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Predator-Prey Relationships

- By Maria Casillas, Devin Morris, John Paul

Phillips, Elly Sarabi, Nernie Tam

Formulation of the Scientific Problem

- There are many instances in nature where one

species of animal feeds on another species of

animal, which in turn feeds on other things. The

first species is called the predator and the

second is called the prey. - Theoretically, the predator can destroy all the

prey so that the latter become extinct. However,

if this happens the predator will also become

extinct since, as we assume, it depends on the

prey for its existence.

- What actually happens in nature is that a cycle

develops where at some time the prey may be

abundant and the predators few. Because of the

abundance of prey, the predator population grows

and reduces the population of prey. This results

in a reduction of predators and consequent

increase of prey and the cycle continues.

-Predator -Prey

An important problem of ecology , the science

which studies the interrelationships of organisms

and their environment, is to investigate the

question of coexistence of the two species. To

this end, it is natural to seek a mathematical

formulation of this predator-prey problem and to

use it to forecast the behavior of populations of

various species at different times.

Risk and Food Availabilty

- Sharks appear to be a major threat to fish
- Availabilty of prey helps animals decide where to

live

Predator-Prey ModelFish Sharks

- We will create a mathmatical model which

describes the relationship between predator and

prey in the ocean. Where the predators are sharks

and the prey are fish. - In order for this model to work we must first

make a few assumptions.

Assumptions

- 1. Fish only die by being eaten by

Sharks, and of natural causes. - 2. Sharks only die from natural causes.
- 3. The interaction between Sharks and Fish can be

described by a function.

Differential Equations and how it Relates to

Predator-Prey

One of the most interesting applications of

systems of differential equations is the

predator-prey problem. In this project we will

consider an environment containing two related

populations-a prey population, such as fish, and

a predator population, such as sharks. Clearly,

it is reasonable to expect that the two

populations react in such a way as to influence

each others size. The differential equations

are very much helpful in many areas of science.

But most of interesting real life problems

involve more than one unknown function.

Therefore, the use of system of differential

equations is very useful. Without loss of

generality, we will concentrate on systems of two

differential equations.

Vito Volterra

- Born
- Ancona, Papal States (now Italy)
- May 3rd, 1860

- Age 2
- Father passed away, family was poor.
- Age 11
- Began studying Legendres Geometry
- Age 13
- Began studying Three Body Problem
- Progress!

- 1878
- Studied under Betti in Pisa
- 1882
- graduated Doctor of Physics
- thesis was on hydrodynamics

- 1883
- Professor of Mechanics at Pisa
- Chair of Mathematical Physics
- conceived idea of a theory of functions

- 1890
- Extended theory of Hamilton and Jacobi
- 1892-1894
- published papers on partial differential equations

- 1896
- published papers on integral equations of the

Volterra type. - WWI
- Air Force
- Scientific collaboration

- Post War
- University of Rome
- Verhulst equation
- logistic curve
- predator-prey equations!

- 1922
- Italian Parliament
- 1930
- Parliament abolished
- 1931
- forced to leave University of Rome

Predator-Prey Populations

- 1926
- published deduction of the nonlinear differential

equation - similar to Lotkas logistic growth equation
- Lotka-Volterra Equation

Crater Volterra

- Lunar Crater
- Location
- Latitude 56.8 degrees North
- Longitude 132.2 degrees East
- Diameter
- 52.0 kilometers!

- 1938
- Offered degree by University of St Andrews!

