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Ch 9.5: Predator-Prey Systems
In Section 9.4 we discussed a model of two species that
interact by competing for a common food supply or other
natural resource.
In this section we investigate the situation in which one
species (the predator) preys on the other species (the prey),
while the prey lives on some other source of food.
For example, foxes and rabbits in a closed forest.
Again we emphasize that a model involving only two species
cannot fully describe the complex relationships among species
that occur in nature.
Nevertheless, the study of simple models is the first step
toward an understanding of more complicated phenomena.
Assumptions
Let x and y be the populations of the prey and predator,
respectively, at time t.
We make the following assumptions:
In the absence of the predator, the prey grows at a rate proportional to
the current population; thus dx/dt = a x, a > 0, when y = 0.
In the absence of the prey, the predator dies out at a rate proportional to
the current population; thus dy/dt = -cy, c > 0, when x = 0.
The number of encounters between predator and prey is proportional to
the product of their populations. Each such encounter tends to promote
the growth of the predator and to inhibit the growth of the prey. Thus
the growth rate of the predator is increased by a term of the form  xy,
while the growth rate of the prey is decreased by a term – xy, where 
and  are positive constants.
Predator-Prey Equations
Thus we have the system of equations
dx / dt  ax   xy  xa   y ,
dy / dt  cy   xy  y  c   x 
The constants a, c, ,  are all positive, where a, c are the
growth rate of prey and death rate of predator, respectively,
and ,  are measures of the effect of the interaction between
the two species.
The predator-prey equations are known as the Lotka-Volterra
equations. Although they are rather simple equations, they do
characterize a wide class of problems.
Our goal here is to determine the qualitative behavior of the
solutions for arbitrary positive initial values x and y.
Example 1: Population Equations
(1 of 8)
Consider the system of equations
dx / dt  x  0.5 xy, dy / dt  0.75 y  0.25 xy,
x, y  0
The critical points are (0,0) and (3,2).
Given below is a direction field for this system of equations,
with the critical points indicated as heavy dots.
The trajectories appear to be closed curves surrounding the
critical point (3,2).
Example 1: Critical Point at (0,0)
(2 of 8)
For the critical point (0,0), the approximating linear system is
0  x 
d  x  1
   
 
dt  y   0  0.75  y 
The eigenvalues and eigenvectors are
r1  1, ξ
(1)
1
   ;
 0
r2  0.75, ξ
( 2)
 0
  
1
and hence the general solution for this linear system is
 x
1 t
 0  0.75t
   c1   e  c2   e
 y
 0
1
Thus (0,0) is an unstable saddle point of both the linear and
nonlinear systems. One trajectory approaches (0,0) along the
y-axis, while all other trajectories depart from (0,0).
Example 1: Critical Point at (3,2)
(3 of 8)
For the critical point (3,2), the approximating linear system is
d  u   0  1.5  u 
   
 ,
0  v 
dt  v   0.5
u   x  3 
   

 v   y  2
The eigenvalues and eigenvectors are
r1 
3i (1)  1 
 ;
, ξ  
2
i / 3
r2  
3i ( 2)  1 

, ξ  
2
i / 3 
Thus (3,2) is stable center point of the linear system, but is
indeterminate for the nonlinear systems, by Theorem 9.3.2.
To find the trajectories for the linear system, we have
dv / dt dv
0.5u
u



du / dt du
1.5v
3v
Example 1: Critical Point at (3,2)
(4 of 8)
Near the critical point (3,2), we thus have
dv / du  u / 3v
The solution to this separable equation is
u 2  3v 2  k , k  0
Thus the trajectories of the linear system are ellipses centered
at the critical point (3,2), and are elongated horizontally.
Returning to the nonlinear system
dx / dt  x  0.5 xy, dy / dt  0.75 y  0.25 xy, x, y  0
we have
dy  0.75 y  0.25 xy

dx
x  0.5 xy
Example 1: Critical Point at (3,2)
(5 of 8)
Thus
dy y 0.75  0.25 x 
1  0.5 y
 0.75  0.25 x


