Bifurcations and Control in a Singular Biological Economic Model

Abstract

In the natural world, there are many species whose individuals have a life history that takes them through two stages, juvenile stage and adult stage . Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.Before 1990, the stage structure has attracted some attention from several authors [6,20,21,23,24,31,48,53,29]. The well-known biological system of single species with stage structure is established by O. Aiello and H. Freedman [1], which represents a milestone in this research field. In the following decades, much research efforts are put into the investigation of biological dynamical systems with stage structure for population. Some related research results can be found in [1,2,3,10,14,26,8,39,34,35,36,37,38,40,44,41,42,43,45,47,51,52,54,25,16,9] and the references therein.Recently, the dynamics of a class of stage-structured prey-predator models with discrete time delay have been studied by several authors [4,55,18,58,5] and the references therein.Arino et al. [4] suggested that the time delay to adulthood should be state dependent and careful formulation of such state dependent time delays can lead to models that produce periodic solutions. Xu et al. [55] studied the persistence and stability of a delayed prey-predator model with stage structure for predator. Gourley et al. [18] formulated a general and robust prey predator model with stage-structure with constant maturation time delay and performed a systematic mathematical and computational Q. Zhang et al.: Complex

Dynamics of a predator-prey ecological system with nonlinear harvesting rate

“…[5][6][7][8] introduce a class of predator-prey ecological economic systems which are described by differential-algebraic equations. They systematically study the issues of singularity induced bifurcation [6] , state feedback control [6] , saddle-node bifurcation [6][7][8] , etc. Recently, Ref.…”

The dynamics and bifurcation control of a singular biological economic model

“…In [7,8], passivity, optimal control, and dynamics analysis were studied, respectively. A class of state feedback controller is presented in [9,10]. In this paper, for (3), using the economic theory of fishery resource [11] , we study the following singular biological economic model:…”

“…Compared with the controllers in [9,10], here we present a class of feedback controllers that has a more generalized form and is more close to the control strategies existing in practice.…”

Hopf bifurcation for a predator–prey biological economic system with Holling type II functional response

“…But then, unreasonable exploitation of biological resources might lead to unfavorable influence on ecological balance. So there has been rapidly growing interest in the analysis and modelling of predator-prey systems [5][6][7][8][9][10][11][12][13][14][15]. Many authors [7][8][9][10][11] have studied the dynamics of predator-prey models with harvesting, and obtained complex dynamic behaviors, such as Hopf bifurcation, direction of periodic solutions bifurcating from Hopf bifurcation, flip www bifurcation, Bogdanov-Takens bifurcation, limit cycle and so on.…”

“…In daily life, economic profit is a very important factor for governments, merchants and even every citizen, so it is necessary to research biological economic systems. Recently, Zhang et al [12][13][14][15] have proposed a class of predator-prey biological economic models which are described by differential-algebraic equations or differential-difference-algebraic equations. Inspired by the modelling method of Zhang et al [12][13][14][15], we gain a biological economic model with Holling type II functional response.…”