In this paper, research of a class of state feedback control model which is mainly used in crop pests management, with Bendixson-Dulac discriminance, proves that this model has an unique and globally stable positive equilibrium under the weak time-delay kernel function. Also, we adopt the subsequent function method in the ordinary differential equation of the geometric theory to prove that a sufficient condition holds for the existence of an order one period solution in the system. At the same time, it also proves that the periodic solution is asymptotically stable.
This paper deals with a diffusive toxin producing phytoplankton‐zooplankton model with maturation delay. By analyzing eigenvalues of the characteristic equation associated with delay parameter, the stability of the positive equilibrium and the existence of Hopf bifurcation are studied. Explicit results are derived for the properties of bifurcating periodic solutions by means of the normal form theory and the center manifold reduction for partial functional differential equations. Numerical simulations not only agree with the theoretical analysis but also exhibit the complex behaviors such as the period‐3, 5, 6, 7, 8, 11, and 12 solutions, cascade of period‐doubling bifurcation in period‐2, 4, quasi‐periodic solutions, and chaos. The key observation is that time delay may control harmful algae blooms (HABs). Moreover, numerical simulations show that the chaotic states induced by the period‐doubling bifurcation are purely temporal, which is stationary in space and oscillatory in time. The investigations may provide some new insights on harmful phytoplankton blooms.
<abstract><p>Unlike conventional methods of pests control, introducing in an appropriate mathematical model can contribute a batter performance on pests control with higher efficiency while lest damage to ecosystem. To fill the research gap on plant root pest control, we propose a plant root pest management model with state pulse feedback control. Firstly, the stability of the equilibrium point of the model (1.3) is analyzed by using the linear approximate equation, given that the only positive equilibrium point of model (1.3) is globally asymptotically stable. Moreover, the existence and uniqueness of order 1 periodic solutions of model (1.3) are discussed in detail according to the geometric theory of semi-continuous dynamical systems, successor functions method and the qualitative theory of differential equations. Finally, with further analysis in different methods, the asymptotic stability of the order 1 periodic solution of model (1.3) is obtained by using Similar Poincare Criterion or interval set theorem. The results show that this model can effectively control the number of pests below the economic level of damage.</p></abstract>
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