- Died
- Rome, Italy
- October 11th, 1940

Alfred James Lotka

- Born
- 1880
- Lviv (Lemberg), Austria (Ukraine)

- Chemist
- Demographer
- Ecologist
- Mathematician

- 1902
- Moved to the United States
- Chemical Oscillations
- 1925
- Wrote Analytical Theory of Biological Populations

- Predator-Prey model
- independent from Volterra
- Analysis of population dynamics
- Metropolitan Life

Population Assoc. of America

- Nonprofit organization
- Scientific organization
- Promoting improvement of human race
- Membership now 3,000!
- Annual meetings

- Famous for Avant La Lettre
- Power Law
- (C/(na))
- Where C is a constant
- If a2, then C(6/(pi2))0.61

Alfred James Lotka

- Died
- 1949
- USA

The Lotka-Volterra Model

- System
- F'(t)aF-bF²-cFS
- S'(t)-kSdSF

- Initial Conditions
- F(0)F0
- S(0)S0

F(t) represents the population of the fish at

time t S(t) represents the population of the

sharks at time t F0 is the initial size of the

fish population S0 is the initial size of the

shark population

Understanding the Model

- F'(t)aF-bF²-cFS
- F(t) the growth rate of the fish population, is

influenced, according to the first differential

equation, by three different terms. - It is positively influenced by the current fish

population size, as shown by the term aF, where a

is a constant, non-negative real number and aF is

the birthrate of the fish. - It is negatively influenced by the natural death

rate of the fish, as shown by the term -bF²,

where b is a constant, non-negative real number

and bF² is the natural death rate of the fish - It is also negatively influenced by the death

rate of the fish due to consumption by sharks as

shown by the term -cFS, where c is a constant

non-negative real number and cFS is the death

rate of the fish due to consumption by sharks.

- S'(t)-kSdSF
- S(t), the growth rate of the Shark population,

is influenced, according to the second

differential equation, by two different terms. - It is negatively influenced by the current shark

population size as shown by the term -kS, where k

is a constant non-negative real number and S is

the shark population. - It is positively influenced by the shark-fish

interactions as shown by the term dSF, where d is

a constant non-negative real number, S is the

shark population and F is the fish population.

Equilibrium Points

- Once the initial equations are understood, the

next step is to find the equilibrium points. - These equilibrium points represent points on the

graph of the function which are significant. - These are shown by the following computations.

- Let X(dF/dt)F(a-bF-cS)
- Let Y (dS/dt)S(-kdF)
- To compute the equilibrium points we solve

(dF/dt)0 and (dS/dt)0 - (dF/dt)0 when F0 or a-bF-cS0
- solution F(a-cS)/b

dS/dt0 when S0 or -kdF0 SolutionF(k/d)

Now we find all the combinations One of our

equilibrium points is (0,0). For F(a-cS)/b

When S0, then F((a-c(0))/b) (a/b) Thus,

one of our equilibrium points is ((a/b),0).

For F((a-cS)/b) and F(k/d)

(k/d)((a-cS)/b), Solution is

S((-kbad)/(dc)) Thus, one of our

equilibrium points is ((k/d),((-kbad)/(dc))).

Our equilibrium points are (0,0), ((a/b),0),

and ((k/d),((-kbad)/(dc))).

Now, to study the stability of the equilibrium

points we first need to find the Jacobian matrix

which is

J(F,S)

To study the stability of (0,0) J(0,0)det

(a- ?)(-k- ?), Solution is ? a,? -k

semi-stable since one eigenvalue is negative and

one is positive.

To study the stability of ((a/b),0)

J((a/b),0)det

(-a-?)(-ka(d/b)-?), Solution is

?-a,?((-kbad)/b)

stable if ? ((-kbad)/b) lt 0 (i.e. ad lt kb)

semi-stable if ? ((-kbad)/b) gt 0 (i.e. ad gt

kb)

To study the stability of ((k/d),((ad-kb)/(cd)))

J((k/d),((ad-kb)/(cd)))det

det

((? kb ? ²d-k²bkad)/d)

Solution is ? (1/(2d))(-kb(k²b²4dk²b-4kad²)1

/2) ? (1/(2d))(-kb-(k²b²4d

k²b-4kad²)1/2) If we simplify a little more,

we get ? (1/(2d))(-kb-(k²b²4dk²b-4kad²)1/2)

-(1/2)((kbi(k)1/2(-kb²-4dkb4ad²)1/2)/d)

? (1/(2d))(-kb(k²b²4dk²b-4kad²)1/2)

-(1/2)((kb-i(k)1/2(-kb²-4dkb4ad²)1/2)/d)

Stable since both of the real parts are

negative. The imaginary numbers tells us that

it will be periodic.

