number of prey and the other axis representing

8787-1726: The quantity V ( x , y ) {\displaystyle V(x,y)} is conserved over time, it plays role of a Hamiltonian function of the system. To see this we can define Poisson bracket as follows { f ( x , y ) , g ( x , y ) } = − x y ( ∂ f ∂ x ∂ g ∂ y − ∂ f ∂ y ∂ g ∂ x ) {\displaystyle \{f(x,y),g(x,y)\}=-xy\left({\frac {\partial f}{\partial x}}{\frac {\partial g}{\partial y}}-{\frac {\partial f}{\partial y}}{\frac {\partial g}{\partial x}}\right)} . Then Hamilton's equations read { x ˙ = { x , V } = α x − β x y , y ˙ = { y , V } = δ x y − γ y . {\displaystyle {\begin{cases}{\dot {x}}=\{x,V\}=\alpha x-\beta xy,\\{\dot {y}}=\{y,V\}=\delta xy-\gamma y.\end{cases}}} The variables x {\displaystyle x} and y {\displaystyle y} are not canonical, since { x , y } = − x y ≠ 1 {\displaystyle \{x,y\}=-xy\neq 1} . However, using transformations p = ln ⁡ ( x ) {\displaystyle p=\ln(x)} and q = ln ⁡ ( y ) {\displaystyle q=\ln(y)} we came up to

8888-445: The rate at which it is preyed upon. The term δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in

8989-445: The relative atomic concentrations of critical nutrients in plankton biomass and is conventionally written "106 C: 16 N: 1 P." This expresses the fact that one atom of phosphorus and 16 of nitrogen are required to " fix " 106 carbon atoms (or 106 molecules of CO 2 ). Research expanded this constant to "106 C: 16 N: 1 P: .001 Fe" signifying that in iron deficient conditions each atom of iron can fix 106,000 atoms of carbon, or on

9090-563: The resulting algal bloom died and sank to the sea floor. Planktos was a US company that abandoned its plans to conduct 6 iron fertilization cruises from 2007 to 2009, each of which would have dissolved up to 100 tons of iron over a 10,000 km area of ocean. Their ship Weatherbird II was refused entry to the port of Las Palmas in the Canary Islands where it was to take on provisions and scientific equipment. In 2007 commercial companies such as Climos and GreenSea Ventures and

9191-604: The sea. He died shortly thereafter during preparations for Ironex I, a proof of concept research voyage, which was successfully carried out near the Galapagos Islands in 1993 by his colleagues at Moss Landing Marine Laboratories . Thereafter 12 international ocean studies examined the phenomenon: John Martin , director of the Moss Landing Marine Laboratories , hypothesized that the low levels of phytoplankton in these regions are due to

9292-855: The second fixed point leads to J ( γ δ , α β ) = [ 0 − β γ δ α δ β 0 ] . {\displaystyle J\left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)={\begin{bmatrix}0&-{\frac {\beta \gamma }{\delta }}\\{\frac {\alpha \delta }{\beta }}&0\end{bmatrix}}.} The eigenvalues of this matrix are λ 1 = i α γ , λ 2 = − i α γ . {\displaystyle \lambda _{1}=i{\sqrt {\alpha \gamma }},\quad \lambda _{2}=-i{\sqrt {\alpha \gamma }}.} As

9393-499: The state of the system at a given point in time may be difficult to describe explicitly. In quantum mechanics , the Schrödinger equation , which describes the continuous time evolution of a system's wave function , is deterministic. However, the relationship between a system's wave function and the observable properties of the system appears to be non-deterministic. The systems studied in chaos theory are deterministic. If

9494-532: The surface can stimulate a bloom using less iron. The iron will be confined within a plastic bag reaching from the surface several kilometers down to the sea bottom. The Centre for Climate Repair at the University of Cambridge, along with India's Institute of Maritime Studies assessed the impact of iron seeding in another experiment. They spread iron-coated rice husks across an area of the Arabian Sea. Iron

9595-544: The surface, where planktons can take it up to grow. It has been shown that reduction in the number of sperm whales in the Southern Ocean has resulted in a 200,000 tonnes/yr decrease in the atmospheric carbon uptake, possibly due to limited phytoplankton growth. Phytoplankton is photosynthetic : it needs sunlight and nutrients to grow, and takes up carbon dioxide in the process. Plankton can take up and sequester atmospheric carbon through generating calcium or silicon-carbonate skeletons. When these organisms die they sink to

9696-409: The years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years and, as prey fish the preferred catch, one would intuitively expect this to increase of prey fish percentage. Volterra developed his model to explain D'Ancona's observation and did this independently from Alfred Lotka. He did credit Lotka's earlier work in his publication, after which

9797-551: Was followed by a larger field experiment (IRONEX I) where 445 kg of iron was added to a patch of ocean near the Galápagos Islands . The levels of phytoplankton increased three times in the experimental area. The success of this experiment and others led to proposals to use this technique to remove carbon dioxide from the atmosphere. In 2000 and 2004, iron sulfate was discharged from the EisenEx. 10 to 20 percent of

9898-478: Was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively the logistic equation , originally derived by Pierre François Verhulst . In 1920 Lotka extended the model, via Andrey Kolmogorov , to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics . The same set of equations

9999-582: Was negligible compared to what naturally enters the ocean. Some environmentalists called the dumping a "blatant violation" of two international moratoria. George said that the Old Massett Village Council and its lawyers approved the effort and at least seven Canadian agencies were aware of it. According to George, the 2013 salmon runs increased from 50 million to 226 million fish. However, many experts contend that changes in fishery stocks since 2012 cannot necessarily be attributed to

10100-508: Was published in 1926 by Vito Volterra , a mathematician and physicist, who had become interested in mathematical biology . Volterra's enquiry was inspired through his interactions with the marine biologist Umberto D'Ancona , who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during

10201-455: Was recorded until the 1980s, when oceanographer John Martin of the Moss Landing Marine Laboratories renewed controversy on the topic with his marine water nutrient analyses. His studies supported Hart's hypothesis. These "desolate" regions came to be called " high-nutrient, low-chlorophyll regions " (HNLC). John Gribbin was the first scientist to publicly suggest that climate change could be reduced by adding large amounts of soluble iron to

#148851