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   . 2012 Apr;9(2):431-44.
   doi: 10.3934/mbe.2012.9.431.

A mutualism-parasitism system modeling host and parasite with mutualism at low density

   [39]Yuanshi Wang^ [40] 1 , [41]Donald L Deangelis
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     * ^1 School of Mathematics and Computational Science, Sun Yat-sen University,
       Guangzhou, China. mcswys@mail.sysu.edu.cn

     * PMID: 22901072
     * DOI: [42]10.3934/mbe.2012.9.431

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A mutualism-parasitism system modeling host and parasite with mutualism at low density

   Yuanshi Wang et al. Math Biosci Eng. 2012 Apr.
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   . 2012 Apr;9(2):431-44.
   doi: 10.3934/mbe.2012.9.431.

Authors

   [46]Yuanshi Wang^ [47] 1 , [48]Donald L Deangelis

Affiliation

     * ^1 School of Mathematics and Computational Science, Sun Yat-sen University,
       Guangzhou, China. mcswys@mail.sysu.edu.cn

     * PMID: 22901072
     * DOI: [49]10.3934/mbe.2012.9.431

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Abstract

   A mutualism-parasitism system of two species is considered, where mutualism is
   the dominant interaction when the predators (parasites) are at low density while
   parasitism is dominant when the predators are at high density. Our aim is to show
   that mutualism at low density promotes coexistence of the species and leads to
   high production of the prey (host). The mutualism-parasitism system presented
   here is a combination of the Lotka-Volterra cooperative model and Lotka-Volterra
   predator-prey model. By comparing dynamics of this system with those of the
   Lotka-Volterra predator-prey model, we present the mechanisms by which the
   mutualism improves the coexistence of the species and production of the prey.
   Then the parameter space is divided into six regions, which correspond to the
   four outcomes of mutualism, commensalism, predation/parasitism and neutralism,
   respectively. When the parameters are varied continuously among the six regions,
   it is shown that the interaction outcomes of the system transition smoothly among
   the four outcomes. By comparing the dynamics of the specific system with those of
   the Lotka-Volterra cooperative model, we show that the parasitism at high density
   promotes stability of the system. A novel aspect of this paper is the simplicity
   of the model, which allows rigorous and thorough analysis and transparency of the
   results.

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