## R Lotka Volterra

- Lotka Volterra Equation Explained
- Lotka Volterra Model Explained
- Lotka Volterra Example
- Lotka Volterra Competition Model Example
- Lotka Volterra R Code
- Lotka Volterra Variables

I have a Lotka-Volterra Model in R; I have been asked to change it from two species to three. Ask Question Asked 11 days ago. Active 11 days ago. The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions.

In this paper, we are concerned with time periodic traveling wave solutions to the following diffusive Lotka-Volterra competition system(1.1)ut=uxx+u(r1(t)−a1(t)u−b1(t)v),vt=d(t)vxx+v(r2(t)−a2(t)u−b2(t)v),where u=u(t,x)and v=v(t,x)are the densities of two competitive species at location x∈Rand time t∈R+, d,ri,ai,bi(i=1,2)are positive T. The Lotka-Volterra model of predator and prey interactions is a classic one, but adds another variable to the 3 constants in the above model. The new variable is a predator-prey encounter rate. In the Lotka-Volterra model, it's easy to give it values that drive predator or prey below zero, which makes no sense, or to drive prey to such small.

### Lotka Volterra Equation Explained

The **predator-prey equations** is a system of two linked equations that models two species that depend on each other: One is the prey, which provides food for the other, the predator. Both prey and predator populations grow if conditions are right. Alfred J. Lotka found these equations in 1925.^{[1]}Vito Volterra found them, independently, in 1926. For this reason, the equations are also called **Lotka-Volterra equations**.^{[2]}

The equations themselves are non-linear differential equations.

There are a number of pre-conditions

- The prey population always finds food
- The size of the predator population only depends on the size of the prey population
- The rate of change of a population depends on its size
- There are no changes in the environment which would favor one species. Genetic changes are not important
- Predators have limitless appetite

In this case the solution of the differential equations is deterministic and continuous. This means that the generations of both the predator and prey are overlapping all the time.^{[3]}

The solutions to the equations lead to the Lotka-Volterra rules:

- There are periodic changes is the populations of predator and prey. The changes in the population of predators follow that of the prey.
- Looking at longer periods of time: the average number of predators, and prey is constant so long as the environment is stable.
- If the number of predators is reduced, the number of prey animals increase.

## Applications[changechange source]

In one of his books, Volterra provided statistics on the number of certain cartilagenous fish caught at some Italian ports in the Mediterranean.^{[4]} Volterra used the term 'sélaciens', which refers to sharks. These often prey on other fish. In the table, Volterra provided numbers for 1905, and 1910 to 1923:

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^{[5]} | – | – | – | – | - | 11,9 | 21,4 | 22,1 | 21,2 | 36,4 | 27,3 | 1600 | 15,9 | 14,8 | 10,7 |

Venice | 21,8 | – | – | – | - | – | – | - | – | – | 30,9 | 25,3 | 25,9 | 26,8 | 26,6 |

He observed that between 1915 and 1920, more of these fish were caught. He expained this as follows: During the First World War, less fishing was done. For this reason, there were more prey animals, so the number of predators increased. After the war, fishing increased again, reducing the number of prey animals. This also led to a decrease in the number of predators, which can be seen in the table above.

The use of the first Lotka-Volerra rule in economics is known as pork cycle.

## Extensions[changechange source]

The predator-prey equations have also been the foundation of other work. Volterra himselft extended it to be able to model intraspecific competition. Intraspecific competition occurs when two aninimals of the same species compete for limited resources. Intraspecific competition is an important factor to regulate population density. It is also important to be able to adapt to a changing environment (and for evoution). Volterra did this by adding new terns to the equation.

Most of Volerra's work is about extending the model to be able to handle more than two classes of animals interacting.