dy 
dx
dx
x1  0.5 y 
y
x
The solution to this separable equation is
0.75 ln x  ln y  0.5 y  0.25 x  c
It can be shown that the graph of this equation, for a fixed c, is
a closed curve about the point (3,2).
Thus the critical point (3,2) is also a center of our nonlinear
system, and hence the predator and prey populations exhibit a
cyclic variation about the equilibrium solution (3,2).
This behavior is seen in the phase portrait on the next slide.
Example 1: Phase Portrait
(6 of 8)
Given below is a phase portrait for our nonlinear system.
For some initial conditions, the trajectories represent small
variations in x and y about (3,2), and are almost elliptical in
shape, as the linear analysis suggests.
For other initial points, the oscillations in x and y are more
pronounced, and the shape of the trajectories are significantly
different from an ellipse.
Note that the trajectories are
traversed counterclockwise.
Example 1: Population Equations
(7 of 8)
A phase portrait along with population graphs x(t) and y(t), for
a typical set of initial conditions, are given below.
Note from both of these figures that the oscillation of the
predator population lags behind that of the prey.
Example 1: Population Equations
(8 of 8)
Starting with a state in which both populations are relatively
small, the prey first increase because of little predation.
Then the predators, with abundant food, increase in population.
This causes heavy predation, and the prey tend to decrease.
Finally, with a diminished food supply, the predator population
also decreases, and the system returns to original state.
General Predator-Prey Equations
(1 of 7)
The general system of equations
dx / dt  ax   xy  xa   y ,
dy / dt  cy   xy  y  c   x 
can be analyzed in the same way as in Example 1.
The critical points are the solutions of the equations
xa   y   0, y c   x  0,
yielding the points (0,0) and (c/, a/).
We next examine the solutions of the corresponding linear
system near each critical point.
General System: Critical Point at (0,0)
(2 of 7)
For the critical point (0,0), the approximating linear system is
0  x 
d  x a
   
 
dt  y   0  c  y 
The eigenvalues and eigenvectors are
r1  a, ξ
(1)
1
   ;
 0
r2  c, ξ
( 2)
0
  
1
and hence the general solution for this linear system is
 x
 1  at
 0  c t
   c1   e  c2   e
 y
 0
1
Thus (0,0) is an unstable saddle point of both the linear and
nonlinear systems. One trajectory approaches (0,0) along the
y-axis, while all other trajectories depart from (0,0).
Critical Point at (c/, a/): Linear System
(3 of 7)
For the critical point (c/, a/), the linear system is
d u   0
   
dt  v    a / 
  c /   u 
 ,
0  v 
u   x  c /  
   

 v   y  a / 
The eigenvalues are
r1  i ac , r2  i ac
Thus (c/, a/) is stable center point of the linear system, but is
indeterminate for the nonlinear systems, by Theorem 9.3.2.
To find the trajectories for the linear system, we have
dv / dt dv
( a /  )u
 2a u


 2
du / dt du
( c /  )v
 cv
Critical Point at (c/, a/): Ellipses
(4 of 7)
Near the critical point (c/, a/), we thus have
dv
 2a u
 2
  2 a u du   2c v dv  0
du
 cv
The solution to this separable equation is
 2 a u 2   2c v 2  k , k  0
Thus the trajectories of the linear system are ellipses centered
at the critical point (c/, a/).
Returning to the nonlinear system
dx / dt  xa   y , dy / dt  y c   x
we have
dy y  c   x 

dx
xa   y 
Critical Point at (c/, a/):
Nonlinear System
(5 of 7)
Thus
dy y  c   x 
a  y
c  x


dy 
dx
dx
xa   y 
y
x
The solution to this separable equation is
a ln y   y  c ln x   x  C
It can be shown that the graph of this equation, for a fixed C, is
a closed curve about the point (c/, a/).
Thus the critical point (c/, a/) is also a center of our
nonlinear system, and hence the predator and prey populations
exhibit a cyclic variation about (c/, a/).
Critical Point at (c/, a/):
Small Deviations and Linear System (6 of 7)
The cyclic variation of the predator and prey populations can
be analyzed in more detail when the deviations from the point
(c/, a/) are small and the linear system can be used.
The solution of the linear system
d u   0
   
dt  v    a / 
  c /   u 
 
0  v 
can be written in the form
u (t ) 
c



K cos act   , v(t ) 
a

c
K sin
a

act  

where the constants K and  are determined by the initial
conditions.
Critical Point at (c/, a/): Small Deviations and
Elliptical Approximation (7 of 7)
The equations below are good approximations to the nearly
elliptical trajectories near (c/, a/).
u (t ) 
c



K cos act   , v(t ) 
a

c
K sin
a

act  

We can use them to draw several conclusions about the cyclic
variation of the predator and prey on such trajectories:
The predator and prey population sizes vary sinusoidally with period
2 /(ac)½. This oscillation period is independent of initial conditions.
The predator and prey populations are out of phase by one-quarter of a
cycle. The prey leads and the predator lags.
The amplitudes of oscillations are Kc/ for the prey, and (c/a)½Ka/ for
the predator, and hence depend on initial conditions and parameters.
The average predator and prey populations over one complete cycle are
c/ and  /a, respectively – same as equilibrium populations.
Modified Predator-Prey Equations
Cyclic variations of predator and prey, as predicted by our
equations, have been observed in nature; see text.
One criticism of the Lotka-Volterra equations is that in the
absence of the predator, the prey will grow without bound.
This can be corrected using a logistic model for x when y = 0.
As a result of this modification, the critical point at (c/, a/)
moves to (c/, a/ -  c/ ) and becomes an asymptotically
stable point. It is either a node or a spiral point, depending on
the parameters in the differential equations.
In either case, other trajectories are no longer closed curves but
approach the critical points as t  .