Lotka-Volterra equations

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The Lotka-Volterra equation , also known as the predator-prey equation , is the first non-linear differential equation of the first order, which is often used to describe the dynamics of the biological system in which two species interacting, one as predator and the other as prey. The population changes over time according to the pair of equations:

dimana

x adalah jumlah mangsa (misalnya, kelinci);

y adalah jumlah beberapa predator (misalnya, rubah);

${\ displaystyle {\ tfrac {dy} {dt}}}  ÂÂ$ dan ${\ displaystyle {\ tfrac {dx} {dt}}}  ÂÂ$ mewakili tingkat pertumbuhan seketika dari dua populasi;

t mewakili waktu;

? , ? , ? , ? adalah parameter nyata positif yang menggambarkan interaksi dua spesies.

The Lotka-Volterra equation system is an example of the Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with the interaction of predators, competition, disease, and mutualism.

Video Lotka-Volterra equations

Histori

The Lotka-Volterra model predators were originally proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively a logistic equation, originally derived by Pierre FranÃÆ'§ois Verhulst. In 1920 Lotka expanded the model, through Andrey Kolmogorov, to "organic systems" using plant species and herbivorous animal species as an example and in 1925 he used the equations to analyze the predator-prey interactions in his book on biomatics. The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who became interested in mathematical biology. The Volterra investigation was inspired through his interaction with marine biologist Umberto D'Ancona, who was after his daughter at the time and later became his son-in-law. D'Ancona studied fish catches in the Adriatic Sea and has noticed that the percentage of caught predatory fish has increased during World War I years (1914-18). This puzzled him, as fishing efforts have greatly diminished during the war years. Volterra developed his model independently of Lotka and used it to explain the observations d'Ancona.

This model was later expanded to include the survival of prey depending on the density and functional response of the form developed by C.Ã, S. Holling; a model that has been known as the Rosenzweig-McArthur model. Both Lotka-Volterra and Rosenzweig-MacArthur models have been used to describe the dynamics of predator and prehistoric natural populations, such as lynx and snowshoe rabbit data from Hudson's Bay Company and deer and wolf populations in Isle Royale National Park.

In the late 1980s, an alternative to the Lotka-Volterra predator-model (and common-general prevalence generalization) appeared, the dependent ratio or the Arditi-Ginzburg model. The validity of models dependent on prey or ratio has been widely debated.

The Lotka-Volterra equation has a long history in economic theory; Their initial application is generally credited to Richard Goodwin in 1965 or 1967.

Maps Lotka-Volterra equations

The physical meaning of the equation

The Lotka-Volterra model makes a number of assumptions, not always realizable in nature, about the environment and the evolution of predator and prey populations:

1. The prey population finds plenty of food all the time.
2. The food supply of the predator population depends entirely on the size of the prey population.
3. The rate of population change is proportional to its size.
4. During the process, the environment does not change to support a species, and genetic adaptation is not important.
5. Predators have unlimited appetite.

When differential equations are used, the solutions are deterministic and continuous. This, in turn, implies that second-generation predators and prey continue to overlap.

Prey

Ketika digandakan, persamaan mangsa menjadi

${\ displaystyle {\ frac {dx} {dt}} = \ alpha x- \ beta xy.}  ÂÂ$

It is assumed to have unlimited supply of food and reproduce exponentially, except subject to predation; this exponential growth is represented in the above equation by the term ? x . The predation rate on the prey is assumed to be proportional to the rate at which predators and prey meet, these are represented above by ? Xy . If either x or y is zero, then there will be no predation.

With both of these terms the above equation can be interpreted as follows: the rate of change in the population of the prey is given by its own growth rate minus the rate at which it is preyed.

Predator

Persamaan predator menjadi

${\ displaystyle {\ frac {dy} {dt}} = \ delta xy- \ gamma y.}  ÂÂ$

In this equation, ? Xy represents the growth of the predator population. (Note the similarity with the predation level; however, different constants are used, because the rate at which the predator population grows is not necessarily the level at which it consumes prey). ? y represents the rate of loss of predators due to either natural or emigration deaths, it leads to exponential decay in the absence of prey.

Therefore, the equation states that the rate of change in the predator population depends on the level of consumption of its prey, minus its intrinsic death rate. Solution

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for the equation

Equations have periodic solutions and do not have simple expressions in terms of ordinary trigonometric functions, although they are quite obedient.

If there are no non-negative parameters ? , ? , ? , ? disappears, three can be absorbed into the normalization of the variable to leave only one parameter: because the first homogeneous equation in x , and the second in y , parameter ? /? and ? /? can be absorbed in normalization y and x respectively, and ? to normalize t , so only ? /? remains arbitrary. These are the only parameters that affect the nature of the solution.

