Lotka-Volterra equations

The Lotka-Volterra equations (also known as predator-prey equations ) are a system of two non-linear, coupled first-order differential equations . They describe the interaction between predator and prey populations. By predator and prey two classes of living beings are meant, one of which feeds on the other. The equations were drawn up in 1925 by Alfred J. Lotka and, independently of this, in 1926 by Vito Volterra . Essential properties of the solutions to these equations are known as Lotka-Volterra rules .

The equations are

${\displaystyle {\frac {\mathrm {d} N_{1}}{\mathrm {d} t}}=N_{1}(\epsilon _{1}-\gamma _{1}N_{2}),\qquad {\frac {\mathrm {d} N_{2}}{\mathrm {d} t}}=-N_{2}(\epsilon _{2}-\gamma _{2}N_{1})}$

with the designations

${\displaystyle N_{1}=N_{1}(t)}$	Number of bag creatures	time-dependent
${\displaystyle \epsilon _{1}>0}$	Reproduction rate of the prey without disturbance and with a large food supply	constant
${\displaystyle \gamma _{1}>0}$	Perdition rate of predators per animal in the pouch = death rate of prey per predator	constant
${\displaystyle N_{2}=N_{2}(t)}$	Number of robbers	time-dependent
${\displaystyle \epsilon _{2}>0}$	Predator death rate when there is no prey	constant
${\displaystyle \gamma _{2}>0}$	Predator reproductive rate per pouch	constant

The Lotka-Volterra equations are an important basis of theoretical biology , and in particular of population dynamics . The predators and the prey need not necessarily be just animals or individual species; in principle, the model can be applied to guilds - see e.g. B. Volterra's fisheries data . The applicability of the Lotka-Volterra equations depends on the extent to which the justification of the mathematical model applies in the individual case.

Justification of the mathematical model

Volterra justifies its system of equations as follows:

• The population numbers of the prey or the predators are denoted by or . ${\displaystyle N_{1}}$ ${\displaystyle N_{2}}$
• Let the undisturbed growth rates per unit of time be and , the signs not yet being fixed. ${\displaystyle \mathrm {d} t}$ ${\displaystyle \lambda _{1}}$ ${\displaystyle \lambda _{2}}$
• The (average) number of encounters between prey and predator per unit of time is a positive real number that is assumed to be constant within a biotope , but generally depends on the biotope. ${\displaystyle \mathrm {d} t}$ ${\displaystyle \alpha N_{1}N_{2}}$ ${\displaystyle \alpha }$
• A sufficiently large number of encounters has an average effect on the population . In the case of the pouch creatures, this is immediately clear: an encounter with a predator leads to a certain probability that the prey will be eaten. On the other hand, the effect of an encounter on the number of robbers is only indirect, but in any case positive; For the modeling, an immediate effect on the population is assumed for the predators. ${\displaystyle n}$ ${\displaystyle \beta _{i}}$ ${\displaystyle N_{i}}$

Taken together, this leads to the equations

${\displaystyle \mathrm {d} N_{1}=\lambda _{1}N_{1}\mathrm {d} t+\alpha N_{1}N_{2}{\frac {\beta _{1}}{n}}\mathrm {d} t\quad {\mbox{ und }}\quad \mathrm {d} N_{2}=\lambda _{2}N_{2}\mathrm {d} t+\alpha N_{1}N_{2}{\frac {\beta _{2}}{n}}\mathrm {d} t.}$

Division by leads to the equations ${\displaystyle \mathrm {d} t}$

${\displaystyle {\frac {\mathrm {d} N_{1}}{\mathrm {d} t}}=\lambda _{1}N_{1}+\alpha N_{1}N_{2}{\frac {\beta _{1}}{n}}\quad {\mbox{ und }}\quad {\frac {\mathrm {d} N_{2}}{\mathrm {d} t}}=\lambda _{2}N_{2}+\alpha N_{1}N_{2}{\frac {\beta _{2}}{n}}.}$

If you set

${\displaystyle \epsilon _{1}=\lambda _{1},\quad \epsilon _{2}=-\lambda _{2},\quad \gamma _{1}=-{\frac {\alpha \beta _{1}}{n}},\quad \gamma _{2}={\frac {\alpha \beta _{2}}{n}},}$

and if you cross the border , you get the Lotka-Volterra equations in the form mentioned at the beginning. ${\displaystyle \mathrm {d} t\to 0}$