Case 1 (a ? gtbk) u(x,y)x(6-2x-4y)

v(x,y)y(-35x)

x(0)1 y(0).5

sharks

fish

x(0)2 y(0)3

sharks

fish

x(0).5 y(0)1.5

sharks

fish

x(0).5 y(0).5

sharks

fish

Case 2 (a ? ltbk) u(x,y)x(2-6x-4y)

v(x,y)y(-35x) x(0)1 y(0).5

sharks

fish

x(0)2 y(0)3

sharks

fish

x(0)0.5 x(0)0.5

sharks

fish

x(0).5 y(0).5

sharks

fish

X(0).1 y(0).1

sharks

fish

Case 3 All constants are equal

u(x,y)x(1-1x-1y) v(x,y)y(-11x)

sharks

fish

X(0)2 y(0)3

sharks

fish

X(0).5 y(0)1.5

sharks

fish

X(0).5 y(0).5

sharks

fish

Case 4 (b0) u(x,y)x(2-0x-1y)

v(x,y)y(-11x)

x(0)1 y(0)0.5

sharks

fish

x(0)2 y(0)3

sharks

fish

x(0)0.5 y(0)1.5

sharks

fish

x(0)0.5 y(0)0.5

sharks

fish

Case 5 ((k/ ?)((a ? -bk)/(c ?)))

u(x,y)x(2-1x-1y) v(x,y)y(-11x)

x(0)1 y(0)0.5

sharks

fish

x(0)2 y(0)3

sharks

fish

x(0)0.5 y(0)1.5

sharks

fish

x(0)0.5 y(0)0.5

sharks

fish

With the eigenvalues -bki(4akk?-4bk?

-bbkk)1/2 and -bk-i(4akk?-4bk? -bbkk)1/2, we

are able to calculate the period of the

oscillations (2?)/(4akk?-4bk? -bbkk)1/2 This

is the rough length of one oscillating cycle for

this model.

The eigenvalues also allow us to describe the

cyclic variation of this model by using their

properties in developing U(t) and V(t) U(t)

(et(-bk/2?))(kK/?)cos(t ((4akk?-4bk ?

-bbkk)1/2)?) V(t) (et(-bk/2?))(kK/c?)((a?-bk)

1/2)sin(t ((4akk?-4bk ? -bbkk)1/2)?) And by

substituting, we get F(t) k/?(1

(et(-bk/2?))Kcost (4akk?-4bk ?

-bbkk)1/2? S(t) (a?-bk)/c?(1

et(-bk/2?)k/(a?-bk)1/2Ksint

(4akk?-4bk ? -bbkk)1/2?)

From those equations, we are able to get the

amplitudes of the oscillations, which are For

F(t) K(k/?)et(-bk/2?) And for S(t)

K(k/c?)(a?-bk)1/2et(-bk/2?) With K and ?

representing the initial conditions F(0),

S(0) And the average number of F(t) is k/? and

S(t)s average number is (a?-bk)/c? Those

numbers are identical to the coordinates of the

critical point.

Both the exponential and trigometrical aspect of

the solutions of F(t) and S(t) tells us that the

graph of the equations will show an infinite

spiraling pattern towards the critical point for

the first case. The first case has a/b greater

than k/?, where a/b is the stable point for the

fish population in a shark-free world, and k/?,

of course is the critical point for the fish

population living with sharks. This case holds

true regardless of the initial conditions, as

long as F(0)gt0, and S(0)gt0.