A linearization of the equation yields a solution similar to a simple harmonic movement with a predatory population lagging that of prey by 90 ° in the cycle.

Simple example

Suppose there are two animal species, baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plan the development of both species over time. Choice of arbitrary time intervals.

One can also plan the solution parametrically as an orbit in the phase space, without representing time, but with one axis representing the number of prey and other axes representing the number of predators for each moment.

Ini sesuai untuk menghilangkan waktu dari dua persamaan diferensial di atas untuk menghasilkan persamaan diferensial tunggal

${\ displaystyle {\ frac {dy} {dx}} = - {\ frac {y} {x}} {\ frac {\ delta x- \ gamma} {\ beta y- \ alpha}}}  ÂÂ$

menghubungkan variabel x dan y . Solusi dari persamaan ini adalah kurva tertutup. Ini setuju untuk pemisahan variabel: mengintegrasikan

${\ displaystyle {\ frac {\ beta y- \ alpha} {y}} \, dy {\ frac {\ delta x- \ gamma} {x}} \, dx = 0}  ÂÂ$

menghasilkan hubungan implisit

${\ displaystyle V = \ delta x- \ gamma \ ln (x) \ beta y- \ alpha \ ln (y),}  ÂÂ$

where V is a constant quantity depending on the initial conditions and is preserved on each curve.

Set aside: This graph represents a serious potential problem with this as a biological model : For this particular parameter selection, in each cycle, the baboon population is reduced to a very low number, but recovered (while the cheetah population remains large at the lowest density of baboons). However, in real-life situations, fluctuations in the chances of the number of discrete individuals, as well as the family structure and life cycle of baboons, can cause baboons to become extinct, and, consequently, cheetahs as well. This modeling problem has been called the "atto-fox problem", an atto-fox being notional 10 ^-18 fox, in the context of rabies modeling in England.

Phase-space plot of next example

Less extreme examples include: ? = 2/3, ? = 4/3, ? = 1 = ? . Assume x , y the number of thousands each. The circle represents the initial condition of prey and predator of x = y = 0.9 to 1.8, in step 0.1. The fixed point is (1, Ã, 1/2).

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System dynamics

In model systems, predators multiply when there are many prey but, ultimately, surpass their supply and decrease their food. Since the predator population is low, the prey population will increase again. This dynamic continues in the cycle of growth and decline.

Equalibrium population

Ekuilibrium populasi terjadi dalam model ketika tidak ada perubahan tingkat populasi, yaitu ketika kedua turunan sama dengan 0:

${\ displaystyle x (\ alpha - \ beta y) = 0,}  ÂÂ$

${\ displaystyle -y (\ gamma - \ delta x) = 0.}  ÂÂ$

Sistem persamaan di atas menghasilkan dua solusi:

${\ displaystyle \ {y = 0, x = 0 \}}  ÂÂ$

dan

${\ displaystyle \ left \ {y = {\ frac {\ alpha} {\ beta}}, x = {\ frac {\ gamma} {\ delta}} \ right \}.}  ÂÂ$

Therefore, there are two equilibrium.

The first solution effectively represents 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 where the two populations maintain the current number, not zero, and, in a simplified model, do so indefinitely. The population level at which this equilibrium is achieved depends on the values ​​of the selected parameter ? , ? , ? , and ?

Fixed point stability

The stability of a fixed point at the origin can be determined by linearizing using a partial derivative.

Matriks Jacobian dari model predator-prey adalah

${\ displaystyle J (x, y) = {\ begin {bmatrix} \ alpha - \ beta y & amp; - \ beta x \\\ delta y & amp; \ delta x- \ gamma \ end {bmatrix}}.}  ÂÂ$

Titik tetap pertama (kepunahan)

Ketika dievaluasi pada kondisi mantap (0, 0), matriks Jacobian J menjadi

${\ displaystyle J (0,0) = {\ begin {bmatrix} \ alpha & amp; 0 \\ 0 & amp; - \ gamma \ end {bmatrix}}.}  ÂÂ$

Nilai eigen dari matriks ini adalah

${\ displaystyle \ lambda _ {1} = \ alpha, \ quad \ lambda _ {2} = - \ gamma.}  ÂÂ$

In the model? and ? is always greater than zero, and thus the mark of the above eigenvalues ​​will always be different. Therefore the fixed point at the origin is the saddle point.

