Download Modeling the Predator-Prey Relationship Michael Olinick

Modeling the Predator-Prey Relationship
Michael Olinick
Middlebury College
MAA Session on Environmental Mathematics
January 12, 2006
A basic assumption of the classic LotkaVolterra predator-prey model is that each species
experiences exponential growth or decay in the
absence of the other,
Recent extensions of this
model investigate logistic growth of one species
when the other is absent, time delays in response
by one species to population changes in the other,
and multiple species interactions. A survey of
consequences and predictions of the original and
the modified models will be presented.
“Nature offers a puzzling variety of interactions
between species.”
– Nino Boccara, Modeling Complex Systems
Mutualism
Competition
Predation
Predation
Parasites: live a major period of their life in a
single host doing nonlethal harm to the host
(tapeworm)
Grazers: attack large numbers of prey during
their lifetimes, consuming part of the host without
causing death ( mosquitos feeding on human blood,
sheep grazing on grass).
True Predators: attack many prey during
their lifetimes, quickly killing the prey (wolves).
Googling “Predator-Prey” returns 609,000 hits.
Today:
A very brief look at a very large and active
area of modeling
•
• Review of the classic Lotka-Volterra:
assumptions, model formulation, analysis and
predictions, successes and failures.
•
Modifications of the Lotka-Volterra Approach.
A Real World Puzzle
Umberto D’Ancona asks Vito Volterra:
After World War I, the proportion of predatory fish caught in the Upper
Adriatic was considerably higher than in the years before the war; the proportion
of prey fish was lower.
The war between Austria and Italy halted commercial fishing.
How did the war benefit the predators more than the prey?
Alfred James
Lotka
1880-1949
Vito Volterra
(1860-1940
Elements of Physical
Biology, Baltimore:
Williams and Wilkins,
1925.
(Dover reprint Elements
of Mathematical Biology,
1956)
“Variazioni e fluttuazioni del
numero d’individui in specie
animali conviventi,” Rendiconti
dell’Academia dei Lincei, 6 (2),
31 – 113 (1926).
(“Fluctuations in the
Abundance of a Species
Considered Mathematically,”
Nature, 118, 558 – 560 (1926).
Francesco M. Scudo and James R. Ziegler, eds., The Golden
Age of Theoretical Ecology, 1923 – 1940, Berlin: SpringerVerlag, 1978.
The Lotka-Volterra Model
Assumptions
A classic formulation of this situation is that of a
population of rabbits, who feed on clover, and a population of
foxes, who feed on rabbits. The assumptions usually made
about the situation are these:
1.
In isolation, the rate of change of population of one species
is proportional to the population of that species. In the absence
of foxes, it is assumed that the rabbits population will exhibit
exponential growth. If there are no rabbits, the fox population
will undergo a pure death process.
2.
There is always so much clover that the rabbits have an
ample supply of food. The only food available to the predatory
foxes are the rabbits.
3. The number of kills of rabbits by foxes is proportional to
the frequency of encounters between the two species. This, in
turn, is proportional to the product of the populations of rabbits
and foxes. Thus there will be few kills if there are few rabbits or
few foxes and many kills only when both populations are
relatively large.
The Lotka-Volterra Model
Mathematical Formulation
x = measure of prey (rabbit) population (population, biomass)
y = measure of predator (fox) population (population, biomass)
dx
= ax bxy
dt
dy
= mxy ny
dt
dx
= ax bxy = x(a by)
dt
dy
= mxy ny = y(mx n)
dt
where a, b, m, n are positive constants.
Can reduce the 4 parameters to 1:
Let X =
Then
m
a
x, Y = y, T = an t, r =
n
b
dX
= rX (1 Y )
dT
dY 1
= Y (X 1)
dT r
a
n
Analyzing the Phase-Plane for Lotka-Volterra
Region
dx dy
dt dt
______________
I
<0
>0
II
<0
<0
III
>0 <0
IV
>0 >0
Signs of dx/dt and dy/dt for the Lotka-Volterra model.
Elementary Observations:
• There are two equilibrium points: (0,0) and (n/m, a/b).
• The positive x-axis and the positive y-axis are each possible
orbits.
• Since L-V is system of autonomous differential equations,
orbits starting in the first quadrant will remain there.
The Volterra mapping technique can be used to find a more
exact orbit to the original predator-prey model. Note that
y (mx - n)
dy
=
x (a - by)
dx
in the original system. After the variables are separated in this
differential equation and integration is completed, the solution
looks like
a log y -by = mx -n log x + C
which may be written as
(ya e-by)(xm e-nx) = K.
y = x m e nx
Volterra Mapping Technique
A typical result.
