Bifurcation analysis of an ecological model with nonlinear state-dependent feedback control by Poincaré map defined in phase set
In this paper, a predator-prey model with state-dependent pulse interventions, including spraying insecticides and releasing natural enemies, in which the implementation of control measures depends on whether the weighted value of pest density and its change rate reaches the action threshold is proposed to develop qualitative analysis techniques. First, the threshold condition for the existence and stability of the boundary periodic solutions is discussed and then define a one-parameter family of discrete (Poincaré) maps. By analyzing the properties of those discrete maps, it is concluded that if there is no interior equilibrium for the system without control measures and the releasing amount of natural enemies is greater than zero, then there is at least one positive periodic solution. While if the system without control measures has an internal equilibrium and only the chemical tactic is applied, then there could be an unstable positive periodic solution near the boundary periodic solution. As a result, backward bifurcation and bi-stability occur. The analytical techniques developed here could be applied to analyze more generalized models and other fields, including infectious disease control. This paper aims to investigate the periodic solutions and the bifurcations associated with the control strategy for the proposed ecological model (\textit{Lotka-Volterra predator-prey model with state-dependent pulse and the signal that triggers pest control action is a convex combination of size and growth speed of the pest population}) (1.2). In Section 2, the relevant definitions and notations of planar impulsive semi-dynamic system and the dynamical behavior of the ODE model, Model (1.2) without control measures, are summarized. In Section 3, the dynamics of impulsive Model (1.2) are discussed when the ODE model has no positive equilibrium, including the exact ranges of impulsive set and phase set and the existence and stability of the boundary periodic solution. In Section 3.2, a Poincaré map for the proposed system is established, and by studying the properties of that map, the existence of an order-1 internal periodic solution is obtained. Furthermore, in Section 4, when the ODE model has a positive equilibrium, the bifurcation of a one-parameter family of discrete maps is analyzed with respect to the key parameters near the trivial fixed point. And the existence and stability of a positive fixed point is obtained, which is the positive periodic solution of the proposed impulsive System (1.2). To sum up the whole paper, the detailed conclusions and discussions are given in the last section.
scientific article; zbMATH DE number 7474623
scientific article