inter specific competition

8/4/2019 Inter Specific Competition

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INTERSPECIFIC COMPETITION: LOTKA-VOLTERRA

Introduction: Interspecific competition refers to the competition between two or more species for somelimiting resource. This limiting resource can be food or nutrients, space, mates, nesting sites-- anything for which demand is greater than supply. When one species is a better competitor, interspecific competitionnegatively influences the other species by reducing population sizes and/or growth rates, which in turn affects

the population dynamics of the competitor. The Lotka-Volterra model of interspecific competition is a simplemathematical model that can be used to understand how different factors affect the outcomes of competitiveinteractions.

Importance: Competitive interactions between organisms can have a great deal of influence on speciesevolution, the structuring of communities (which species coexist, which don't, relative abundances, etc.), andthe distributions of species (where they occur). Modeling these interactions provides a useful framework for

predicting outcomes.

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

Variables:

N population size

t time

K carrying capacity

r intrinsic rate of increase

E competition coefficient

Methods: The logistic equation below models a 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 population size[ N ]) describes a population's growth in the absence of competition. The second term ([ K - N ] / K ) incorporatesintraspecific competition, or density-dependence, into the model, and takes a value between 0 and 1. As

population size ( N ) approaches carrying capacity ( K ), the numerator ( K - N ) becomes smaller but thedenominator ( K ) stays the same and the second term decreases. The addition of this term describes a rate of

population growth that slows down as population size increases, until the population reaches its carryingcapacity. In other words, 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 include the effects of interspecific competition as well as intraspecificcompetition. The Lotka-Volterra model of interspecific competition is comprised of the following equationsfor population 1 and population 2, respectively:


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The big difference (other than the subscripts denoting populations 1 and 2) is the addition of a term involvingthe competition coefficient, E . The competition coefficient represents the effect that one species has on theother: E 12represents the effect of species 2 on species 1, and E 21represents the effect of species 1 on species 2(the first number of the subscript always refers to the species being affected). In the first equation of the Lotka-Volterra model of interspecific competition, the effect that species 2 has on species 1 ( E 12) is multiplied by the

population size of species 2 ( N 2). When E 12 is < 1 the effect of species 2 on species 1 is less than the effect of species 1 on its own members. Conversely, when E 12is > 1 the effect of species 2 on species 1 is greater thanthe effect of species 1 on its own members. The product of the competition coefficient, E 12, and the populationsize of species 2, N 2, therefore represents the effect of an equivalent number of individuals of species 1, and isincluded in the intraspecific competition, or density-dependence, term. The E 21 N 1 term in the second equationis interpreted in the same way.

To understand the predictions of the model it is helpful to look at graphs that show how the size of each population increases or decreases when we start with different combinations of species abundances (i.e.,low N 1 low N 2, high N 1 low N 2, etc.). These graphs are called state-space graphs, in which the abundance of species 1 is plotted on the x-axis and the abundance of species 2 is plotted on the y-axis. Each point in a state-space graph represents a combination of abundances of the two species. For each species there is a straight lineon the graph called a zero isocline. Any given point along, for example, species 1's zero isocline represents acombination of abundances of the two 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 graphs below show the zero isoclines for species 1 (left, solid yellow line) and species 2 (right, dashed

pink line). (All graphs adapted from Begon et al. [1996] and Gotelli [1998])


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Note that the zero isoclines divide each graph into two parts. Below and to the left of the isocline the population size increases because the combined abundances of both species are less than the carrying capacityof the one, while above and to the right the population size decreases because the combined abundances aregreater than the carrying capacity. For the graph of the isocline of species 1, the isocline intersects the graph onthe x-axis when N 1 reaches its carrying capacity ( K 1) and no individuals of species 2 are present. The isoclineintersects the graph on the y-axis at K 1/E 12, when the carrying capacity of species 1 is filled by the equivalent


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number of individuals of species 2 and no individuals of species 1 are present. The intersections of the isoclinefor species 2 are essentially the same, but on different axes.