Mathematical treatment

Of course, it was also clear to Volterra that the time-dependent population numbers and can only assume integer values ​​and therefore are functions of either constant or indifferentifiable. But with large population numbers the relative error made by switching to the continuous model is small. The advantage of the two-dimensional Lotka-Volterra equation, however, is that some statements can be mathematically proven that have an interesting relationship to real data, as described below. ${\displaystyle N_{1}(t)}$ ${\displaystyle N_{2}(t)}$ ${\displaystyle t}$

For the mathematical treatment of Lotka-Volterra systems, the somewhat simpler notation is usually used today

${\displaystyle {\frac {\mathrm {d} N}{\mathrm {d} t}}=N(a-bP),\quad {\frac {\mathrm {d} P}{\mathrm {d} t}}=P(cN-d),}$

wherein positive constants and the number of prey and the number of predators ( predators call). ${\displaystyle a,b,c,d}$ ${\displaystyle N(t)}$ ${\displaystyle P(t)}$

Constant solutions

The constant solutions (also called equilibrium points or critical points ) are obtained by setting the right-hand sides of the Lotka-Volterra equations equal to zero:

${\displaystyle N(a-bP)=0,\qquad P(cN-d)=0.}$

So there are exactly two constant solutions, namely the trivial equilibrium point and the inner equilibrium point ${\displaystyle (0,0)}$

${\displaystyle (N^{*},P^{*})=\left({\frac {d}{c}},{\frac {a}{b}}\right).}$

A first integral

One method for finding non-constant solutions is to look for a first integral , i.e. an invariant of the motion . Volterra finds one in the following way: If you multiply the first basic equation by and the second by , and then add the two equations, the terms with the product vanish and you get ${\displaystyle c}$ ${\displaystyle b}$ ${\displaystyle NP}$

${\displaystyle c\,{\frac {\mathrm {d} N}{\mathrm {d} t}}+b\,{\frac {\mathrm {d} P}{\mathrm {d} t}}=ac\,N-bd\,P}$.

By multiplying the first basic equation with and the second with and then adding, you get to ${\displaystyle d/N}$ ${\displaystyle a/P}$

${\displaystyle {\frac {d}{N}}\,{\frac {\mathrm {d} N}{\mathrm {d} t}}+{\frac {a}{P}}\,{\frac {\mathrm {d} P}{\mathrm {d} t}}=ac\,N-bd\,P}$.

The numerically calculated phase space trajectories show a fixed point around which predator and prey populations fluctuate cyclically. The solutions move counterclockwise on the level lines of . ${\displaystyle V}$

Subtracting these two equations gives

${\displaystyle c\,{\frac {\mathrm {d} N}{\mathrm {d} t}}-{\frac {d}{N}}\,{\frac {\mathrm {d} N}{\mathrm {d} t}}+b\,{\frac {\mathrm {d} P}{\mathrm {d} t}}-{\frac {a}{P}}\,{\frac {\mathrm {d} P}{\mathrm {d} t}}=0}$.

By integrating this last equation, one finally reaches the relationship

${\displaystyle V(N,P):=c\,N-d\,\ln N+b\,P-a\,\ln P=}$ const.

Conversely, one can calculate the total derivative of the function defined in this way according to : ${\displaystyle V}$ ${\displaystyle t}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}V(N,P)&={\frac {\partial V(N,P)}{\partial N}}\cdot {\frac {\mathrm {d} N}{\mathrm {d} t}}+{\frac {\partial V(N,P)}{\partial P}}\cdot {\frac {\mathrm {d} P}{\mathrm {d} t}}\\&=\left(c-{\frac {d}{N}}\right)\,N(a-bP)+\left(b-{\frac {a}{P}}\right)\,P(cN-d)\\&=0;\end{aligned}}}$

this also leads to the statement that is constant ( invariant ) on the solutions of the basic equations ; a solution of the Lotka-Volterra equation can not leave its level lines from . ${\displaystyle V}$ ${\displaystyle V}$

Another way to find an invariant of the motion is to transform the Lotka-Volterra equations into an exact differential equation with the help of an Euler's multiplier and then to integrate this.

stability

Since there is also a Lyapunov function as the first integral , and since it has a strict local minimum at the inner equilibrium point, it follows from Lyapunov's first criterion that this equilibrium point is stable. ${\displaystyle V}$ ${\displaystyle V}$

The Lotka-Volterra Laws

With the help of the first integral , Volterra proves three mathematical properties of the solutions ("laws") of the Lotka-Volterra equations, the biological interpretations of which have found widespread use as Lotka-Volterra rules . ${\displaystyle V}$

From the boundary behavior of the function one can conclude that there is no trajectory which has a point in the first quadrant ${\displaystyle V}$

