Dynamic analysis of a phytoplankton-fish model with the impulsive feedback control depending on the fish density and its changing rate

Abstract

<abstract><p>This paper proposes and studies a comprehensive control model that considers fish population density and its current growth rate, providing new ideas for fishing strategies. First, we established a phytoplankton-fish model with state-impulse feedback control based on fish density and rate of change. Secondly, the complex phase sets and impulse sets of the model are divided into three cases, then the Poincar$ \acute{\mbox{e}} $ map of the model is defined and its complex dynamic properties are deeply studied. Furthermore, some necessary and sufficient conditions for the global stability of the fixed point (order-$ 1 $ limit cycle) have been provided even for the Poincar$ \acute{\mbox{e}} $ map. The existence conditions for periodic solutions of order-$ k $($ k \ge 2 $) are discussed, and the influence of dynamic thresholds on system dynamics is shown. Dynamic thresholds depend on fish density and rate of change, i.e., the form of control employed is more in line with the evolution of biological populations than in earlier studies. The analytical method presented in this paper also plays an important role in analyzing impulse models with complex phase sets or impulse sets.</p></abstract>