These two graphs illustrate what happens to a population when it is below or above its isocline, but they onlyaccount for one isocline at a time. The following four graphs include both species' isoclines, and illustrate the

possible outcomes of interspecific competition depending on where each species' isocline lies in relation to the

other. In each graph, the solid yellow line represents the isocline of species 1, and the dashed pink linerepresents the isocline of species 2. The thick black arrows represent the joint trajectory of the two populations,and the thinner colored arrows indicate the trajectories of the individual populations.


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Interpretation: The first scenario is one in which the isocline for species 1 is above and to the right of theisocline for species two. For any point in the lower left corner of the graph (i.e., any combination of speciesabundances), both populations are below their respective isoclines and both increase. For any point in theupper right corner of the graph, both species are above their respective isoclines and both decrease. For any

point in between the two isoclines, species 1 is still below its isocline and increases, while species 2 is aboveits isocline and decreases. The joint movement of the two populations (thick black arrows) is down and to theright, so species 2 is driven to extinction and species 1 increases until it reaches carrying capacity ( K 1). Theopen circle at this point represents a stable equilibrium. In this scenario, species 1 always outcompetes species

2, and is referred to as the competitive exclusion of species 2 by species 1.

The second scenario is the opposite of the first; the isocline of species 2 is above and to the right of the isoclinefor species 1. This graph can be interpreted in much the same way as the previous one, except that the jointtrajectory of the two populations when starting in between the isoclines is up and to the left. In this casespecies 2 always outcompetes species 1, and species 1 is competitively excluded by species 2.


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In the third scenario, the isoclines of the two species cross one another. Here, the carrying capacity of species 1( K 1) is higher than the carrying capacity of species 2 divided by the competition coefficient ( K 2/E 21), and thecarrying capacity of species 2 ( K 2) is higher than the carrying capacity of species 1 divided by the competitioncoefficient ( K 1/E 12). Below both isoclines and above both isoclines the populations increase or decrease as inthe first two scenarios, and there is an unstable equilibrium point (closed circle) where the isoclines intersect.For points above the dashed pink line (species 2 isocline) and below the solid yellow line (species 1 isocline),


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the outcome is the same as in the first scenario: competitive exclusion of species 2 by species 1. On the other hand, for points above the solid yellow line (species 1 isocline) and below the dashed pink line (species 2isocline), the outcome is the same as in the second scenario: competitive exclusion of species 1 by species 2.The two stable equilibrium points are again represented by open circles. In this scenario, the outcome dependson the initial abundances of the two species.

Finally, in the fourth scenario we can see that the isoclines cross one another, but in this case both species'carrying capacities are lower than the other's carrying capacity divided by the competition coefficient. Again,

below both isoclines the populations increase and above both isoclines the populations decrease. In this case,however, when the populations of the two species are between the isoclines their joint trajectories always headtoward the intersection of the isoclines. Rather than outcompeting one another, the two species are able tocoexist at this stable equilibrium point (open circle). This is the outcome regardless of the initial abundances.

Conclusions: The Lotka-Volterra model of interspecific competition has been a useful starting point for biologists thinking about the outcomes of competitive interactions between species. The assumptions of themodel (e.g., there can be no migration and the carrying capacities and competition coefficients for both speciesare 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 affecting the dynamics of one or both

populations. Environmental change, disease, and chance are just a few of these factors.

Additional Question:

1. The Lotka-Volterra model predicts that stable coexistence of two species is possible onlywhen intraspecific competition has a greater effect than interspecific competition. Why would this be the case?

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

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

The Lotka-Volterra Model of Interspecific Competition:

The model is based on logistic growth. Remember that logistic growth models intraspecific (within onespecies) competition. The Lotka-Volterra model of interspecific (between species) competition includes theeffects of intraspecific competition, but adds the competitive effect of another species.

The model is based on two equations of population growth; one for each of two competing species. Thespecies are referred to (unimaginatively enough) as species 1 and species 2; subscripts 1 and 2 are used on thesymbols to indicate the species to which they refer. The equations follow; the symbols are defined below theequations.

