Dynamic Properties of a Differential-Algebraic Biological Economic System
We analyze a differential-algebraic biological economic system with time delay. The model has two different Holling functional responses. By considering time delay as bifurcation parameter, we find that there exists stability switches when delay varies, and the Hopf bifurcation occurs when delay passes through a sequence of critical values. Furthermore, we also consider the stability and direction of the Hopf bifurcation by applying the normal form theory and the center manifold theorem. Finally, using Matlab software, we do some numerical simulations to illustrate the effectiveness of our results.
Recently, dynamics of differential-algebraic equations have become a subject of intense research activities because differential-algebraic equation models can describe many kinds of practical systems such as power system, biological economic system, and singular prey-predator economic model [1, 2]. The dynamic relationship between predators and their preys with harvesting has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Harvesting has a strong impact on the dynamic evolution of a population. Depending on the nature of applied harvesting strategy, the long-run stationary density of the population may be significantly smaller than the long-run stationary density of a population in the absence of harvesting. Time delays of one type or another have been incorporated into mathematical models of population dynamics due to maturation time, capturing time, or other reasons. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the populations to fluctuate. Recently, a number of researches have been devoted to the dynamic properties for delayed predator-prey systems with harvesting. For further details, refer to [3–7].
Kar and Pahari  have investigated the dynamical behavior of an exploited system consisting of a prey and a predator that are harvested: where and respect prey and predator population densities at time and is the growth rate of prey when is zero. is the response function, is the death rate of and is the conversion factor. and represent harvesting of and .
In system (1.1) and have the same Holling function. But in real life they may have different adaptation for the environment. So we formulate the aforementioned problem with different Holling functions as follows:
In 1954, Gordon analyzed 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).
Based on the previous economic theory, we let represent the harvest effort for at time . and . represents harvesting reward per unit harvesting effort for unit weight of prey. represents the cost per unit harvest effort for prey. Then a differential-algebraic model which consists of two differential equations and an algebraic equation can be established as follows: Define From the standpoint of biology, we are only interested in the dynamics of model (1.3) in the region
2. Stability Analysis
Firstly, we consider the location and number of the equilibria of model (1.3) in . The interior equilibrium point is a root of the following equation: It is obvious that (2.1) has an only real solution where and , . In order to analyze the local stability of the positive equilibrium point for the system (1.3) we use the linear transformation Then we have for which system (1.3) yields
Now we derive the formula for determining the properties of the positive equilibrium point of the system (2.5). Firstly, we consider the local parametric  , which is defined as follows: where , , , and , . Introducing perturbations and defining , , and , then we can obtain the parametric system of system (2.5) as follows: Neglecting 2nd- and higher-order products of and , we get The characteristic equation of system (2.8) at takes the form
First of all, we know that the roots of (2.9) with have always negative real parts when , where
Next, we will investigate the distribution of the roots of (2.9). If is a root of (2.9), then we have Separating the real and imaginary parts, we have which lead to When and hold, we can get that (2.13) has two positive roots and .
According to the analysis previous, we have the following lemma 2.1:
Differentiating the two sides of (2.9) and noticing that is a function of , we have
and then we can obtain
Therefore, we can obtain the following results about the stability of the positive equilibrium and the Hopf bifurcation of (2.7).
Theorem 2.2. Assume that . (1)If and are satisfied, then there exists an integer such that the zero solution of system (2.7) is asymptotically stable when and unstable when .(2)When , there are Hopf bifurcations at the equilibrium point of system (2.7).
3. Direction and the Stability of Hopf Bifurcation
In the previous section, we have already obtained the conditions which ensure that model (2.7) undergoes the Hopf bifurcation at the critical values . In this section, we will study the direction and stability of the Hopf bifurcation based on the normal form approach theory and center manifold theory introduced by Hassard et al. .
First of all, by the transformation , , , , the parametric system (2.7) is equivalent to the following Functional Differential Equation (FDE) system in where , for , and . Let . Then with By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions such that In fact, we can choose where for . Define Then the system (3.1) is equivalent to for , the adjoint operator of is and a bilinear inner product is given by
By the discussion in Section 2, we know that are eigenvalues of . Thus, they are also eigenvalues of . It is easy to calculate that where Let be the solution of (3.9) when . Define On the center manifold , we have Note that is real. When we have Then where From (3.9) and (3.16), we have Rewrite where We can obtain Now we have , , , and : where
At last, we can calculate
Now we have a conclusion.
Conclusion. (1) The sign of determines the direction of Hopf bifurcation. When , the Hopf bifurcation is supercritical; when , the Hopf bifurcation is subcritical.
(2) determines the stability of bifurcated periodic solutions. When , the periodic solutions are stable; when , the periodic solutions is unstable.
(3) determines the period of bifurcated periodic solutions. When , the period increases; when , the period decreases.
4. Numerical Simulations
In this section, we use some numerical simulations to illustrate the analytical results we obtained in previous sections.
Let , , , , and . The system (1.3) is
The only positive equilibrium point of (4.1) is . By simple computation we have and . Substituting these parameters into (2.14) gives , , , and . Applying Theorem (2.5), we have that the positive equilibrium point of (4.1) is asymptotically stable when . The results can be illustrated in Figures 1 and 3.
When and is sufficiently near , the bifurcating periodic solution from positive equilibrium point of (4.1) occurs. Furthermore, we consider the properties of the Hopf bifurcation at . Based on Theorem (2.5) and conclusion 1, we get , and . This implies that
We conclude that the bifurcating periodic solution is subcritical and unstable. The results are illustrated in Figure 2.
This paper was supported by the National Natural Science Foundations of China (10871056).
H. S. Gordon, “The economic theory of a common property resource: the fishery,” Journal of Political Economy, vol. 62, no. 2, pp. 124–142, 1954.View at: Google Scholar
B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981.