In this paper, we consider a differential–algebraic biological economic system with time delay and harvesting where the dynamics is logistic with the carrying capacity proportional to prey population. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the differential–algebraic biological economic system based on the normal form approach and the center manifold theory. At last, numerical simulations are performed to illustrate the analytical results.
In recent years, great attention has been paid to the management of renewable resources. The reason is that mankind is facing the problems about shortage of resource and worsening environment. Concerning the conservation for the long-term benefits of humanity, there is a wide-range of interest in analysis and modelling of biological systems, especially on predator–prey systems with or without time delay. The inclusion of delays in these systems has illustrated more complicated and richer dynamics in terms of stability, bifurcation, periodic solutions and so on. For references see [1], [2], [3], [4], [5], [6], [7], [8], [9] and [10]. Most of these discussions on biological models are based on normal systems governed by differential equations or difference equations.
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, which can be described by differential–algebraic equations or differential–difference–algebraic equations. In 1954, Gordon [11] studied the effect of harvest effort on ecosystem from an economic perspective and proposed the following economic theory:
Net Economic Revenue (NER) = Total Revenue (TR) − Total Cost (TC).
This provides theoretical evidence for the establishment of differential algebraic equation. Recently, based on the economic theory as mentioned above, Zhang et al. [12] and [13] have established a class of differential–algebraic biological economic models and they investigate the dynamical behavior of the system with harvesting on predator. However, they do not derive the formula for determining the properties of Hopf bifurcation of the differential–algebraic biological economic models. In this paper, we investigate the stability and direction of the Hopf bifurcation of the differential–algebraic biological economic model system with time delay and harvesting on predator. In what follows, we introduce the new biological economic system.
The basic model we consider is based on the following ratio-dependent predator–prey system with harvest
where r1,r2r1,r2 represent growth rate of the prey and predator, respectively, εε and θ1θ1 are positive constants, and ττ denotes the delay time for the prey density, x(t)x(t) and y(t)y(t) can be interpreted as the densities of prey and predator populations at time t, respectively, E(t)E(t) represents harvesting effort, yE(t)yE(t) indicate that the harvests for prey and predator population are proportional to their densities at time t. When there is no harvesting was considered by Çelik [7] in detail. In this model, predator density is logistic with time delay and the carrying capacity proportional to prey density. Notice that, in model (1) the carrying capacity is not constant but proportional to prey density that changes the dynamical structure in different aspects.
According to Gordon [10] economic theory equation as mentioned above, we let NER = m, TR=pE(t)y(t)TR=pE(t)y(t) and TC=cE(t)TC=cE(t), where p>0p>0 is harvesting reward per unit harvesting effort for unit weight predator, c>0c>0 is harvesting cost per unit harvesting effort for predator, m>0m>0 is the economic profit, then, we can obtain a biological economic system expressed by differential–algebraic equation as follows:
The organization of this paper is as follows: regarding ττ as bifurcation parameter, we study the stability of the equilibrium point of the system (2) and Hopf bifurcation of the positive equilibrium depending on ττ where we show that the positive equilibrium loses its stability and the system (2) exhibits Hopf bifurcation in the second section. Then, based on the new normal form of the differential–algebraic system introduced by Chen et al. [15] and the normal form approach theory and center manifold theory introduced by Hassard et al. [18], we derive the formula for determining the properties of Hopf bifurcation of the system in the third section. Numerical simulations aimed at justifying the theoretical analysis will be reported in Section 4. Finally, this paper ends with a discussion.
Nowadays, biological resources in the predator–prey system are mostly harvested and sold with the purpose of achieving the economic profit, and economic profit is a very important factor for governments, merchants and even every citizen, so it is necessary to research biological economic systems, which motivates the introduction of harvesting in the predator–prey system, and these systems can be described by differential–algebraic equations or differential-difference-algebraic equations. In fact, as was pointed by Kuang [17] that any model of species dynamics without delays is an approximation at best. So, in this paper, we provide a new and efficacious method for the qualitative analysis of the Hopf bifurcation of a differential–algebraic biological economic system with time delay, it is believed that the method may be extended to analyze some other more complicated algebraic models.
From the above results, we can conclude that the stability properties of the system could switch with parameter ττ that is incorporated on the time delay on predator density in the differential–algebraic biological economic system (2). From an economic perspective, the persistence and sustainable development of the predator–prey ecosystem will be very important, so with the purpose of maintaining the sustainable development of the biological resources in practice and application, and in future works, the following topics about the nonlinear dynamics of differential–algebraic biological economic system will be discussed:
(i)
More precise mathematic or physical architectures of the differential–algebraic biological economic system are proposed, demonstrating a mature strategy rather than a concept.
(ii)
Analysis dynamic property of differential–algebraic biological economic system in practice or from experimental point of views.
