Abstract
This paper brings up the idea of a biological economic system with time delay in a polluted environment. Firstly, by proper linear transformation and parametric method, the singular time-delay systems are transformed to differential time-delay systems. Then, using center manifold theory and Poincare normal form method, the direction of Hopf bifurcation and the stability and period of its periodic orbits are analysed. At last, we have performed numerical simulation to support the analytical results.
Highlights
Environmental pollution has been increasingly influencing the biological systems
When τ = 0.2 < 0.6043, the dynamical responses of system (70) are shown by Figure 1, system (70) is stable at P, and the population and economic profits develop sustainably in this case; when τ = 1 > 0.6043, the dynamical responses of system (70) are shown by Figure 2, system (70) is unstable, and the population and economic profits cannot develop sustainably in this case; Figure 3 shows that the dynamic behavior of the population changes with time delay and the Hopf bifurcation exists when τ > 0.6043
Based on the mathematical biology theory, the Hopf bifurcation theory of differential system, and the singular system theory, this paper considers a singular biological economic system with time delay in a polluted environment
Summary
Environmental pollution has been increasingly influencing the biological systems. In order to investigate the development and dynamics of population of the biological systems, it is necessary to consider the factor of pollution when establishing a mathematical model. Being always an important mathematical means to investigate the bifurcation problems with parameter and the qualitative theory of differential equations, more attention has been paid to the Poincare normal form method for a long time, home and abroad. In [12], the author lays the foundation of the center manifold standard method by combining the normal form theory and the center manifold theorem, and the method was used on the investigation of Hopf bifurcation When it comes to related properties of the Hopf bifurcation, the center manifold standard method is usually used to reduce the dimension of high-dimension system, which isolates the asymptotic behaviors of complex systems, so that we can investigate the original system in a center. This paper takes a singular biological economy system with timedelay in a polluted environment and analyses it using the stability theory of singular system, the theory of economic system, the theory of the Hopf bifurcation of delay differential system, and so forth
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