## References[changechange source]

### Lotka Volterra Model Explained

- ↑Lotka A.J. 1925.
*Elements of physical biology*. Williams and Wilkins. - ↑Leigh E.R. 1968. The ecological role of Volterra's equations, in
*Some Mathematical Problems in Biology*. A modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903. - ↑Cooke, D.; Hiorns, R. W.; et al. (1981).
*The Mathematical Theory of the Dynamics of Biological Populations*.**II**. Academic Press. - ↑Volterra, Vito (1931).
*Leçons sur la Théorie Mathématique de la Lutte pour la Vie., 1931*. Gauthier-Villars. - ↑Fiume is called Rijeka today. From 1805 to 1945 it was part of Italy. At the end of World War II, about 80% of its inhabitants were Italians.

**PREDATOR-PREYDYNAMICS: LOTKA-VOLTERRA**

**Introduction:** The Lotka-Volterra model is composed of a pairof differential equations that describe predator-prey (or herbivore-plant,or parasitoid-host) dynamics in their simplest case (one predator population,one prey population). It was developed independently by Alfred Lotka andVito Volterra in the 1920's, and is characterized by oscillations in thepopulation size of both predator and prey, with the peak of the predator'soscillation lagging slightly behind the peak of the prey's oscillation.The model makes several simplifying assumptions: 1) the prey populationwill grow exponentially when the predator is absent; 2) the predator populationwill starve in the absence of the prey population (as opposed to switchingto another type of prey); 3) predators can consume infinite quantitiesof prey; and 4) there is no environmental complexity (in other words, bothpopulations are moving randomly through a homogeneous environment).

**Importance:** Predators and prey can influenceone another's evolution. Traits that enhance a predator's ability to findand capture prey will be selected for in the predator, while traits thatenhance the prey's ability to avoid being eaten will be selected for inthe prey. The 'goals' of these traits are not compatible, and it is theinteraction of these selective pressures that influences the dynamics ofthe predator and prey populations. Predicting the outcome of species interactionsis also of interest to biologists trying to understand how communitiesare structured and sustained.

### Lotka Volterra Example

**Question:** What are the predictions of theLotka-Volterra model? Are they supported by empirical evidence?

**Variables:**

P | number of predators or consumers |

N | number of prey or biomassof plants |

t | time |

r | growth rate of prey |

a' | searching efficiency/attackrate |

q | predator or consumer mortalityrate |

c | predatorís or consumerísefficiency at turning food into offspring (conversion efficiency) |

**Methods: **We begin by looking at whathappens to the predator population in the absence of prey; without foodresources, their numbers are expected to decline exponentially, as describedby the following equation:

.

This equation uses the product of the number of predators(*P*) and the predator mortality rate (

*q*) to describe the rateof decrease (because of the minus sign on the right-hand side of the equation)of the predator population (

*P*) with respect to time (

*t*).In the presence of prey, however, this decline is opposed by the predatorbirth rate,

*caíPN*, which is determined by the consumption rate (

*aíPN*,which is the attack rate[

*a'*] multiplied by the product of the numberof predators [

*P*] times the number of prey [

*N*]) and by thepredatorís ability to turn food into offspring (

*c*). As predatorand prey numbers (

*P*and

*N*, respectively) increase, theirencounters become more frequent, but the actual rate of consumption willdepend on the attack rate (

*aí*). The equation describing the predatorpopulation dynamics becomes

### Lotka Volterra Competition Model Example

.

*ca'P*is the predator's numericalresponse, or the per capita increase as a function of prey abundance. Theentire term,

*ca'PN*, tells us that increases in the predator populationare proportional to the product of predator and prey abundance.

Turning to the prey population, we would expect thatwithout predation, the numbers of prey would increase exponentially. Thefollowing equation describes the rate of increase of the prey populationwith respect to time, where *r* is the growth rate of the prey population,and *N* is the abundance of the prey population:

.

In the presence of predators, however, the prey populationis prevented from increasing exponentially. The term for consumption ratefrom above (*aíPN*) describes prey mortality, and the population dynamicsof the prey can be described by the equation

.