Fixed point stability is important. If stable, the nonzero population may be attracted towards it, and thus the dynamics of the system may lead to the extinction of both species for many initial population level cases. However, since the fixed point at the point of origin is the saddle point, and hence unstable, the extinction of both species is difficult in the model. (In fact, this can only happen if the prey is artificially completely eradicated, causing the predator to die of starvation.If the predator is eradicated, the prey population will grow unattached in this simple model.) The population of prey and predator can get very close to zero and still recovering.

Secondary fixes (oscillations)

Mengevaluasi J pada titik tetap kedua mengarah ke

Nilai eigen dari matriks ini adalah

$            {\displaystyle         {?}_{            1          }        =        \mathrm{saya}                  \sqrt{            ?            ?          }                ,                {?}_{            2          }        =        -        \mathrm{saya}                  \sqrt{            ?            ?          }                .      }        \{\backslash \; displaystyle\; \backslash \; lambda\; \_\; \{1\}\; =\; i\; \{\backslash \; sqrt\; \{\backslash \; alpha\; \backslash \; gamma\}\},\; \backslash \; quad\; \backslash \; lambda\; \_\; \{2\}\; =\; -\; i\; \{\backslash \; sqrt\; \{\backslash \; alpha\; \backslash \; gamma\}\}.\}\;  ÂÂ$

Karena nilai eigen keduanya murni imajiner dan berkonjugasi satu sama lain, titik tetap ini berbentuk bulat panjang, sehingga solusinya bersifat periodik, berosilasi pada elips kecil di sekitar titik tetap, dengan periode ${\ displaystyle \ omega = {\ sqrt {\ lambda _ {1} \ lambda _ {2}}} = {\ sqrt {\ alpha \ gamma}}}  ÂÂ$ .

As illustrated in the oscillation circulating in the figure above, the level curve is closed orbits around fixed points: the rate of the population of the predator and prey population and oscillates without damping around fixed points with the period ${\ displaystyle \ omega = {\ sqrt {\ alpha \ gamma}}}$ .

Nilai konstanta gerak V , atau, ekuivalen, K Â = Â exp ( V ), < math xmlns = "http://www.w3.org/1998/Math/MathML" alttext = "{\ displaystyle K = y ^ {\ alpha} e ^ {- \ beta y} x ^ {\ gamma} e ^ {- \ delta x}} ">                         K          =                     y                        ?                                         e                         -             ?              y                                         x                        ?                                         e                         -             ?              x                                      {\ displaystyle K = y ^ {\ alpha} e ^ {- \ beta y} x ^ {\ gamma} e ^ {- \ delta x}}    , dapat ditemukan untuk orbit tertutup di dekat titik tetap.

Meningkatkan K memindahkan orbit tertutup lebih dekat ke titik tetap. Nilai terbesar dari konstanta K diperoleh dengan menyelesaikan masalah optimasi

${\ displaystyle y ^ {\ alpha} e ^ {- \ beta y} x ^ {\ gamma} e ^ {- \ delta x} = {\ frac {y ^ {\ alpha} x ^ {\ gamma}} {e ^ {\ delta x \ beta y}}} \ longrightarrow \ max \ limits _ {x, y & gt; 0}.}  ÂÂ$

Nilai maksimal K demikian tercapai pada titik stasioner (tetap) ${\ displaystyle \ left ({\ frac {\ gamma} {\ delta}}, {\ frac {\ alpha} {\ beta}} \ right)}  ÂÂ$ dan berjumlah

$            {\displaystyle         K^{            *          }        =        {            \left(                          \frac{?}{                  ?                  e                }                        \right)          }^{            ?          }        {            \left(                          \frac{?}{                  ?                  e                }                        \right)          }^{            ?          }        ,      }        \{\backslash \; displaystyle\; K\; ^\; \{*\}\; =\; \backslash \; kiri\; (\{\backslash \; frac\; \{\backslash \; alpha\}\; \{\backslash \; beta\; e\}\}\; \backslash \; right)\; ^\; \{\backslash \; alpha\}\; \backslash \; kiri\; (\{\backslash \; frac\; \{\backslash \; gamma\}\; \{\backslash \; delta\; e\}\}\; \backslash \; right)\; ^\; \{\backslash \; gamma\},\}\;  ÂÂ$

where e is the Euler number.

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See also

• Lotka-Volterra Competitive Equation
• Generalized Lotka-Volterra equation
• Lotka-Volterra mutualism and equation
• Community Matrix
• Population dynamics
• Fishery population dynamics
• Nicholson-Bailey Model
• The reaction-diffusion system
• Enrichment paradox

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Note

Source of the article : Wikipedia