Dynamical Behavior Of Stochastic Type K Monotone Lotka-Volterra Systems

In 1986,Smith studied general Kolmogorov-type models of competition between sub-communities.He considered models for which,after suitable permutations of species indices,the community consists of two disjoint complementary sub-communities.The interactions between every pair of species in same sub-community are mutualistic.On the other hand,the interaction between species in different sub-community is competitive.Smith called such a system a type K monotone system.In fact,the classical and deterministic type K monotone systems have been widely studied because of the significance of theory and practice.However,in the real ecosystem,biological growth process will inevitably be affected by various forms of random perturbations.Ignoring the randomness of the system may lead to deviations in description and prediction of system behavior.It's therefore important to introduce the influence of environmental noise on parameters in the population systems.In this paper,two stochastic type K monotone Lotka-Volterra systems are proposed and investigated.For non-autonomous system,firstly we show a sufficient condition for that there is a unique positive solution to the model,and prove the ultimate boundedness in mean of the system on this condition.Moreover,sufficient conditions for stochastic permanence and global attractivity are established by applying It?'s formula and constructing auxiliary functions.In addition,by virtue of the knowledge of stochastic differential equations such as Burkholder-Davis-Gundy's inequality,Chebyshev's inequality and Borel-Cantelli lemma,the growth rates of the positive solution are investigated.For autonomous system,we first separate the situation to discuss the non-persistence on average of each species according to the characteristics of the type K monotone matrix.Then we analysis how the environmental noise of the outside world will affect the development of populations in the system.The results reveal that,the stochastic noise of one population is unfavorable for the persistence of all species in the system.The corresponding numerical examples are also given to verify the correctness of the theoretical results.