Equation for population growth of species 1:

Equation for population growth of species 2:


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The symbols are largely the same ones that were used in logistic growth; note that the r values refer to rmvalues for species 1 and 2. The main symbol that is added here is alpha ( E ).

y E 12 refers to the impact on population growth of species 1 of an individual of species 2, relative to theimpact of an individual of species 1.

y E refers to the impact on population growth of species 2 of an individual of species 1, relative to theimpact of an individual of species 2.

Thus if the negative impact of an individual of one species on the other is the same for the different species,alpha would be equal to 1. If alpha is greater than 1, it means that the impact of an individual of the other species is greater than an individual of one's own species; if alpha is less than one it means that the impact of an individual of the other species is less than the impact of one's own species.

These equations can be used to predict the outcome of competition over time. To do this, we

determine equilibria: population sizes for species 1 and 2 for which population growth of both species will bezero. If population growth is zero, then the population sizes do not change over time, and we have anequilibrium (a situation in which conditions remain the same over time.)

To find the equilibria, we will first determine combinations of population sizes for species 1 and species 2 for which species 1 population growth is zero. Then we will determine combinations of population sizes for species 1 and species 2 for which species 2 population growth is zero . Finally, we will put these conditionstogether to find combinations of population sizes of both species for which population growth of both speciesis zero.

Population sizes for which species 1 growth is zero:

To find the population sizes for which species 1 growth is zero, we set the growth equation (given above) for species 1 to equal zero. That is:

0 = r 1 N1(K 1-N 1-E 12 N2)

This equation will equal zero if any of the three multiplied terms is equal to zero. The three multiplied termsare: r 1, N 1, and (K 1-N 1-E 12 N2).

If r 1 were zero, then the MAXIMUM population growth rate for species 1 would be zero -- such a specieswould never grow and would not exist, so this is not ecologically interesting, and we won't worry about thissituation.

If N 1 were zero, it would mean that there are no individuals of species 1 in the population. In other words, the population of species 1 does not exist. That's not very interesting either -- we want to look at situations wherethe species DOES exist to see what happens when it competes with species 2. So we won't worry about thissituation either.

The ecologically interesting situation occurs when:

K 1-N 1-E 12 N2=0


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Solving this, we find:

N1=K 1-E 12 N2

Now, look at the form of this equation. If we were to plot N 1 on the Y axis and N 2 on the X axis of a graph, sothat Y represents N 1 and X represents N 2, then note that this is in the form:

Y=mX + b

You should recognize this form: it is a straight line, with slope m and intercept b. This means if we plot N1 versus N 2, we get a line with Y intercept K 1 and slope of negative alpha 12. We can also determine the Xintercept of this line by setting Y=0 (that is, N1=0):

0=K 1-E 12 N2

N2=K 1/E 12 So when we draw this graph, it looks like this:

The line is called the zero growth isocline for species 1: it represents all combinations of N 1 and N 2for whichgrowth of N 1 is zero.

Suppose we had a combination of population sizes of species 1 and species 2 that falls below this line. Thiswould represent a situation with fewer individuals of the two species than the numbers required to causegrowth to be zero. In this situation, because there are few individuals of the two species, there would be plentyof resources for species 1, and the population of species 1 would increase in size. In contrast, if we have acombination of population sizes of species 1 and species 2 that falls above the zero growth isocline for species1, this means there are more individuals of the two species than what would cause growth to be zero -- in the

presence of so many individuals, resources would be depleted and species 1 population size would decrease.We represent these areas of increase or decrease on the graph by drawing arrows, like this:


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An arrow pointing up represents growth of species 1 since moving a point up on this graph means that species1, plotted on the vertical (Y) axis, gets bigger. An arrow pointing down represents decrease in population sizeof species 1 since moving a point down on this graph means that species 1, plotted on the vertical (Y) axis,gets smaller.

Combinations of population sizes of species 1 and 2 for which growth of species 2 is zero:

Note that the growth equation for species 2 looks just like the growth equation for species 1, but with all the 1'sand 2's reversed. This means if we go through all the steps we did above for species 1, but apply them to theequation for species 2, we're going to get the same result (but with all the 1's and 2's reversed.) This meanswe're going to get a zero growth isocline for species 2 , with intercept on the N2 axis at K2 and the intercepton the N1 axis at K1/alpha21. If you don't believe me, work out the equation following the steps we usedabove (this would be a good way to practice and make sure you understand how this model works!)

