I-optimal curve for impulsive Lotka—Volterra predator-prey model

For the classical Lotka—Volterra predator-prey system, new notion I-optimal curve ξI is introduced. This curve is disposed in the phase space of the system. The curve ξI intersects each trajectory γc of Lotka—Volterra system at least once. The points of ξI possess the following optimal property: if (m, M) ξI ∩ γc0, then after a “jump” with magnitude I to the origin of coordinates, it hits a trajectory γc1 and c1 is minimal; i.e., γc1 is the “nearest” to the stable centre. The minimality concerns the rest points of initial trajectory γc0, from which the “impulsive jumps” (subtractings) with magnitude I to (0,0) are realized. The monotonicity, continuity, and linear asymptotical behaviour of ξI curve are proved.