Math 19. Lecture 6First-Order SystemsT. JudsonFall 20041 Competing S peciesLet x and y be the populations of two species at time t. We will assumethat each species, in absence of the other, grows logistically:x′= x(a1− b1x)y′= y(a2− b2y),where a1, a2are the growth rates of the two populations and a1/b1, a2/b2are the carrying capacities. The simplest expression for reducing the growthrate of species x due to the presence of species y is to replace the growthfactor a1− b1x with a1− b1x − α1, where is a measure of the degree to whichspecies y interferes with species x. The new system is n owx′= x(a1− b1x) − α1xyy′= y(a2− b2y) − α2xy.2 Left and Right-Curling SnailsLet L(t) denote the number (in millions) of left curling snails at time tand R(t) the number of right-curling snails. The two populations competefor the same resources and might be governed by the following system ofdifferential equations.dRdt= R − (R2+ aRL)dLdt= L − (L2+ aLR).1The Case a > 1Suppose a = 2. ThendRdt= R − (R2+ 2RL)dLdt= L − (L2+ 2LR).The Case a < 1Suppose a = 1/2. ThendRdt= R − (R2+ RL/2)dLdt= L − (L2+ LR/2).23 The Lotka-Volterra EquationSuppose we have a population of rabbits, R, and foxes, F . The systemdRdt= (a − bR − cF )RdFdt= (−d + eR)F.models the predator-prey relationship between the foxes and rabbits.Consider the following systems.•dRdt= (2 − 1.2F )RdFdt= (−1 + 0.9R)F .•dRdt= (2 − R − 1.2F )RdFdt= (−1 + 0.9R)F .3Homework• Chapter 5. Exercises 1; p. 86.Readings and References• C. Taubes. Modeling Differential Equations in Biology. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 5.• “Left S nails and Right Minds,” pp.
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