Two types of permanence of a stochastic mutualism model
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RESEARCH
Open Access
Two types of permanence of a stochastic mutualism model Hong Qiu1,2 , Jingliang Lv1* and Ke Wang1,3 *
Correspondence: [email protected] 1 Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, P.R. China Full list of author information is available at the end of the article
Abstract A stochastic mutualism model is proposed and investigated in this paper. We show that there is a unique solution to the model for any positive initial value. Moreover, we show that the solution is stochastically bounded, uniformly continuous and globally attractive. Under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean. Finally, we introduce some figures to illustrate our main results. Keywords: global attractivity; stochastic permanence; persistence in mean; extinction
1 Introduction Population systems have long been an important theme in mathematical biology due to their universal existence and importance. As far as mutualism system is concerned, lots of proofs have been found in many types of communities. Mutualism occurs when one species provides some benefit in exchange for some benefit. One of the simplest models is the classical Lotka-Volterra two-species mutualism model, which reads ⎧ ⎨ dN (t) = N (t)[a – b N (t) + c N (t)], dt ⎩ dN (t) = N (t)[a – b N (t) + c N (t)].
()
dt
There are many excellent results on the two-species mutualism model (). It is well known that in nature, with the restriction of resources, it is impossible for one species to survive if its density is too high. Thus the above model is not so good in describing the mutualism of two species (see []). Gopalsamy [] proposed the mutualism model as follows: ⎧ ⎨ dN (t) = r (t)N (t)[ K (t)+α (t)N (t) – N (t)], dt +N (t) ⎩ dN (t) = r (t)N (t)[ K (t)+α (t)N (t) – N (t)], dt
()
+N (t)
where N (t) and N (t) denote population densities of each species at time t, ri denotes the intrinsic growth rate of species Ni and αi > Ki , i = , . The carrying capacity of species Ni is Ki in the absence of other species, while with the help of the other species, the carrying capacity becomes (Ki (t) + αi (t)N–i (t))/( + N–i (t)), i = , . It is assumed that the coefficients of the system are all continuous and bounded. Li and Xu [] obtained sufficient conditions for the existence of positive periodic solutions. Chen and You [] gave the sufficient conditions for the permanence of the model. Chen et al. [] considered the permanence of a © 2013 Qiu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Qiu et al. Advances in Difference Equations 2013, 2013:37 http://www.advancesindifferenceequations.com/content/2013/1/37
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delayed discrete mutualism model with feedback control
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