Permanence of a discrete amensalism system with refuge and saturation effect
A discrete amensalism system with refuge and saturation effects is proposed,and a set of sufficient conditions to ensure the permanence of the system are obtained.Research results showed that the refuge and the second population played an important role in the permanence of the system.Under the condition of system permanence,there may be a unique globally stable positive equilibrium and there may also be periodic positive solutions;There is a threshold k0.If k0<0,the system is permanence and the numerical simulation shows that the refuge will not affect the permanence of the system,but with the increase of the refuge,the upper and lower bounds of the permanence domain of the first population gradually increase.If k0>0and k as tends to k0,the lower bound of the permanence of the first population also tends to 0,which means that when k tends to k0,the population number is small,which means that the anti-interference ability of the population is weakened and the risk of extinction is easy to occur;Although the system is permanence,the dynamics of the first population is complex and has many possibilities,when k>k0.The dynamic behavior of the discrete system is much more complex than that of the corresponding continuous system.