Predator-Prey Relationships

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Predator-Prey Relationships

• By Maria Casillas, Devin Morris, John Paul
  Phillips, Elly Sarabi, Nernie Tam

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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.

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• 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
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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.
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Risk and Food Availabilty

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

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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.

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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.

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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.
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Vito Volterra

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

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

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

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

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

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• 1896
• published papers on integral equations of the
  Volterra type.
• WWI
• Air Force
• Scientific collaboration

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

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

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Predator-Prey Populations

• 1926
• published deduction of the nonlinear differential
  equation
• similar to Lotkas logistic growth equation
• Lotka-Volterra Equation

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Crater Volterra

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

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• 1938
• Offered degree by University of St Andrews!

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

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Alfred James Lotka

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

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• Chemist
• Demographer
• Ecologist
• Mathematician

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

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

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Population Assoc. of America

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

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

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Alfred James Lotka

• Died
• 1949
• USA

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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
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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.

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• 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.

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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.

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• 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

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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))).
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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.
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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)
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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.
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Case 1 (a ? gtbk) u(x,y)x(6-2x-4y)
v(x,y)y(-35x)
x(0)1 y(0).5
sharks
fish
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x(0)2 y(0)3
sharks
fish
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x(0).5 y(0)1.5
sharks
fish
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x(0).5 y(0).5
sharks
fish
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Case 2 (a ? ltbk) u(x,y)x(2-6x-4y)
v(x,y)y(-35x) x(0)1 y(0).5
sharks
fish
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x(0)2 y(0)3
sharks
fish
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x(0)0.5 x(0)0.5
sharks
fish
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x(0).5 y(0).5
sharks
fish
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X(0).1 y(0).1
sharks
fish
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Case 3 All constants are equal
u(x,y)x(1-1x-1y) v(x,y)y(-11x)
sharks
fish
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X(0)2 y(0)3
sharks
fish
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X(0).5 y(0)1.5
sharks
fish
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X(0).5 y(0).5
sharks
fish
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Case 4 (b0) u(x,y)x(2-0x-1y)
v(x,y)y(-11x)
x(0)1 y(0)0.5
sharks
fish
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x(0)2 y(0)3
sharks
fish
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x(0)0.5 y(0)1.5
sharks
fish
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x(0)0.5 y(0)0.5
sharks
fish
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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
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x(0)2 y(0)3
sharks
fish
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x(0)0.5 y(0)1.5
sharks
fish
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x(0)0.5 y(0)0.5
sharks
fish
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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.
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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?)
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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.
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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.
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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.
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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!)
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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.
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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.
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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.
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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.
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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.
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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!
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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