${\displaystyle Q_{+}=\{(N,P)\;:\;N>0,P>0\}=\{(N_{1},N_{2})\;:\;N_{1}>0,N_{2}>0\}}$

owns, leaves it: the first quadrant is invariant. The Lotka-Volterra laws apply generally to maximal solutions of the Lotka-Volterra equations in this quadrant; if one of the two animal classes dies out, this quadrant is left and the Lotka-Volterra laws lose their validity.

periodicity

The populations plotted against time give the image of a sine- like oscillation with a phase shift between the hunter and the prey population

Since the function in the quadrant is strictly convex and takes its minimum in the inner equilibrium point, the level lines of form closed curves in phase space. Since every solution must be contained in a level line of , the periodicity of the solutions follows from the uniqueness and a consideration of the local directional field. ${\displaystyle V}$ ${\displaystyle Q_{+}}$ ${\displaystyle V}$ ${\displaystyle V}$

Loi du cycle périodique. - The fluctuations des deux espèces sont périodique.

(Law of periodic cycles: the fluctuations of the two classes are periodic.)

Preservation of the mean values

This follows from the periodicity of the solutions with a few lines of calculation

Loi de la conservation des moyennes. - Les moyennes pendant une période des nombres des individus des deux espèces sont independent des conditions initiales, et égales aux nombres qui correspondent à l'ètat stationnaire, pour les valeurs données des “coefficients d'accroissement” , et des “coefficients de voracité” . ${\displaystyle \epsilon _{1},\epsilon _{2}}$ ${\displaystyle \gamma _{1},\gamma _{2}}$

(Law of conservation of mean values: the time mean values ​​over a period of the individual numbers of the two classes do not depend on the initial conditions, and are equal to the numbers of the equilibrium state for the given "growth coefficients" and the "feeding coefficients" .) ${\displaystyle \epsilon _{1},\epsilon _{2}}$ ${\displaystyle \gamma _{1},\gamma _{2}}$

This means that the mean values ​​over time satisfy the equations

${\displaystyle {\overline {N_{1}}}={\frac {\epsilon _{2}}{\gamma _{2}}}\quad {\mbox{und}}\quad {\overline {N_{2}}}={\frac {\epsilon _{1}}{\gamma _{1}}}.}$

At first glance, it is confusing that the mean value of the prey population depends only on the death and feeding rate of the predator population and not on the reproduction rate of the prey animals. In contrast, the mean value of the predator population depends only on the reproduction and death rate of the prey population and not on the predation rate and death rate of the predators. The equilibrium number of prey is higher, the more unfavorable the parameters are for the predators. The equilibrium number of predators, on the other hand, is higher, the more favorable the parameters are for the prey animals. ${\displaystyle N_{1}(t)}$ ${\displaystyle N_{2}(t)}$

Naturally, this property is the Lotka-Volterra model when you look at the in mathematical reasoning coming for application modeling look: control over the population number of a class of animals this is the responsibility solely of the other class.

Disturbance of the mean values

The most interesting of these laws because of its biological interpretation is that

Loi de la perturbation des moyennes. - Si l'on détruit les deux espèces uniformément et proportionnellement aux nombres de leurs individus (assez peu pour que les fluctuations subsistent), la moyenne du nombres des individus de l'espèce dévorée croît et celles de l'espèce devorant diminue.

(Law of the disturbance of the mean values: If the two animal classes are decimated equally and proportionally to the population size, and if the rate of prey decimation is less than its reproduction rate, the mean value of the prey population increases and the mean value of the predator population decreases.)

In fact, Volterra proves a quantitative version: if the rate of destruction of the pouch creatures and the rate of destruction of the predators, then the mean values for the solutions of the perturbed Lotka-Volterra equations are ${\displaystyle \alpha }$ ${\displaystyle \beta }$

${\displaystyle {\overline {N_{1}}}={\frac {\epsilon _{2}+\beta }{\gamma _{2}}}\quad {\mbox{und}}\quad {\overline {N_{2}}}={\frac {\epsilon _{1}-\alpha }{\gamma _{1}}}.}$

This means: the number of pouch creatures averaged over a Lotka-Volterra period increases precisely when the predators are decimated - pretty much independently of a decimation of the prey, as long as it is not exterminated. Conversely, the mean number of predators always decreases when the pouch creatures are decimated, and this decrease does not depend on how much the predators are additionally decimated (as long as they are not exterminated).

Extensions

In theoretical ecology , the Lotka-Volterra equations form the starting point for developing more complex models, some of which are already described in Volterra's book.