If we plot this graph on axes like the ones above where N1 is on the Y axis and N2 is on the X axis, the graphlooks like this:

As we did for species 1, we can determine the regions of the graph for which species 2 increases or decreasesin size. Below the isocline, there are few individuals of species 1 and species 2, and therefore plenty of resources. In this area of the graph, species 2 increases. Above the isocline, there are many individuals of species 1 and 2, and resources are depleted. In this area of the graph, species 2 decreases. We representincrease and decrease in species 2 by arrows pointing to the right (increase) or left (decrease), as shown here:


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Note that an arrow to the right represents increase in species 2 (rather than an arrow pointing up, as we usedfor species 1) because species 2 is plotted on the horizontal (X) axis so moving a point to the right meansspecies 2 gets bigger and moving a point to the left means species 2 gets smaller.

Finding Equilibrium Situations for Species 1 and 2:

We now need to determine whether there are situations for which population growth for both species (1 and 2)is zero -- these will be situations for which neither population changes, so they will represent equilibria. To dothis, we are going to plot both isoclines on the same graph. It turns out that there are four ways to plot thesetwo lines relative to each other. These are shown here:


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We'll consider each case.

Case 1: The species 1 isocline is above the species 2 isocline.

Consider what happens in each section of the graph. Below both isoclines, species 1 and 2 both increase. Inthe range of the graph between the two isoclines, we are above the species 2 isocline so it decreases, but belowthe species 1 isocline so it continues to increase. The following graph shows these changes in species 1 andspecies 2 with arrows (vertical arrows for species 1 since it is on the vertical axis and horixontal arrows for species 2 since it is on the horizontal axis.)


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The result is that species 2 declines to zero and species 1 increases to its carrying capacity. In this case species1 has competitively excluded species 2.

We can look at the K and alpha values to understand this in terms of the strengths of interspecific competitionand intraspecific competition. Note, on the N 1 axis, that the K 1 term falls above the K 2/alpha 21term. Thus:

K 1>K 2/E 21

so K 1E 21>K 2

This means that when species 1 is at its carring capacity, its impact on species 2 (measured by K 1 times a 21) isgreater than the impact of K 2 individuals of species 2. Thus, species 1 is affecting species 2 more negativelythan species 2 affects itself. Interspecific competition regulates species 2 more than species 2 is regulated byintraspecific competition.

From the N 2 axis, note that K 2 falls below the K 1/alpha1 12 term. Thus:

K 2


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The result is that species 1 declines to zero and species 2 increases to its carrying capacity. In this case species2 has competitively excluded species 1.

As we did above, we will look at the K and alpha values to understand this in terms of the strengths of interspecific competition and intraspecific competition. From the N 1 axis, we see that:

K 2/E 21>K 1

so K 2 > K 1E 21

So, using the same reasoning as above, intraspecific competiton impacts species 1 more than does interspecificcompetiton with species 1.

And from the N2 axis, we see that:

K 1/E 12 < K 2

so K 1 < K 2E 12

indicating that species 1 is more impacted by interspecific competition with species 1 than it is by intraspecificcompetition.

From cases 1 and 2, we see that that competitive exclusion is occurring if one species is more regulated byintraspecific competition, the other more by interspecific competition.

Case 3: Isoclines for the two species cross; the K values on each axis are lower than the K/ E values

This time, there are four sections of the graph to consider. Below both isoclines, both species increase. Abovethe isoclines, both decrease. In the triangular sections of the graph combinations of species are above theisocline for one species but not the other. In the upper left, the combination is above the species 1 isocline sospecies 1 declines, but above the species 2 isocline so species 2 increases, and the populations move toward the

point where the lines cross. In the lower right, the combination is above the species 2 isocline so species 2declines, but below the species 1 isocline so species 1 increases. The populations move toward the pointwhere the lines cross.


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In this case, there is a stable equilibrium where both species coexist.

