Edited with the trial version of
Foxit Advanced PDF Editor
To remove this notice, visit:
www.foxitsoftware.com/shopping
236 4. Stability Theory
1
* *
r(1-N /K)/acN
*
1-exp(-r(1-H /K))
*
H
N2*
N1*
FIGURE 4.30.
The nontrivial equilibrium points are given by
ln  1
"
H" = , P = ln .
( - 1) ac a
"
It can be shown by linearization that (H", P ) is unstable. Thus this model
is too simple for any practical applications except possibly under contrived
laboratory conditions.
It is reasonable to modify the H(n) equation (4.7.4) to incorporate some
saturation of the prey population, or, in terms of predator encounters, a
prey-limiting model. Hence a more realistic model is given by

H(n)
H(n +1) = H(n)exp r 1 - - aP (n) , r > 0,
k
P (n +1) = cH(n)(1 - exp(-aP (n))). (4.7.21)
The equilibrium points are solutions of

H"
" " "
1 = exp r 1 - - aP , P = cH"(1 - exp(-aP )).
K
Hence

"
r H" r P
"
P = 1 - = (1 - q), H" = . (4.7.22)
"
a K a (1 - 
ap )
Thus

H"
r(1 - )
H"
K
= 1 - exp -r 1 - . (4.7.23)
acH" K
" "
Clearly, H1 = K, P1 = 0 is an equilibrium state. The other equilibrium
point may be obtained by plotting the left- and right-hand sides of (4.7.19)
against H". From Figure 4.30 we see that there is another equilibrium point
" "
with 0 < H2 < K. Then we may find P2 from (4.7.18).


Wyszukiwarka

Podobne podstrony:
Applications of linear algebra to differential equation [sharethefiles com]
An introduction to the Analytical Writing Section of the GRE
An Introduction to the Kabalah
Geiss An Introduction to Probability Theory
An Introduction To Options Trading Success with James Bittman
An Introduction to Yang Mills Theory
An Introduction to the Study of the Tarot
21 Wstęp do temperacji 07 10 2009 An introduction to temperament
Surviving Childhood An Introduction to the Impact of Trauma
Introduction to Ceremonial Magic by Phil Hansford
Ball An Elementary Introduction to Modern Convex Geometry
CSharp Introduction to C# Programming for the Microsoft NET Platform (Prerelease)
IMiR NM2 Introduction to MATLAB
Introduction to multivariate calibration in analytical chemistry
Smirnov, A V Introduction to tensor calculus (2004)
Introduction to Lean for Poland

więcej podobnych podstron