*a'*and

*P*is the predator'sfunctional response, or rate of prey capture as a function of prey abundance(see TYPE I or TYPEII functional response modules). Here the term

*a'PN*reflectsthe fact that losses from the prey population due to predation are proportionalto the product of predator and prey abundances.

Equations (2) and (4) describe predator and preypopulation dynamics in the presence of one another, and together make upthe Lotka-Volterra predator-prey model. The model predicts a cyclical relationshipbetween predator and prey numbers: as the number of predators (*P*)increases so does the consumption rate (*a'PN*), tending to reinforcethe increase in *P*. Increase in consumption rate, however, has anobvious consequence-- a decrease in the number of prey (*N*), whichin turn causes *P* (and therefore *a'PN*) to decrease. As *a'PN*decreases the prey population is able to recover, and *N* increases.Now *P* can increase, and the cycle begins again. This graph showsthe cyclical relationship predicted by the model for hypothetical predatorand prey populations.

Huffaker (1958) reared two species of mites to demonstratethese coupled oscillations of predator and prey densities in the laboratory.Using *Typhlodromus occidentalis* as the predator and the six-spottedmite (*Eotetranychus sexmaculatus*) as the prey, Huffaker constructedenvironments composed of varying numbers of oranges (fed on by the prey)and rubber balls on trays. The oranges were partially covered with waxto control the amount of feeding area available to *E. sexmaculatus*,and dispersed among the rubber balls. The results of one of the many permutationsof his experiments are graphed below. Note that the prey population sizeis on the left vertical axis and the predator population is on the rightvertical axis, and that the scales of the two are different (after Huffaker,1958 [fig.18]).

**Interpretation: **It is apparent from the graphthat both populations showed cyclical behavior, and that the predator populationgenerally tracked the peaks in the prey population. However, there is someinformation about this experiment that we need to consider before concludingthat the experimental results truly support the predictions made by theLotka-Volterra model. To achieve the results graphed here, Huffaker addedconsiderable complexity to the environment. Food resources for *E. sexmaculatus*(the oranges), were spread further apart than in previous experiments,which meant that food resources for *T. occidentalis* (i.e., *E.sexmaculatus*) were also spread further apart. Additionally, the orangeswere partially isolated with vaseline barriers, but the prey's abilityto disperse was facilitated by the presence of upright sticks from whichthey could ride air currents to other parts of the environment. Slots gokkasten. In otherwords, predator and prey were not encountering one another randomly inthe environment (see assumption 4 from the Introduction).

**Conclusions:** A good model must be simple enoughto be mathematically tractable, but complex enough to represent a systemrealistically. Realism is often sacrificed for simplicity, and one of theshortcomings of the Lotka-Volterra model is its reliance on unrealisticassumptions. For example, prey populations are limited by food resourcesand not just by predation, and no predator can consume infinite quantitiesof prey. Many other examples of cyclical relationships between predatorand prey populations have been demonstrated in the laboratory or observedin nature, but in general these are better fit by models incorporatingterms that represent carrying capacity (the maximum population size thata given environment can support) for the prey population, realistic functionalresponses (how a predator's consumption rate changes as prey densitieschange) for the predator population, and complexity in the environment.

**Additional Question:**

1) How would increases or decreases in any of theparameters *r*, *q*, *a*', or *c* affect the rate ofchange of either the predator or prey populations? How would the shapeof the graph change?

### Lotka Volterra R Code

**Sources:** Begon, M., J. L. Harper, and C. R.Townsend. 1996. __Ecology: Individuals, Populations, and Communities__,3rd edition. Blackwell Science Ltd. Cambridge, MA.

### Lotka Volterra Variables

Gotelli, N. J. 1998. __A Primer of Ecology__,2nd edition. Sinauer Associates, Inc. Sunderland, MA.

Huffaker, C. B. 1958. Experimental studies on predation:dispersion factors and predator-prey oscillations. Hilgardia **27**(14):343-383.

*copyright 1999, M. Beals, L. Gross, and S. Harrell*