Examining K and K/alpha values from the axes, we find from the N1 axis that:

K 2/E 21>K 1

soK 2>K 1E 21

Indicating that species 2 is regulated more by intraspecific competition than by interspecific competition

And from the N2 axis we see that:

K 1/E 12>K 2

So

K 1 > K 2E 12

Indicating that species 1 is regulated more by intraspecific competition than by interspecific competition.

So we can see that when each species is regulated more by intraspecific competition than by competition withthe other species, the two species coexist.

Case 4: Isoclines for the two species cross; the K values on each axis are higher than the K/ E values

As with case 3, we need to consider all four sections of the graph. When both populations are small, bothincrease. When both are large, both decrease. In the upper left, species 1 is below its isocline and species 2 isabove its isocline, so species 1 increases and species 2 decreases; this leads to competitive exclusion of species2 by species 1. In the lower right, species 2 is below its isocline and species 1 is above its isocline, so species2 increases and species 2 decreases; this leads to competitive exclusion of species 1 by species 2. The pointwhere the lines cross is not a stable equilibrium because populations tend to move away from it, so it will not

be observed in natural populations.


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This situation is unlike the others in that there are two possible outcomes. If the populations grow into theupper left triangle, species 1 "wins." If the populations grow into the lower right triangle, species 2 "wins."

Situations in which a population of a species is large give that species the edge -- make it more likely to "win."This could occur if one species established a population in an area before the other species, or if one specieshad a higher per capita growth rate than the other.

Once again we can examine the K/alpha and K terms to understand this situation in terms of inter andintraspecific competition. In this situation, from the N 1 axis:

K 1>K 2/E 21

so

K 1E 21>K 2

Indicating higher impact of interspecific competition than intraspecific competition on species 2

From the N 2 axis:

K 2>K 1/E 12

so K 2E 12>K 1

Indicating higher impact of interspecific competition than intraspecific competition on species 1.

Thus, the situation in which either species could exclude the other occurs when, for each species, interspecificcompetition is stronger than is intraspecific competition. This situation is termed mutual antagonism .

The LotkaVolterra equations , also known as the predatorprey equations , are a pair of first-order, non-linear , differential equations frequently used

to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They evolve in time according to the pair

of equations:


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where,

y is the number of some predator (for example, wolves );

x is the number of its prey (for example, rabbits );

and represent the growth of the two populations against time;

t represents the time; and

, , and are parameters representing the interaction of the two species .

The LotkaVolterra system of equations is an example of a Kolmogorov model, [1][2][3] which is a more general framework that can model

the dynamics of ecological systems with predator-prey interactions, competition, disease, and mutualism.

History

The LotkaVolterra predatorprey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. [4][5] This

was effectively the logistic equation ,[6] which was originally derived by Pierre Franois Verhulst .[7] In 1920 Lotka extended, via Kolmogorov (see above),

the model to "organic systems" using a plant species and a herbivorous animal species as an example [8] and in 1925 he utilised t he equations to

analyse predator-prey interactions in his book on biomathematics [9] arriving at the equations that we know today. Vito Volterra , who made a statistical

analysis of fish catches in the Adriatic [5] independently investigated the equations in 1926. [10][11]

C.S. Holling extended this model yet again, in two 1959 papers, in which he proposed the idea of functional response .[12][13] Both the Lotka-Volterra

model and Holling's extensions have been used to model the moose and wolf populations in Isle Royale National Park , [14] which with over 50 published

papers is one of the best studied predator-prey relationships.

[edit ]In economics

The LotkaVolterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in

1965 [15] or 1967. [16][17] In economics, links are between many if not all industries; a proposed way to model the dynamics of various industries has been

by introducing trophic functions between various sectors, [18] and ignoring smaller sectors by considering the interactions of only two industrial

sectors. [19]

[edit ]Physical meanings of the equations

The Lotka-Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:

1. The prey population finds ample food at all times.

2. The food supply of the predator population depends entirely on the prey populations.

3. The rate of change of population is proportional to its size.


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4. During the process, the environment does not change in favour of one species and the genetic adaptation is sufficiently slow.

