具避难所和毒素作用的捕食系统动力学行为

研究一类有避难所、毒素作用和外界捕获影响的捕食系统,通过构造恰当的Dulac甬数,证明系统在第一象限没有极限环,从而证明了平衡点的全局稳定性;分析了避难所、毒素、捕获作用对该系统最终种群数量的影响,发现:①当捕食者的死亡率大或其被捕获得多时,两种群的绝灭与避难所的存在无关;当捕食者的死亡率小、被捕获得多且避难所足够小时,随着食饵避难所的增加,捕食者种群数量也随之增加;②当毒素d_2=0时,毒素d_1的增加对食饵种群最终数量没有影响,而使捕食者种群数量下降.

Dynamic behaviors of a predator model incoporating refuge and toxicity

A predator model with refuge and toxicity is studied. By constructing a suitable Dulac function, it is shown that there is no limit cycle in the first quadrant, thus the globally stability of the equilibrium is proved. Further more, the paper analyzes the influence of refuge, toxicity and harvesting. It is amazing that ① if the death rate of predator is big or the harvesting effort is big,the existence of the predator is unrelated with refuge; if the death rate of predator and the harves-ting effort are small, and the refuge is small enough, the population of both species is increased as the increase of refuge; ② if d_2=0, the increase of d_1 has no influence on the prey population, while the population of predator is decreased.