1. Lotka Volterra Competition Model Example
2. Lotka Volterra Model Explained
3. Lotka Volterra R Code
4. Lotka Volterra In R
5. Lotka Volterra Graphs

3New Lotka–Volterra function Fig. 4Source on save command Sourcing can be thought of as a way of making the R environ - ment reevaluate the file/script in question. By sourcing the file/ script, you are telling the R environment to execute the file/script, which can either result in running a program or in this case, updat - ing a function. Aim: I am trying to numerically solve a Lotka-Volterra ODE in R, using de sde.sim function in the sde package. I would like to use the sde.sim function in order to eventually transform this system into an SDE. So initially, I started with an simple ODE system (Lotka Volterra model) without a noise term. The Lotka-Volterra ODE system.

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.

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

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:

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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:

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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'PNdecreases 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. 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?

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

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

Introduction: Interspecific competition refers to the competitionbetween two or more species for some limiting resource. This limiting resourcecan be food or nutrients, space, mates, nesting sites-- anything for whichdemand is greater than supply. When one species is a better competitor,interspecific competition negatively influences the other species by reducingpopulation sizes and/or growth rates, which in turn affects the populationdynamics of the competitor. The Lotka-Volterra model of interspecific competitionis a simple mathematical model that can be used to understand how differentfactors affect the outcomes of competitive interactions.

Importance: Competitive interactions between organisms can havea great deal of influence on species evolution, the structuring of communities(which species coexist, which don't, relative abundances, etc.), and thedistributions of species (where they occur). Modeling these interactionsprovides a useful framework for predicting outcomes.

Question: Under what circumstances can two species coexist? Underwhat circumstances does one species outcompete another?

Methods: The logistic equation below modelsa rate of population increase that is limited by intraspecific competition(i.e., members of the same species competing with one another).

The first term on the right side of the equation(rN, the intrinsic rate of increase [r] times the populationsize [N]) describes a population's growth in the absence of competition.The second term ([K-N] / K) incorporates intraspecificcompetition, or density-dependence, into the model, and takes a value between0 and 1. As population size (N) approaches carrying capacity (K),the numerator (K-N) becomes smaller but the denominator (K)stays the same and the second term decreases. The addition of this termdescribes a rate of population growth that slows down as population sizeincreases, until the population reaches its carrying capacity. In otherwords, the growth curve described by the logistic equation is sigmoidal,and the rate of growth depends on the density of the population.

The logistic equation can be modified to includethe effects of interspecific competition as well as intraspecificcompetition. The Lotka-Volterra model of interspecific competition is comprisedof the following equations for population 1 and population 2, respectively:

Lotka Volterra Competition Model Example

The big difference (other than the subscripts denotingpopulations 1 and 2) is the addition of a term involving the competitioncoefficient, a. Thecompetition coefficient represents the effect that one species has on theother: a₁₂representsthe effect of species 2 on species 1, and a₂₁representsthe effect of species 1 on species 2 (the first number of the subscriptalways refers to the species being affected). In the first equation ofthe Lotka-Volterra model of interspecific competition, the effect thatspecies 2 has on species 1 (a₁₂)is multiplied by the population size of species 2 (N₂).When a₁₂is< 1 the effect of species 2 on species 1 is less than the effect ofspecies 1 on its own members. Conversely, when a₁₂is> 1 the effect of species 2 on species 1 is greater than the effect ofspecies 1 on its own members. The product of the competition coefficient,a₁₂,and the population size of species 2, N₂, therefore representsthe effect of an equivalent number of individuals of species 1, and isincluded in the intraspecific competition, or density-dependence, term.The a₂₁N₁termin the second equation is interpreted in the same way.

To understand the predictions of the model it is helpful to look atgraphs that show how the size of each population increases or decreaseswhen we start with different combinations of species abundances (i.e.,low N1 low N2, high N1 low N2, etc.). Thesegraphs are called state-space graphs, in which the abundance of species1 is plotted on the x-axis and the abundance of species 2 is plottedon the y-axis. Each point in a state-space graph represents a combinationof abundances of the two species. For each species there is a straightline on the graph called a zero isocline. Any given point along, for example,species 1's zero isocline represents a combination of abundances of thetwo species where the species 1 population does not increase or decrease(the zero isocline for a species is calculated by setting dN/dt,the growth rate, equal to zero and solving for N). The two graphsbelow show the zero isoclines for species 1 (left, solid yellow line) andspecies 2 (right, dashed pink line). (All graphs adapted from Begon etal. [1996] and Gotelli [1998])

Note that the zero isoclines divide each graph into two parts. Belowand to the left of the isocline the population size increases because thecombined abundances of both species are less than the carrying capacityof the one, while above and to the right the population size decreasesbecause the combined abundances are greater than the carrying capacity.For the graph of the isocline of species 1, the isocline intersects thegraph on the x-axis when N1 reaches its carrying capacity(K₁) and no individuals of species 2 are present. Theisocline intersects the graph on the y-axis at K₁/a₁₂,when the carrying capacity of species 1 is filled by the equivalent numberof individuals of species 2 and no individuals of species 1 are present.The intersections of the isocline for species 2 are essentially the same,but on different axes.

