454 Ordinary Differential Equations
Assumption 3 If one species is not present, there is no change in its
population.
Stated mathematically, this assumption is (i) if x = 0, then f(O, y) = 0 and(ii) if y = 0, g(x, 0) = 0. Setting x = 0 and f(O, y) = 0 in equation (2) leads
to the requirement 0 = A+ Dy+ Ey^2 for ally. Thus, A = D = E = 0.
Likewise, setting y = 0 and g(x, 0) = 0 in equation (3), yields the requirement
0 = K + L x + Mx^2 for all x. Thus, K = L = M = 0.
Summarizing to this point, assumptions 1, 2, and 3 result in a system of
the form
(4)(5)dx
dtBx+ Cx^2 + Hxy = f(x, y)
dy = Ny+ Py^2 + Qxy = g(x, y).
dt
Assumption 4 An increase in the prey population increases the growth
rate of the predators.
Mathematically, this assumption is gx (x, y) > 0 for all x, y. Since from
equation (5) gx = Qy, this assumption means Q is positive for y positive.
Assumption 5 An increase in the predator population decreases the
growth rate of the prey.
This assumption is fy(x, y) < 0 for all x, y. Since from equation (4) fy =
Hx, this assumption mea ns His negative for x positive.
Assumption 6 If there are no predators, then the prey population in-
creases according to the Malthusian law.
Thus, if y = 0, then f(x, 0) > 0 and f (x, 0) = rx where r is positive. From
equation (4), this assumption leads to the requirement
f(x, 0) = Bx+ C x^2 = rx.
Hence, B = r, a positive constant, and C = 0.
Assumption 7 If there are no prey, then the predator population decreases
according to the Malthusian law.
Mathematically, this assumption is: if x = 0, then g(O, y) < 0 and g(O, y) =
-sy, where s is positive. From equation (5), this assumption leads to the
requirement
g(O, y) = Ny+ Py^2 = -sy.
Hence, N = -s, where s is positive, and P = 0.
These seven assumptions yield the Volterra-Lotka prey-predator model(6)dx
dtrx - H x y = f(x, y)
dy
dt = -sy + Q x y = g(x, y)