For the second case, when a/b is less than k/?,

we arrive at an interesting conclusion, which is

supported by simple algebra. When a/b lt k/?,

then for the shark equation, the critical point

becomes a negative number! a/b lt k/? gt a? lt

bk so S(t) (a?-bk)/c? results in a negative

number. Therefore in the second case, the shark

population will die out REGARDLESS of the initial

conditions! So the solution would converge to the

shark-free stable point.

For the third case, what if all of the constants

were the same? A simple glance at the equations

tells us that this would be similar to the second

case we get a/bk/?1, yet the critical point

would be (1,0) which is on the y-axis (S) and

identical to the stable point for the fish

population in a shark-free world. So here the

sharks die out again. (But its hard to feel

sorry for sharks!)

For the fourth case, we make b0 which turns the

model into the simplest form of the predator/prey

model. The new equations look like this F

F(a-0F-cS) F(a-cS) S S(-k?F) So the

critical point becomes (k/?, a/c) and we get an

ellipse around the critical point, the shape and

size depending on the constants and initial

conditions. So both the fish and shark

populations wax and wane in a cyclic pattern with

the sharks lagging behind the fish.

Now for the fifth case, we pose the question

What happens when F(t) S(t), i.e. k/?

(a?-bk)/c? ? Answer This is pretty much similar

to the first case, since a/b gt k/?, with a

simpler spiral as the result. The only

significant impact is the location of the

critical point.

There are other cases that we have yet to explore

here, such as a0, c0, k0, ?0, or a

combination of those, but those would render the

model meaningless, as they would cancel the

relationship between the fish and the sharks or

eliminate the fishs growth rate or the sharks

death rate.

In Conclusion, This Lotka-Volterra Predator-Prey

Model is a rudimentary model of the complex

ecology of this world. It assumes just one prey

for the predator, and vice versa. It also assumes

no outside influences like disease, changing

conditions, pollution, and so on. However, the

model can be expanded to include other variables,

and we have Lotka-Volterra Competition Model,

which models two competing species and the

resources that they need to survive. We can

polish the equations by adding more variables and

get a better picture of the ecology. But with

more variables, the model becomes more complex

and would require more brains or computer

resources.

This model is an excellent tool to teach the

principles involved in ecology, and to show some

rather counter-initiative results. It also shows

a special relationship between biology and

mathematics. Now, what does this has to do with

orbital mechanics? Simple this model is similar

to the models of orbits with those spirals,

contours and curves. We can apply this model with

constants representing gravitational pulls and

speeds of bodies.

Conclusion

- Hopefully, you now have a little insight into the

thinking that was behind the creation of the

Lotka-Volterra model for predator-prey

interaction!

Thank you, and this has been a fun project!

Work Cited

- Boyce, William. Elementary Differential

Equations. New York John Wiley Sons, Inc.,

1986 - Cullen, Michael Zill, Dennis. Differential

Equations with Boundary-Value Problems. Boston

PWS-Kent Publishing Company, 1993 - Zill, Dennis. A First Course in Differential

Equations The Classic Fifth Edition. California

Brooks/Cole, 2001 - Neuhauser, Claudia. Calculus for Biology and

Medicine New Jersey, 2000 - Intoduction to the Predator Prey Problem.

http//www.messiah.edu/hpages/facstaff/deroos/CSC1

71/PredPrey/PPIntro.htm 8/20/02 - Mathematical Formulation. http//www.pa.uky.edu/s

orokin/stuff/cs685S/analyt/node1.html 8/20/02 - Lotka Volterra Model. http//www.ento.vt.edu/sharo

v/PopEcol/lec10/lotka.html 8/20/02 - Predator-Prey Modeling. http//www-rohan.sdsu.edu/

jmahaffy/courses/bridges/bridges00.htm 8/22/02 - Predator Prey Model. http//www.enm.bris.ac.uk/sta

ff/hinke/courses/CDS280/predprey.html 8/20/02

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