As differential equations are used, the solution is deterministic and continuous . This, in turn, implies that the generations of both the predator and prey

are continually overlapping. [20]

[edit ]P rey

When multiplied out, the prey equation becomes:

The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this exponential growth is

represented in the equation above by the term x . The rate of predation upon the prey is assumed to be proportional to the rate at which the

predators and the prey meet; this is represented above by x y . If either x or y is zero then there can be no predation.

With these two terms the equation above can be interpreted as: the change in t he prey's numbers is given by its own growth minus the rate at

which it is preyed upon.

[edit ]P redators

The predator equation becomes:

In this equation, x y represents the growth of the predator population. (Note the simil arity 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). y represents the loss rate of the predators due to either natural death or emigration; it l eads to an exponential decay in the

absence of prey.

Hence the equation expresses the change in the predator population as growth fueled by the food supply, minus natural death.

[edit ]Solutions to the equations

The equations have periodic solutions which do not have a simple expression in terms of the usual trigonometric functions . However,

a linearization of the equations yields a solution similar to simple harmonic motion [21] with the population of predators following that of

prey by 90.


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[edit ]A n example problem

Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40

cheetahs, one can plot the progression of the two species over time. The choice of time interval is arbitrary.

One can also plot a solution which corresponds to the oscillatory nature of the population of the two species. This solution is in a state

of dynamic equilibrium. At any given time in this phase plane , the system is in a limit cycle and lies somewhere on the inside of these

elliptical solutions. There is no particular r equirement on the system to begin within a limit cycle and thus in a stable solution, however,

it will always reach one eventually.


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These graphs clearly illustrate a serious problem with this as a biological model: in each c ycle, the baboon population is reduced to

extremely low numbers yet recovers (while t he cheetah population remains sizeable at the lowest baboon density). Given chance

fluctuations, discrete numbers of individuals, and the family structure and lifecycle of baboons, the baboons actually go extinct and by

consequence the cheetahs as well. This modelling problem has been called the "atto-fox problem", [22] an atto- fox being an imaginary

10 18 of a fox, in relation to rabies modelling in the UK.

[edit ]Dynamics of the system

In the model system, the predators thrive when there are plentiful prey 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 cycle of growth and decline.

[edit ]P opulation equilibrium

Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the d erivatives are equal

to 0.

When solved for x and y the above system of equations yields


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and

Hence, there are two equilibria.

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 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, , , , and .

[edit ]S tability of the fixed points

The stability of the fixed point at the origin can be determined by performing a linearization using partial

derivatives , while the other fixed point requires a slightly more sophisticated method.

The Jacobian matrix of the predator-prey model is

[edit ]First fixed point

When evaluated at the steady state of (0, 0) the Jacobian matrix J becomes

The eigenvalues of this matrix are

In the model and are always greater than zero, and as such the s ign of the

eigenvalues above will always differ. Hence the fixed point at the origin is

a saddle point .

The stability of this fixed point is of importance. If it were s table, non-zero

populations might be attracted towards it, and as such the dynamics of t he

system might lead towards the extinction of both species for many cases of initial

population levels. However, as the fixed point at the origin is a saddle point, and

hence unstable, we find that the extinction of both species is difficult in the

model. (In fact, this can only occur if t he prey are artificially completely

eradicated, causing the predators to die of starvation. If the predators are

eradicated, the prey population grows without bound in this simple model).


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[edit ]S econd fixed point

Evaluating J at the second fixed point we get

The eigenvalues of this matrix are

As the eigenvalues are both purely imaginary, this fixed point is

not hyperbolic , so no conclusions can be drawn from the linear

analysis. However, the system admits a constant of motion

and the level curves, where K = const, are closed

trajectories surrounding the fixed point. Consequently, the

levels of the predator and prey populations cycle, and

oscillate around this fixed point.

The largest value of the constant K can be obtained by

solving the optimization problem

The maximal value of K is attained at the

stationary point and it is given by

where e is Euler's Number .

1. In general, protozoans are referred to as animal-like protists because of movement (motile).Protozoans are unicellular organisms and are often called the animal-like protists because they subsistentirely on other organisms for food. Most protozoans can move about on their own. Amoebas, Paramecia,and Trypanosomes are all examples of animal-like Protists.

inter specific competition

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