生态环境系统中一类相互作用生态种群的动态优化与控制(英文)

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AbstractEcological environment system is a complex information system requiring the interdisciplinary employment of sciences of biology， environment， information， mathematics as well as the intensive study of computer science. The study of decision on macro optimistic control of ecological system has in recent years become a major subject that mathematics and ecologists in and out of China explore in depth. Based on the needs of the current population ecology research to the development of macroscopic and microscopic， extending and the number of population ecology of complex system modeling， by discussing the analyticity and macro optimistic control of two kinds of interacting ecological population models with competitive mechanism and local stability， this paper makes more rigorous the control of ecological environment system and provides valuable methods for analytical analysis and control optimization of a kind of ecological system. It not only helps significantly with the modeling and analyzing of two kinds of competing and reciprocal ecological system， but also guides the analytical analysis and macro-control of more complex ecological environment system.

Key wordsEcological population model；Isocline equation；Density restraint；Analytical analysis；Optimization and control

CLC numberO 175Document codeA

The stability， bound and existence of limit cycle solution of the interactive kolmogorov model of two kinds of groups， namely，

2The Bound of the Kolmogorov Model

The two-species competition and predator-prey model

（1） F12< 0 （The bait is restrained by the hunter）

（2） F21> 0 （The hunter gets the provision given by the bait）

（3） When x2= 0， F11< 0 （Without the hunter， the bait is of density restraint）

（4） F22< 0 （The hunter’s increasing is of density restraint）

（5） There is a constant number K > 0 which makes F1（0，K） = 0 （K is the hunter’s above critical density without the bait）

（6） There is a constant number L > 0 which makes F1（L，0） = 0 （L is the bait loaded capacity without the hunter）

（7） There is a constant number M > 0 which makes F2（M，0） = 0 （M is the bait below critical density without the hunter）

Finally， let’s suppose that the hunter’s increasing only relies on the provision of the bait， then，

Therefore， the trend that the integral curve in EN runs through is shown in Figure 3.

3The Existence of the Limit Cycle of the Kolmogorov Model

In order to study the existence of the limit cycle of the kolmogorov model （1）， let’s suppose the following：

（9） L > M.

（10） Through the isocline equation F1（x1，x2） = 0 of the prey， we can solve that x2 is written as a function by x1 and the solution recorded as x2 = f（x1） is the only one， f is defined in the region [0，L] and f is continuously differentiable and monotone decreasing， f（0） = K and f（L） = 0 exist.

（11） The hunter’s isocline equation F2（x1，x2） = 0 can work out the only one x1 function shown by x2marked x1= g（x2）， the definition of g is in the region [0，∞]， and it is monotone increasing and continuously differentiable function， moreover， g（0） = M exists[11].

The above suppositions are the kolmogorov original conditions， but the following condition （9′） presented by Rosenzwoig in 1972 can be used to substitute them.

（9′） Suppose the isocline of the hunter and the prey can be shown in Figure 4， the point singularity plies at the rising part of the prey isocline.

4Prospect

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