Intraspecific competition terms

A first extension of the Lotka-Volterra equations is created by subtracting terms proportional to or that model the intraspecific competition . There are various ways of justifying the form and the newly added terms: ${\displaystyle N_{1}^{2}}$ ${\displaystyle N_{2}^{2}}$ ${\displaystyle -\alpha _{1}N_{1}^{2}}$ ${\displaystyle -\alpha _{2}N_{2}^{2}}$

• With the empirical studies on population development according to Pierre-François Verhulst , see logistic equation .
• By assuming that the (undisturbed) growth rate of a population is proportional to the difference between a capacity limit and the actual population number.
• By analyzing the influence of intraspecific encounters on population numbers similar to Volterra's justification of the term for modeling interspecific competition: An intraspecific encounter is with a certain probability a competition for a resource in which one individual loses out.

The resulting Lotka-Volterra competing equations of theoretical biology are a classic approach to describe the dynamics of a greatly simplified biocenosis , consisting of a renewable resource and at least 2 competing species: ${\displaystyle r}$

${\displaystyle {\frac {\mathrm {d} N_{1}}{\mathrm {d} t}}=N_{1}ar-m_{1}N_{1}}$

${\displaystyle {\frac {\mathrm {d} N_{2}}{\mathrm {d} t}}=N_{2}br-m_{2}N_{2}}$,

where a, b are exponential growth rates and the m (mortality rate) represent death rates. The amount of resources available at a time is assumed to be: ${\displaystyle r_{\mathrm {max} }-(pN_{1}+qN_{2})}$

Numerically calculated trajectory for a Lotka-Volterra system with intraspecific competition. The system converges towards a fixed point.

This results in:

${\displaystyle {\frac {\mathrm {d} N_{1}}{\mathrm {d} t}}=N_{1}a(r_{\mathrm {max} }-(pN_{1}+qN_{2}))-m_{1}N_{1}}$

${\displaystyle {\frac {\mathrm {d} N_{2}}{\mathrm {d} t}}=N_{2}b(r_{\mathrm {max} }-(pN_{1}+qN_{2}))-m_{2}N_{2}}$

by multiplying and replacing the coefficients

${\displaystyle \epsilon _{1}=ar_{\mathrm {max} }-m_{1}}$

${\displaystyle \gamma _{1}=aq}$

${\displaystyle \alpha _{1}=ap}$

${\displaystyle \epsilon _{2}=m_{2}-br_{\mathrm {max} }}$

${\displaystyle \gamma _{2}=-bp}$

${\displaystyle \alpha _{2}=-bq}$

one arrives at equations of the form

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} N_{1}}{\mathrm {d} t}}&=N_{1}(\epsilon _{1}-\gamma _{1}N_{2}-\alpha _{1}N_{1}),\\{\frac {\mathrm {d} N_{2}}{\mathrm {d} t}}&=N_{2}(-\epsilon _{2}+\gamma _{2}N_{1}-\alpha _{2}N_{2}),\end{aligned}}}$

which in turn allow two equilibrium positions: the trivial equilibrium point and the inner equilibrium point , which is given by a linear system of equations: ${\displaystyle (0,0)}$ ${\displaystyle (N_{1}^{*},N_{2}^{*})}$

${\displaystyle {\begin{pmatrix}\alpha _{1}&\gamma _{1}\\-\gamma _{2}&\alpha _{2}\end{pmatrix}}{\begin{pmatrix}N_{1}^{*}\\N_{2}^{*}\end{pmatrix}}={\begin{pmatrix}\epsilon _{1}\\-\epsilon _{2}\end{pmatrix}}}$

The equilibrium point can be found by solving this system of equations

${\displaystyle {\begin{pmatrix}N_{1}^{*}\\N_{2}^{*}\end{pmatrix}}={\frac {1}{\alpha _{1}\alpha _{2}+\gamma _{1}\gamma _{2}}}{\begin{pmatrix}\alpha _{2}\epsilon _{1}+\gamma _{1}\epsilon _{2}\\\gamma _{2}\epsilon _{1}-\alpha _{1}\epsilon _{2}\end{pmatrix}},}$

which is in the first quadrant under the condition . ${\displaystyle \gamma _{2}\epsilon _{1}>\alpha _{1}\epsilon _{2}}$

There is also a Lyapunov function for this extended Lotka-Volterra system:

${\displaystyle V(N_{1},N_{2})=\gamma _{2}(N_{1}-N_{1}^{*}\ln N_{1})+\gamma _{1}(N_{2}-N_{2}^{*}\ln N_{2}),}$

with which the requirements of Lyapunov's Second Criterion for the equilibrium point are fulfilled. It follows that this equilibrium point is now asymptotically stable . ${\displaystyle (N_{1}^{*},N_{2}^{*})}$

More than two classes of living beings

Much of Volterra's book refers to extensions of his system to include more than two classes of living things that interact with one another in different ways.

