Transcript Predator-Prey Relationships By: Maria Casillas, Devin Morris, John Paul Phillips, Elly Sarabi, & Nernie Tam.

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 Model:
Fish & 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 other’s 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 Legendre’s 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 Lotka’s 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/(n^{a}))
– Where C is a constant
– If a=2, then C=(6/(pi^{2}))=0.61
Alfred James Lotka

Died:
– 1949
– USA
The Lotka-Volterra Model

System

Initial Conditions

F'(t)=aF-bF²-cFS

F(0)=F0

S'(t)=-kS+dSF

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
F is the initial size of the fish population
0
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)=-kS+dSF
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(-k+dF)

To compute the equilibrium points we
solve (dF/dt)=0 and (dS/dt)=0
 (dF/dt)=0 when F=0 or a-bF-cS=0
 solution: F=(a-cS)/b

dS/dt=0 when S=0 or -k+dF=0
Solution:{F=(k/d)}
Now we find all the combinations:
One of our equilibrium points is (0,0).
For F=(a-cS)/b: When S=0, 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=((-kb+ad)/(dc))}
Thus, one of our equilibrium points is ((k/d),((-kb+ad)/(dc))).
Our equilibrium points are (0,0), ((a/b),0), and ((k/d),((-kb+ad)/(dc))).
Now, to study the stability of the equilibrium points we first need to
find the Jacobian matrix which is:
dx/dF
J(F,S)=
dy/dF
dx/dS
dy/dS
a-2bF-cS -cF
=
d*S
-k+d*F
To study the stability of (0,0):
a-λ
0
J(0,0)=det
0
-k-λ
= (a- λ)(-k- λ),
Solution is: {λ =a},{λ =-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-λ
((-ca)/b)
0
-k+((ad)/b)-λ
= (-a-λ)(-k+a(d/b)-λ),
Solution is: {λ=-a},{λ=((-kb+ad)/b)}
stable if λ =((-kb+ad)/b) < 0 (i.e. ad < kb)
semi-stable if λ =((-kb+ad)/b) > 0 (i.e. ad > kb)
To study the stability of ((k/d),((ad-kb)/(cd))):
J((k/d),((ad-kb)/(cd)))=det
a-2b(k/d)((-ck)/d)
c((ad-kb)/(cd))λ
((ad-kb)/c)
-b(k/d)-λ
((-ck)/d)
=det
((ad-kb)/c)
-λ
-λ
= ((λ kb+ λ ²d-k²b+kad)/d)
Solution is: {λ =(1/(2d))(-kb+(k²b²+4dk²b-4kad²)^1/2)}
{λ =(1/(2d))(-kb-(k²b²+4dk²b-4kad²)^1/2)}
If we simplify a little more, we get:
λ =(1/(2d))(-kb-(k²b²+4dk²b-4kad²)^1/2)
= -(1/2)((kb+i(k)^1/2(-kb²-4dkb+4ad²)^1/2)/d)
λ =(1/(2d))(-kb+(k²b²+4dk²b-4kad²)^1/2)
= -(1/2)((kb-i(k)^1/2(-kb²-4dkb+4ad²)^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 λ >bk)
x(0)=1
y(0)=.5
u(x,y)=x(6-2x-4y) v(x,y)=y(-3+5x)
1. 8
1. 6
1. 4
sharks 1. 2
1
0. 8
0. 6
0. 4
0. 6
0. 8
fish
1
1. 2
x(0)=2
y(0)=3
4
3. 5
3
sharks
2. 5
2
1. 5
1
0. 5
0. 5
1
fish
1. 5
2
x(0)=.5
y(0)=1.5
1. 5
1. 4
1. 3
sharks 1. 2
1. 1
1
0. 45
0. 5
0. 55
0. 6
fish
0. 65
0. 7
x(0)=.5
y(0)=.5
1. 6
1. 4
1. 2
sharks
1
0. 8
0. 6
0. 4
0. 5
0. 6
0. 7
fish
0. 8
0. 9
1
1. 1
Case 2: (a λ <bk)
u(x,y)=x(2-6x-4y) v(x,y)=y(-3+5x)
x(0)=1 y(0)=.5
0. 5
0. 4
sharks
0. 3
0. 2
0. 1
0
0. 4
0. 6
fish
0. 8
1
x(0)=2
y(0)=3
3. 5
3
2. 5
2
sharks 1. 5
1
0. 5
0
0. 5
1
fish
