Phase Portraits of a Lotka-Volterra Model with Discontinuous Harvesting Action

In this work we study a Lotka–Volterra predator-prey system in which the predator population is subjected to a discontinuous harvesting action that depends on the prey abundance. In this case the harvesting is activated when the prey level falls below a critical value xᵣ and deactivated otherwise, giving rise to a two-dimensional piecewise-smooth differential system whose switching boundary is x = xᵣ . Assuming constant growth rates, with g(x) = a ∈ R⁺ for the prey and f(x) = f ∈ R⁺ for the predator, and a Holling type II functional response, we perform a complete analysis of the phase portraits in the Poincaré disc. The qualitative behavior of the trajectories depends on the parameters a and μ = f - d (where $$d>0$$ is the predator mortality). The adopted harvesting strategy does not allow the stabilization of the prey and predator populations at the desired value x = xᵣ. Moreover it is proven that the system does not exhibit limit cycles. Finally numerical simulations illustrate the distinct global configurations.