Applications

Fishing data

In the introduction to Volterra's book there is a table that contains the percentage of cartilaginous fish ( Sélaciens ), in particular sharks, of the total catch of the fishing port for the years 1905 and 1910–1923 and for three fishing ports:

	1905	1910	1911	1912	1913	1914	1915	1916	1917	1918	1919	1920	1921	1922	1923
Trieste	-	5.7	8.8	9.5	15.7	14.6	7.6	16.2	15.4	-	19.9	15.8	13.3	10.7	10.2
Fiume	-	-	-	-	-	11.9	21.4	22.1	21.2	36.4	27.3	1600	15.9	14.8	10.7
Venise	21.8	-	-	-	-	-	-	-	-	-	30.9	25.3	25.9	26.8	26.6

«Cela prouve pendant la période 1915–1920, où la pêche était moins intense à cause de la guerre, un accroissement relatif de la classe des Sélaciens qui, particulièrement voraces, se nourissent d'autres poissons. Les statistique inclinent donc à penser qu'une diminution dans l'intensité de la destruction favorise les espèces les plus voraces. »

These statistics show an increased proportion of predatory fish in the years 1915 to 1920, when fishing in the Mediterranean was less intensive because of the First World War , which then declined again with the intensification of fishing after 1920. The third Lotka-Volterra law, the shift in mean values, offers a plausible explanation for this.

Medical epidemiology

In theoretical biology as well as in medical epidemiology , models of the Lotka-Volterra type are used to describe the processes of spread of diseases. Some examples can be found in the SI model , SIR model and SIS model .

Economics

The Goodwin model for explaining business cycle fluctuations is based on Lotka-Volterra equations, with the wage share playing the role of predator and the employment share playing the role of prey.

Gerold Blümle developed an economic model in which (mathematically) the investment quota plays the role of predators and the spread or variance of profits plays the role of prey. For Frank Schohl, the variance in the company's changes in returns plays the role of predators, and the variance in the company's changes in supply plays the role of prey.

literature

• Alfred J. Lotka : Analytical Theory of Biological Populations. Plenum Press, New York NY u. a. 1998, ISBN 0-306-45927-2 , English translation of the two volumes
  • Théorie analytique des associations biologiques (= Exposés de biométrie et de statistique biologique. Vol. 4 = Actualités scientifiques et industrielles. Vol. 187). Première partie: Principes. Hermann, Paris 1934.
  • Théorie analytique des associations biologiques. Deuxième partie: Analysis démographique avec application particulière à l'espèce humaine (= Exposés de biométrie et de statistique biologique. Vol. 12 = Actualités scientifiques et industrielles. Vol. 780). Hermann, Paris 1939.

Individual evidence

1. ^ Elements of Physical Biology . 1925, p. 115
2. ↑ Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. In: Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2, 31-113.
3. ↑ Gerold Blümle: Growth and business cycle with differential profits - A Schumpeter model of economic development. In: HJ Ramser, Hajo Riese (ed.) Contributions to applied economic research. Gottfried Bombach on his 70th birthday. Berlin 1989, pp. 13-37. Also shown in Frank Schohl: The market-theoretical explanation of the economy. Writings on applied economic research. Tübingen 1999.
4. ↑ Frank Schohl: The market-theoretical explanation of the economy. Writings on applied economic research. Tübingen 1999, p. 232.

• Vito Volterra: Leçons sur la Théorie Mathématique de la Lutte pour la Vie. Gauthier-Villars, 1931; Authorized reprint: Éditions Jaques Gabay, 1990, ISBN 2-87647-066-7

1. ↑ chap. I, sec. II. Deux espèces dont l'une dévore l'autre
2. ↑ p. 14
3. ↑ p. 14f
4. ↑ p. 15
5. ↑ pp. 15-27
6. ↑ pp. 15-19
7. ↑ p. 2 ff.
8. ↑ p. 2

• James D. Murray: Mathematical Biology I: An Introduction. 3. Edition. Springer, 2008, ISBN 978-0-387-95223-9

1. ↑ p. 79

• Günther J. Wirsching: Ordinary differential equations. An introduction with examples, exercises and sample solutions. Teubner, Wiesbaden 2006, ISBN 978-3-519-00515-5

1. ↑ Example 4.2, p. 65 ff.
2. ↑ pp. 67-70
3. ↑ p. 80 ff.
4. ↑ p. 82