A fusion plasma is modeled by a one-dimensional Vlasov–Maxwell system of equations. Particle annihilation is considered by a dedicated factor in the Vlasov equation. A term describing background damping mechanisms is considered in the equation relating the electric field to the current (Berk–Breizman model). Results are presented for simulations in a one-dimensional velocity space, the extension by a further velocity dimension is sketched. As an initial condition, a bump-on-tail distribution is assumed. The time evolution of the system is studied and characterized with respect to different points in parameter space. A fractional steps method using cubic splines interpolation is applied for time-integrating this system.