Yunfei Lv

Hopf bifurcation and global stability of a diffusive Gause-type predator–prey models

Abstract This paper mainly provides Hopf bifurcation formulas for a general Gause type predator–prey system with diffusion and Neumann boundary condition by using the center manifold theory and normal form method, where the spectral and stability analysis around an equilibrium is addressed, and our results can be applied to the case without diffusion. As an application of these results, we give a complete and rigorous analysis of the global dynamics of a diffusive predator–prey model with herd behavior, especially, the Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions.

EFFECT OF HARVESTING AND PREY REFUGE IN A PREY–PREDATOR SYSTEM

Considering that the ecological system is often deeply perturbed by human exploiting activities, this paper deals with a prey–predator model with prey refuge in which both species are independently harvested. First, some sufficient conditions for global stability of equilibria are obtained, and the existence and uniqueness of limit cycles are established. Our results indicate that over-exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population, which is in line with reality. Furthermore, the existence of bionomic equilibrium is discussed. Finally, the influences of prey refuge and harvesting efforts on equilibrium density values are considered and some numerical simulations are given to illustrate our results.

Modeling and analysis of a predator–prey model with state-dependent delay

We propose and study a predator–prey model with state-dependent delay where the prey population is assumed to have an age structure. The state-dependent delay appears due to the mature condition that the prey must spend an amount of time in the immature stage sufficient to accumulate a threshold amount of food. We perform a qualitative analysis of the solutions, which includes studying positivity and boundedness, existence and local stability of equilibria. For the global dynamics of the system, we discuss an attracting region which is determined by solutions, and the region collapses to the interior equilibrium in the constant delay case.

Optimal control problem in an epidemic disease SIS model with stages and delays

Based on literature [J. Q. Li, Z. E. Ma and F. Q. Zhang, Stability analysis for an epidemic model with stage structure, J. Appl. Math. Comput. 9 (2008) 1672–1679], incorporating the recovery of the infected population with the length of the infectious periods, a modified epidemic disease SIS model with delay and stage was investigated. First, the criteria keeping stability with delay were given. Next, in order to lower the level of the infected individuals and minimize the cost of treatment, mixed, early and late therapeutic strategies were introduced into our model, respectively. Then we investigated the existence and uniqueness of optimal controls. And then, we expressed the unique optimal control in terms of the solution of the optimality systems. Finally, by numerical simulations, several important results were acquired: (1) The terminal time influenced the early optimal control largely. In detail, for a shorter terminal time it was optimal to initiate treatment with maximal effort at the start of the...

Multiple kinds of optimal impulse control strategies on plant-pest-predator model with eco-epidemiology

Yongzhen etźal. (2010) describe a mathematical model of a scenario where a plant population is imported to a pest-predator system with an infected pest. Thus a plant-pest-predator eco-epidemiological model disturbed by an impulsive effect is proposed. First of all, the stability conditions of the susceptible pest-eradication periodic solution for eradicating the susceptible pest are investigated. Compared with the results in (Yongzhen etźal., 2010), the presence of the plant population increases the cost of natural enemies as well as the demand for insecticide. In addition, we study the effect of the death rate of the infected pest on pest control in terms of evolution of virulence and the basic reproductive number. Results show that larger mortalities of the infected pest will lead to the frustrated invasion or the instability of susceptible pest-eradication periodic solutions. Next, we focus on the four kinds of optimal impulsive control strategies, biological control, chemical control, and integrated control with fixed period or variable period, to maximize the yields of plants at the terminal time with minimum efforts. All the optimal control problems are solved via a time scaling technique and a gradient-based optimization method. Our results show that two parameters, the amount of sprayed infective pest and the kill fraction of the susceptible pest, play a key role in improving the yield of the plants. In addition, for the four kinds of control strategies, our results also show that biological control is more effective than chemical control to achieve an optimal solution, and the last two strategies can produce higher yields than the first two control strategies.

A complete analysis of the global dynamics of a diffusive predator and toxic prey model

Considering many species can release toxic substances to protect themselves against predators, a diffusive predator and toxic prey system with spatial heterogeneity in predator and prey populations has been investigated. For this system, we give a complete and rigorous analysis of the global dynamics with the boundedness, globally asymptotical stability, transcritical bifurcation, Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions. Meanwhile, we consider the effects of toxins produced by the prey on the dynamic behavior. The consequence of the global stability of the coexistence equilibrium is that the toxins intrinsic characteristic will not change the stability of the system irreversibly. Our results show that the toxin-produced by the prey (phytoplankton) may be used as a bio-control agent for the Harmful Algal Bloom problems.