These two graphs illustrate what happens to a population when it isbelow or above its isocline, but they only account for one isocline ata time. The following four graphs include both species' isoclines, andillustrate the possible outcomes of interspecific competition dependingon where each species' isocline lies in relation to the other. In eachgraph, the solid yellow line represents the isocline of species 1, andthe dashed pink line represents the isocline of species 2. The thick blackarrows represent the joint trajectory of the two populations, and the thinnercolored arrows indicate the trajectories of the individual populations.

Interpretation: The first scenario is one in which the isoclinefor species 1 is above and to the right of the isocline for species two.For any point in the lower left corner of the graph (i.e., any combinationof species abundances), both populations are below their respective isoclinesand both increase. For any point in the upper right corner of the graph,both species are above their respective isoclines and both decrease. Forany point in between the two isoclines, species 1 is still below its isoclineand increases, while species 2 is above its isocline and decreases. Thejoint movement of the two populations (thick black arrows) is down andto the right, so species 2 is driven to extinction and species 1 increasesuntil it reaches carrying capacity (K₁). The open circleat this point represents a stable equilibrium. In this scenario, species1 always outcompetes species 2, and is referred to as the competitive exclusionof species 2 by species 1.

The second scenario is the opposite of the first; the isocline of species2 is above and to the right of the isocline for species 1. This graph canbe interpreted in much the same way as the previous one, except that thejoint trajectory of the two populations when starting in between the isoclinesis up and to the left. In this case species 2 always outcompetes species1, and species 1 is competitively excluded by species 2.

In the third scenario, the isoclines of the two species cross one another.Here, the carrying capacity of species 1 (K₁) is higherthan the carrying capacity of species 2 divided by the competition coefficient(K₂/a₂₁),and the carrying capacity of species 2 (K₂) is higherthan the carrying capacity of species 1 divided by the competition coefficient(K₁/a₁₂).Below both isoclines and above both isoclines the populations increaseor decrease as in the first two scenarios, and there is an unstable equilibriumpoint (closed circle) where the isoclines intersect. For points above thedashed pink line (species 2 isocline) and below the solid yellow line (species1 isocline), the outcome is the same as in the first scenario: competitiveexclusion of species 2 by species 1. On the other hand, for points abovethe solid yellow line (species 1 isocline) and below the dashed pink line(species 2 isocline), the outcome is the same as in the second scenario:competitive exclusion of species 1 by species 2. The two stable equilibriumpoints are again represented by open circles. In this scenario, the outcomedepends on the initial abundances of the two species.

Finally, in the fourth scenario we can see that the isoclines crossone another, but in this case both species' carrying capacities are lowerthan the other's carrying capacity divided by the competition coefficient.Again, below both isoclines the populations increase and above both isoclinesthe populations decrease. In this case, however, when the populations ofthe two species are between the isoclines their joint trajectories alwayshead toward the intersection of the isoclines. Rather than outcompetingone another, the two species are able to coexist at this stable equilibriumpoint (open circle). This is the outcome regardless of the initial abundances.

Conclusions: The Lotka-Volterra model of interspecific competitionhas been a useful starting point for biologists thinking about the outcomesof competitive interactions between species. The assumptions of the model(e.g., there can be no migration and the carrying capacities and competitioncoefficients for both species are constants) may not be very realistic,but are necessary simplifications. A variety of factors not included inthe model can affect the outcome of competitive interactions by affectingthe dynamics of one or both populations. Environmental change, disease,and chance are just a few of these factors.

Lotka Volterra Model Explained

Additional Question:

1. The Lotka-Volterra model predicts that stable coexistence of twospecies is possible only when intraspecific competition has a greatereffect than interspecific competition. Why would this be the case?

Lotka Volterra R Code

Sources: Begon, M., J. L. Harper, and C. R. Townsend. 1996.Ecology:Individuals, Populations, and Communities, 3rd edition. Blackwell ScienceLtd. Cambridge, MA.

Lotka Volterra In R

Gotelli, N. J. 1998. A Primer of Ecology, 2nd edition. SinauerAssociates, Inc. Sunderland, MA.

Lotka Volterra Graphs

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