Volume 34, Issue 4
Stability Analysis of a Lotka-Volterra Commensal Symbiosis Model Involving Allee Effect

Xinyu Guan

Ann. Appl. Math., 34 (2018), pp. 364-375.

Published online: 2022-06

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  • Abstract

In this paper, we present a stability analysis of a Lotka-Volterra commensal symbiosis model subject to Allee effect on the unaffected population which occurs at low population density. By analyzing the Jacobian matrix about the positive equilibrium, we show that the positive equilibrium is locally asymptotically stable. By applying the differential inequality theory, we show that the system is permanent, consequently, the boundary equilibria of the system is unstable. Finally, by using the Dulac criterion, we show that the positive equilibrium is globally stable. Although Allee effect has no influence on the final densities of the predator and prey species, numeric simulations show that the system subject to an Allee effect takes much longer time to reach its stable steady-state solution, in this sense that Allee effect has unstable effect on the system, however, such an effect is controllable. Such an finding is greatly different to that of the predator-prey model.

  • AMS Subject Headings

34C25, 92D25, 34D20, 34D40

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COPYRIGHT: © Global Science Press

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@Article{AAM-34-364, author = {Guan , Xinyu}, title = {Stability Analysis of a Lotka-Volterra Commensal Symbiosis Model Involving Allee Effect}, journal = {Annals of Applied Mathematics}, year = {2022}, volume = {34}, number = {4}, pages = {364--375}, abstract = {

In this paper, we present a stability analysis of a Lotka-Volterra commensal symbiosis model subject to Allee effect on the unaffected population which occurs at low population density. By analyzing the Jacobian matrix about the positive equilibrium, we show that the positive equilibrium is locally asymptotically stable. By applying the differential inequality theory, we show that the system is permanent, consequently, the boundary equilibria of the system is unstable. Finally, by using the Dulac criterion, we show that the positive equilibrium is globally stable. Although Allee effect has no influence on the final densities of the predator and prey species, numeric simulations show that the system subject to an Allee effect takes much longer time to reach its stable steady-state solution, in this sense that Allee effect has unstable effect on the system, however, such an effect is controllable. Such an finding is greatly different to that of the predator-prey model.

}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/20584.html} }
TY - JOUR T1 - Stability Analysis of a Lotka-Volterra Commensal Symbiosis Model Involving Allee Effect AU - Guan , Xinyu JO - Annals of Applied Mathematics VL - 4 SP - 364 EP - 375 PY - 2022 DA - 2022/06 SN - 34 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/aam/20584.html KW - Lotka-Volterra commensal symbiosis model, Allee effect, global stability. AB -

In this paper, we present a stability analysis of a Lotka-Volterra commensal symbiosis model subject to Allee effect on the unaffected population which occurs at low population density. By analyzing the Jacobian matrix about the positive equilibrium, we show that the positive equilibrium is locally asymptotically stable. By applying the differential inequality theory, we show that the system is permanent, consequently, the boundary equilibria of the system is unstable. Finally, by using the Dulac criterion, we show that the positive equilibrium is globally stable. Although Allee effect has no influence on the final densities of the predator and prey species, numeric simulations show that the system subject to an Allee effect takes much longer time to reach its stable steady-state solution, in this sense that Allee effect has unstable effect on the system, however, such an effect is controllable. Such an finding is greatly different to that of the predator-prey model.

Xinyu Guan. (2022). Stability Analysis of a Lotka-Volterra Commensal Symbiosis Model Involving Allee Effect. Annals of Applied Mathematics. 34 (4). 364-375. doi:
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