1. 5
2
x(0)=0.5
x(0)=0.5
1. 4
1. 2
1
0. 8
sharks
0. 6
0. 4
0. 2
0
0. 1
0. 2
0. 3
fish
0. 4
0. 5
x(0)=.5
y(0)=.5
0. 5
0. 4
0. 3
sharks
0. 2
0. 1
0
0. 25
0. 3
0. 35
fish
0. 4
0. 45
0. 5
X(0)=.1
y(0)=.1
0. 1
0. 08
sharks
0. 06
0. 04
0. 02
0
0. 1
0. 15
0. 2
fish
0. 25
0. 3
Case 3: All constants are equal
u(x,y)=x(1-1x-1y)
v(x,y)=y(-1+1x)
0. 5
0. 4
0. 3
sharks
0. 2
0. 1
0. 7
0. 75
0. 8
0. 85
fish
0. 9
0. 95
1
X(0)=2 y(0)=3
3
2.5
2
sharks
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
1.2
fish
1.4
1.6
1.8
2
X(0)=.5 y(0)=1.5
1.4
1.2
1
sharks
0.8
0.6
0.4
0.2
0.3
0.4
0.5
0.6
fish
0.7
0.8
0.9
X(0)=.5 y(0)=.5
0.5
0.4
sharks
0.3
0.2
0.1
0.5
0.6
0.7
fish
0.8
0.9
Case #4: (b=0)
u(x,y)=x(2-0x-1y) v(x,y)=y(-1+1x)
x(0)=1
y(0)=0.5
5
4
3
sharks
2
1
0. 5
1
1. 5
2
fish
2. 5
3
3. 5
x(0)=2
y(0)=3
3. 5
3
2. 5
sharks
2
1. 5
1
0. 5
1
1. 5
fish
2
x(0)=0.5
y(0)=1.5
3
2. 5
sharks
2
1. 5
0. 6
0. 8
1
1. 2
fish
1. 4
1. 6
1. 8
x(0)=0.5
y(0)=0.5
5
4
sharks
3
2
1
0. 5
1
1. 5
2
fish
2. 5
3
3. 5
Case 5: ((k/ λ)=((a λ -bk)/(c λ))) u(x,y)=x(2-1x-1y) v(x,y)=y(-1+1x)
x(0)=1
y(0)=0.5
1
0. 9
0. 8
sharks
0. 7
0. 6
0. 5
1
1. 05
1. 1
1. 15
fish
1. 2
1. 25
1. 3
x(0)=2
y(0)=3
3
2. 5
2
sharks
1. 5
1
0. 4
0. 6
0. 8
1
1. 2
fish
1. 4
1. 6
1. 8
2
x(0)=0.5
y(0)=1.5
1. 5
1. 4
1. 3
1. 2
sharks
1. 1
1
0. 9
0. 8
0. 5
0. 6
0. 7
0. 8
fish
0. 9
1
1. 1
x(0)=0.5
y(0)=0.5
1
0. 9
0. 8
sharks
0. 7
0. 6
0. 5
0. 4
0. 6
0. 8
1
fish
1. 2
1. 4
With the eigenvalues
-bk+i(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) = (e^t(-bk/2))(kK/)cos(t ((4akk-4bk  -bbkk)^1/2)+)
V(t) = (e^t(-bk/2))(kK/c)((a-bk)^1/2)*sin(t ((4akk-4bk  bbkk)^1/2)+)
And by substituting, we get:
F(t) = k/(1+ (e^t(-bk/2))Kcos[t {(4akk-4bk  -bbkk)^1/2}+]
S(t) = {(a-bk)/c}*(1+ {e^t(-bk/2)}*[{k/(a-bk)}^1/2]*Ksin[t
{(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/){e^t(-bk/2)}
And for S(t); K(k/c){(a-bk)^1/2}{e^t(-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)>0, and S(0)>0.
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 < k/, then for the shark equation, the critical point
becomes a negative number!
a/b < k/ ==> a < 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/b=k/=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 it’s hard to feel sorry
for sharks!)
For the fourth case, we make b=0 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 > 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 a=0, c=0, k=0, =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 fish’s growth rate or
the shark’s 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/CSC171/PredPrey/PPIntro.htm;
8/20/02
Mathematical Formulation.
http://www.pa.uky.edu/~sorokin/stuff/cs685S/analyt/node1.html; 8/20/02
Lotka Volterra Model. http://www.ento.vt.edu/sharov/PopEcol/lec10/lotka.html;
8/20/02
Predator-Prey Modeling. http://wwwrohan.sdsu.edu/~jmahaffy/courses/bridges/bridges00.htm; 8/22/02
Predator Prey Model.
http://www.enm.bris.ac.uk/staff/hinke/courses/CDS280/predprey.html; 8/20/02