The Lotka-Volterra predator-prey model

Proposition 1 (Boukal and Krivan 1999, Krivan and Vrkoc 2007) When
predators behave optimally, all trajectories with positive initial conditions converge
to a global atractor (shown in blue). The atractor is in the plane e 1 R 1 =
e 2 R 2 (shown in cyan) and it is formed by the solutions of differential equation
(
dR 1
r1 λ 2 + r 2 λ 1
= R 1 − λ )
1λ 2
C
dt
λ 1 + λ 2 λ 1 + λ 2
dC
= C(e 1 λ 1 R 1 − m)
dt
that satisfy C(t)≥ r 1−r 2
λ 1
(i. e., they are above the dashed line). Preferences of
the predator on the attractor satisfy
u 1 = r 1− r 2 + λ 2 C
(λ 1 + λ 2 )C
C
R 1
R 2