Abstract
A two species stage-structured commensalism model is proposed and studied in this paper. Local and global stability property of the boundary equilibrium and the positive equilibrium are investigated, respectively. If the stage-structured species is extinct, then depending on the intensity of cooperation, the species may still be extinct or become persistent. If the stage-structured species is permanent, then the final system is always globally asymptotically stable, which means the species is always permanent. Our study shows that increasing the intensity of the cooperation between the species is one of very useful methods to avoid extinction of the endangered species. Such a finding may be useful in protecting the endangered species. An example together with its numeric simulations is presented to verify our main results.
Highlights
1 Introduction The aim of this paper is to investigate the dynamic behaviors of the following stagestructured commensalism system: dx1 dt
Chen et al [34] proposed and studied the following May type stage-structured cooperation model: x 1 (t) b1e–d11τ1 x1(t a11x21(t) c1 + f1x2(t) a12x21(t), y1(t) = b1x1(t) – d11y1(t) – b1e–d11τ1 x1(t – τ1), x 2 (t) b2e–d22τ2 x2(t a22x22(t) c2 + f2x1(t) a21x22(t), y2(t) = b2x2(t) – d22y2(t) – b2e–d22τ2 x2(t – τ2). They showed that with introduction of the stage structure, the May type cooperative system may admit partial survival property, that is, despite the cooperation between the species, the species may still be driven to extinction due to the stage structure
5 Conclusion Recently, many scholars investigated the dynamic behaviors of the mutualism and commensalism model [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]
Summary
The aim of this paper is to investigate the dynamic behaviors of the following stagestructured commensalism system: dx dt. Aiello and Freedman [42] for the first time proposed the following stage-structured single species model: αx2(t). They showed that system (1.8) admits a unique positive equilibrium which is globally asymptotically stable. Under assumption (2.12), three characteristic roots of matrix are all negative; A1(0, is locally asymptotically stable. This ends the proof of Theorem 2.2. A4(x∗1∗, x∗2∗, y∗∗) is locally asymptotically stable This ends the proof of Theorem 2.4. Theorem 2.4 shows that if the positive equilibrium exists, it is locally asymptotically stable
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