Download Lecture #3 Continuous Lotka-Voltera Model 1

TAIBAH UNIVERSITY
College of Science
Department of Mathematics
Mathematical Modeling, MATH 200
Summer Term 1434/2013
Dr. Mostafa Zahri
Lecture #3
Continuous Lotka-Voltera Model
Reference:
1. R. Aris,
Mathematical Modeling Techniques. Dover, New York, 1994.
2. M. Meerschaert,
Mathematical Modeling. 3rd Edition, 2007, Academic Press/Elsevier, 2007.
3. S. Horton, William P. Fox, F. R. Giordano,
First Course in Mathematical Modeling. Brooks/ Cole, 2008.
1
Continuous Lotka-Voltera Model
The basic assumptions used in the two species model system are stated below:
• The foxes eat rabbits and breed.
• The rabbits eat grass and breed.
• The area is limited, but the grass grows faster than the rabbits can eat.
The discrete Lotka-Voltera model(two species model) is given as:
x0 , y 0
are a given initial values
xn+1 = axn − bxn yn
yn+1 = cyn + dxn yn
(1)
where
a is the natural growth rate of rabbits in the absence of predation,
b is the death rate per encounter of rabbits due to predation,
c is the efficiency of turning predated rabbits into foxes,
d is the natural death rate of foxes in the absence of food (rabbits).
Let us replace the number of rabbits xn by R(t) and the number of foxes yn by F (t). Moreover,
we write the differential change of the numbers of Rabbits and Foxes as
xn+1 = R(t + dt)
yn+1 = F (t + dt)
So the model (1) can be written in the following form
R(0), F (0)
are a given initial values
R(t + dt) = aR(t) − bR(t)F (t)
F (t + dt) = cF (t) + dR(t)F (t)
1
(2)
TAIBAH UNIVERSITY
College of Science
Department of Mathematics
Mathematical Modeling, MATH 200
Summer Term 1434/2013
Dr. Mostafa Zahri
which is equivalent to
R(0), F (0)
are a given initial values
R(t + dt) − R(t) = (a − 1)R(t) − bR(t)F (t)
F (t + dt) − F (t) = (c − 1)F (t) + dR(t)F (t)
(3)
Then, we get by passage to the limit the continuous Lotka-Voltera Model (Ordinary Differential System ODE)
R(0), F (0)
are a given initial values
dR(t)
= (a − 1)R(t) − bR(t)F (t)
dt
dF (t)
= (c − 1)F (t) + dR(t)F (t)
dt
2
Questions
2.1
Explain the behavior of the differential system above for following critical cases:
• x0 = 100 and y0 = 0.
• x0 = 0 and y0 = 10.
• a = 1.
• b = 0.
• c = 1.
• d = 0.
2
(4)