Fig. 4.12 Orbits for the predator-prey system dxldt = ax -bxy, dyldt =
mxy -ny obtained from the Volterra mapping technique. Here a = 5, b =
3, m = 2, and n = 6. Critical points are (0,0) and (3, 513). All orbits
move in a counterclockwise direction. The constant K is determined by
x0n y0a
initial conditions (x0,y0) and is equal to by0+ mx0 At (3,5/3), K= .156.
e
Elliptical orbit of the linearized version of the predator-prey model.
Volterra Solves the Puzzle
After The War:
dx
= x(a by)
dt
dy
= y(mx n)
dt
Average Prey Population is
n
m
Average Predator Population is
a
b
Before the War
dx
= x(a c by)
dt
dy
= y(mx n d)
dt
n+d
m
ac
Average Predator Population was
b
Average Prey Population was
Linearizing
dx
= f (x, y)
dt
dy
= g(x, y)
dt
If (x*,y*) is an equilibrium point, set
u = x – x*, v = y – y*.
In the neighborhood of this equilibrium point,
neglecting quadratic and higher degree terms, we have
f
du f
=
(x*, y*)u + (x*, y*)v
dt x
y
g
dv g
=
(x*, y*)u +
(x*, y*)v
dt x
y
The general solution of the linear system is
u (0)
u (t) = exp(D f(x*)t)u
where u (t) = (u(t), v(t)), f = (f , g), x * = (x*, y*)
and D f(x*) is the Jacobian matrix of f at x* (often called the
community matrix in ecology)
f
x (x*, y*)
D f(x*) = g
x (x*, y*)
f
(x*, y*)
y
g
(x*, y*)
y
Lotka-Volterra
Equilibrium
Jacobian
Eigenvalues
a 0 0 n bn 0
m
ma
0 b
a,-n
Point
(0,0)
n a
m , b unstable
neutrally
i an , i an stable
Does the Lotka-Volterra model predict patterns
observed in measured real-world data?
Changes in the abundance of the lynx and the snowshoe hare, as
indicated in the number of pelts received by the Hudson’s Bay
Company from 1845 to 1934. “This is a classic case of cyclic oscillation
in population density.”
BUT… In some cycles, the maximum of the lynx
population led rather than lagged behind the hare
population.
The cycles in these two animals as observed in the Hudson Bay
data are not described by predator–prey interactions of the
Lotka–Volterra type.
Michael Gilpin, “Do Hares Eat Lynx?” American Naturalist,
Volume 107 (1973), 727 – 730.
C. Modifying the Model
Several variations of the Lotka- Volterra predator-prey model
have been proposed that offer more realistic descriptions of the
interactions of the populations.
1.
If the population of rabbits is always much larger than the
number of foxes, then the considerations that entered into the
development of the logistic equation may come into play. If the number
of rabbits becomes sufficiently great, then the rabbits may be
interfering with each other in their quest for food and space. One way
to describe this effect mathematically is to replace the original model by
the more complicated system,
dx
x
= ax(1 ) bxy
dt
k
dy
= mxy ny
dt
where the prey population grows logistically in the absence of
predators, with a carrying capacity of k.
Modified Lotka-Volterra
(Carrying Capacity for Prey)
Equilibrium
Jacobian
Eigenvalues
Point
(0,0)
(k,0)
n a
m , b a 0 a,-n
0 n a
bk ( a, mk n)
0 mk n an
bn Complex with
mk
m negative real
ma(mk n)
0 mkb
part if
k>
n
an
+
m 4m2
unstable
Unstable
If k > m/n
Asymptot
ically
stable
EXERCISE:
Most predators feed on more than one type of food. If
the foxes can survive on an alternative resource, although the presence
of their natural prey (rabbits) favors growth, a possible alternative
model is the system
dx
= ax -bx2 -cxy
dt
dy
= mxy + ny - py2
dt
where a, b, c, m, n, p are positive constants.
3.
P. H. Leslie and J. C. Gower studied a third variation, the system
of equations
dx
= ax - bxy
dt
dy
y
=(c - e
dt
x
)y
where the parameters a, b, c, e are again positive constants. Here the
term y/x arises from the fact that this ratio ought to affect the growth
of the predator. When foxes are numerous and rabbits are scarce, y/x
is large and the growth of fox population will be small. Conversely,
when the supply of rabbits is ample for the foxes, y/x is small and there
is slight restriction on the increase of the predators.
