The global stability of coexisting equilibria for three models of mutualism

Georgescu, Paul; Zhang, Hong; Maxin, Daniel

doi:10.3934/mbe.2016.13.101

Cited by 8 publications

(16 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We can also have a combination of the situations shown in the models (9) and (13), in which x 2 reduces the death rate of x 1 and x 1 increases the birth rate of x 2 , which allows for the modelling of interacting species with very different vital parameters, as follows:…”

Section: Models With Asymmetric Mutualism Effectsmentioning

confidence: 99%

“…Here, the assumptions on a 1 and f 1 are those used in the analysis of model (9) while for f 2 and a 2 we consider the assumptions from Equation (13). Again, the bounding theorems presented above hold with very similar proofs and the Dulac function is ϕ(…”

Section: Models With Asymmetric Mutualism Effectsmentioning

confidence: 99%

“…implying that the solutions of model (20) are bounded. Since there is always a unique positive equilibrium of model (20) (see, for instance [13]), this equilibrium is globally stable. The same results hold also for a Holling type II formulation of the saturating functional response [19,31].…”

Section: Models With Saturating Functional Response For Mutualistic Imentioning

confidence: 99%

“…Lyapunov functionals extending those of [30] were used to explore global stability in this case. Georgescu et al [13] further enlarged applicability of the abstract functionals introduced in [12] to a model of mutualism with restricted growth rates proposed in [14] and to the versions of models (1) and (2) with the logistic growth rates replaced by the Richards ones. Despite a variety of proposed models of mutualism, there is still potential for developing others.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Maxin

Georgescu

Sega

et al. 2017

Journal of Biological Dynamics

View full text Add to dashboard Cite

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.

show abstract

Section: Models With Asymmetric Mutualism Effectsmentioning

confidence: 99%

Section: Models With Asymmetric Mutualism Effectsmentioning

confidence: 99%

Section: Models With Saturating Functional Response For Mutualistic Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Maxin

Georgescu

Sega

et al. 2017

Journal of Biological Dynamics

View full text Add to dashboard Cite

show abstract

“…Recently, Georgescu et al (2016) have analyzed the dynamics of three models of mutualism and establishing the global stability via the method of Lyapunov functional. The authors found that the use of higher-order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.…”

Section: Introductionmentioning

confidence: 99%

Global stability of two-species mutualism model with proportional harvesting

Ahmad

2017

Int. j. adv. appl. sci.

View full text Add to dashboard Cite

This paper deals with the study on a mathematical model consisting of mutualistic interactions among two-species with proportional harvesting. Harvesting function is introduced to describe the rate of removal of the species. The local stability analysis shows that the unique positive equilibrium point is asymptotically stable when certain conditions are satisfied. Global stability is discussed by constructing Lyapunov function. It has been shown that the unique positive equilibrium point is globally asymptotically stable. Finally, numerical simulations supporting theoretical results are also included.

show abstract