### Population Dynamics

```Population Dynamics
Application of Eigenvalues &
Eigenvectors
• Consider the system of equations
 dx
 dt  x(1  x  y)
 dy
  y(0.5  0.25y  0.75x)
 dt
• The critical points are (0,0), (1,0), (0,2) &
(.5,.5). These critical points correspond to
equilibrium solutions
Linearization for critical point (0,0)
• For this critical point the approximating linear
system is
d  x   1 0  x 
   
 
dt  y   0 .5  y 
• The eigenvalues and eigenvectors are
1
 0
1  1, v1    and 2  .5, v2   
 0
1
• Thus (0,0) is an unstable node for both the linear
and nonlinear systems
Linearization for critical point (1,0)
• For this critical point the approximating linear
system is
d  x    1  1  x 
   
 
dt  y   0  .25 y 
• The eigenvalues and eigenvectors are
1
 4
1  1, v1    and 2  .25, v2   
 0
  3
• Thus (1,0) is an asymptotically stable node of
both the linear and nonlinear systems
Linearization for critical point (0,2)
• For this critical point the approximating linear
system is
0  x 
d  x   1
   
 
dt  y    1.5  .5  y 
• The eigenvalues and eigenvectors are
 1
 0
1  1, v1    and 2  .5, v2   
 3
1
• Thus (0,2) is an asymptotically stable node for
both the linear and nonlinear systems
Linearization for critical point (.5,.5)
• For this critical point the approximating linear
system is
 .5  x 
d  x    .5
   
 
dt  y    .375  .125 y 
• The eigenvalues and eigenvectors are
 1 
 1 
 and 2  .78, v2   
1  .16, v1  
  1.32
 .57
• Thus (0,2) is a unstable saddle node for both the
linear and nonlinear systems
Phase Portrait & Direction Field
Trajectories starting above the separatrix
approach the node at
(0,2), while those below approach the node at
(1,0).
If initial state lies on separatrix, then the
solution will approach
the saddle point, but the slightest perturbation
will send the
trajectory to one of the nodes instead.
Thus in practice, one species will
survive the competition and the
other species will not.
```