The orbits associated with the Leslie-Gower model are curves that
spiral in toward the critical point. We illustrate a typical situation of
this stable equilibrium.
Results of the Leslie-Gower model for a predator-prey system. Here
dx/dt = ax -bxy, dy/dt = (c -e(y/x) )y. The curves illustrated are for a =
1, b = .1, c = 1, e = 2.5 and initial populations x0 = 80, y0 = 20. The
critical point is (25, 10). The top graph shows the orbit of a solution of
the system of differential equations; it spirals in toward the critical
point. The bottom graph shows x and y as functions of t. From E.
Pielou, An Introduction to Mathematical Ecology, New York: John
Wiley, 1969.
Gary Harrison’s Models
Each of the models we have seen makes the assumption
of insatiable predators. Limits on gut size and time
available for hunting dictate that the predators' kill rate
will approach an upper bound as the density of prey
increases.
Gary Harrison, “Comparing Predator_Prey Models to
Luckinbill’s Experiment with Didinium and Paramecium,”
Ecology Volume 76 (1995), 357 – 374.
dx
x
= ax(1 ) bf (x)y
dt
k
dy
= mf (x)y ny
dt
where f(x) is a measure of the functional response of the
predator.
Many predators exhibit a “saturation” effect in their
functional response when prey is abundant. One way to model
this is to use the function
f (x) =
x
c+ x
which is called a Holling Type 2 functional response.
Leo S. Luckinbill, “Coexistence in Laboratory Populations of
Paramecium Aurelia and its Predator Didinium Nasutum,”
Ecology Volume 54 (1973), 1320 – 1327.
The Model (aka Rosenzweig-MacArthur model)
becomes
dx
x
bxy
= ax(1 )
dt
k c+ x
dy mxy
=
ny
dt c + x
M. L. Rosenzweig and R. H. MacArthur , “Graphical Representation
and Stability Conditions of Predator-Prey Interactions,” American
Naturalist, Volume 97 (1963), 209-223.
Two trajectories in the (Prey,Predtaor)-plane converging to a
stable limit cycle in Harrison’s model.
Other Modifications/Approaches
•
Fluctuating Environments
Carrying Capacity may vary with time.
Replace k by k(t)
Example:
•
k(t) = k(1+ qsin t)
Discrete-Time Models
The discrete version of
dX
= rX(1 Y )
dT
dY 1
= Y (X 1)
dT r
is
X(T + 1) = X(T ) + rX(t)(1 Y (T ))
1
Y (T + 1) = Y (T ) + Y (T )(X(T ) 1)
r
More generally,
X(T + 1) = X(T ) + f (X(T ),Y (t))
Y (T + 1) = Y (T ) + g(X(t),Y (t))
In the neighborhood of the equilibrium point (x*,y*), neglecting
quadratic and higher degree terms, we have
f
f
(x*, y*)u(T ) + (x*, y*)v(T )
x
y
g
g
v(T + 1) = v(T ) + (x*, y*)u(T ) + (x*, y*)v(T )
y
x
u(T + 1) = u(T ) +
General Solution of Linear System is
u(T ) = ( I + Df (x*, y*)) u(0)
T
The equilibrium point (x*,y*) is asymptotically stable if
the absolute values of the eigenvalues of I + Df(x*,y*)
are less than 1.
The discrete time analogue of Lotka-Volterra has the
nontrivial equilibrium point (1,1). The eigenvalues of I
+ Df(1,1) are 1 + i and 1 –i. The neutrally stable
equilibrium point of the continuous time model is
unstable for the discrete time model.
•
Time-Delay Models
K. Gopalsamy, Stability And Oscillations In Delay
Differential Equations Of Population Dynamics, Kluwer
Academic, Dordrecht/Norwell, MA, 1992.
Yang Kuang Delay Differential Equations with
Applications in Population Dynamics, Academic Press,
Boston, 1993.
•
Lattice Models
Nino Boccara, O. Roblin, and Morgan Roger,
“An Automata Network Predator-Prey Model with
Pursuit and Evasion,” Physical Review Volume 50
(1994), 4531 – 4541.
•
Multiple Species Models
Werner Krebs, “A General Predator-Prey
Model,” Mathematical and Computer Modeling of
Dynamical Systems, Volume 9 (2003), 387 – 401.