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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)