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

Abstract 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 $$x_r$$ and deactivated otherwise, giving rise to a two-dimensional piecewise-smooth differential system whose switching boundary is $$x = x_r$$ . Assuming constant growth rates, with $$g(x) = a \in \mathbb {R}^+$$ for the prey and $$f(x) = f \in \mathbb {R}^+$$ 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 $$\mu = f - d$$ (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 $$x = x_r.$$ Moreover it is proven that the system does not exhibit limit cycles. Finally numerical simulations illustrate the distinct global configurations.