In this paper, the bifurcation control of a fractional-order mosaic virus infection model for Jatropha curcas with farming awareness and an execution delay is investigated. By analyzing the associated characteristic equation, Hopf bifurcation induced by the execution delay is studied for the uncontrolled system. Then, a time-delayed controller is introduced to control the occurrence of Hopf bifurcation. Our study implies that bifurcation dynamics is significantly affected by the change of the fractional order, the feedback gain and the extended feedback delay provided that the other parameters are fixed. A series of numerical simulations is performed, which not only verifies our theoretical results but also reveals some specific features. Numerically, we find that the Hopf bifurcation gradually occurs in advance with the increase of the fractional order, and there exist extreme points for the feedback gain and the extended feedback delay which can minimize the bifurcation value.

In this paper, mathematical models for the management of biological resources based on a given predator-prey relationship are proposed, and two types of control strategies, unilateral and bilateral control with impulsive state feedback, are studied. The existence of the order-1 homoclinic orbit, order-1 periodic solution, and bifurcation of homoclinic of the unilateral control system are obtained, and the attraction region of this system is also discussed. Besides, sufficient conditions for the existence and stability of the order-1 and order-2 periodic solutions of the bilateral control system are also gained. A series of numerical simulations including bifurcation diagrams of periodic solution are performed, which not only verify the theoretical results we get but also reveal some peculiar dynamical phenomena, such as the appearance of high-order periodic solutions and existence of parameter intervals with drastic order change of periodic solution. By comparing the two management strategies, our study encourages bilateral control rather than unilateral control for the risk of predator extinction.

To study the biological control strategy of aphids, in this paper we propose host-parasitoid-predator models for the interactions among aphids, parasitic wasps and aphidophagous Coccinellids incorporating impulsive releases of Coccinellids, and then study the long-term control and limited time optimal control of aphids by adjusting release amount and release timing of Coccinellids. For the long-term control, the existence and stability of the aphid-eradication periodic solution are investigated and threshold conditions about the release amount and release period to ensure the ultimate extinction of the aphid population are obtained. For the limited-time control, three different optimal impulsive control problems are studied. A time rescaling technique and an optimization algorithm based on gradient are applied, and the optimal release amounts and timings of natural enemies are gained. Our simulations indicate that in the limited-time control, the optimal selection of release timing should be given higher priority compared with the release amount.

To study the finite-time control of plasma glucose for diabetic patients with impulsive injections of insulin, we propose an impulsive differential equation model with initial and boundary conditions. The goal of glucose control is supposed to be achieved if the system has a solution, otherwise the goal cannot be achieved. By constructing two comparison systems and using a comparison principle, several conditions under which the system has a solution are obtained. Furthermore, some numerical simulations are given. The results show that a relatively higher initial insulin level is beneficial for the glucose control. For a predefined finite time, injection dose and injection period are two important adjustable factors which can guarantee the achievement of the control goal.