Lyapunov functions for two-species cooperative systems

Vargas-De-León, Cruz

doi:10.1016/j.amc.2012.08.084

Cited by 20 publications

(8 citation statements)

References 10 publications

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“…It may be shown that, for all endemic equilibrium points that form the upper curve, since * > * 0 , from (18) it results in the fact that Det( ) > 0; thereby, in (16) it follows that both eigenvalues 1 and 2 are negative, and therefore, these equilibrium points are asymptotically stable. On the other hand, it can also be shown that for endemic equilibrium points that are part of the lower curve, since * < * 0 , from (18) it occurs that Det( ) < 0; in this way, again from (16), it is obtained that 1 > 0 and 2 < 0, and consequently such equilibrium points are unstable. Note that in the latter case (in order that * ∈ Δ), quadratic branch is delimited by the point (1, 0) in which, by the way, Det( ) = 0 and therefore 1 < 0 and 2 = 0; that is, the said point is nonhyperbolic and in it a bifurcation could also occur.…”

Section: Equilibria Local Stability and Backward Bifurcationmentioning

confidence: 94%

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Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

Lozano-Ochoa¹,

Camacho²,

Vargas-De-León³

2017

Discrete Dynamics in Nature and Society

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We study an epidemiological mathematical model formulated in terms of an ODE system taking into account both social and nonsocial contagion risks of obesity. Analyzing first the case in which the model presents only the effect due to social contagion and using qualitative methods of the stability analysis, we prove that such system has at the most three equilibrium points, one diseasefree equilibrium and two endemic equilibria, and also that it has no periodic orbits. Particularly, we found that when considering 0 (the basic reproductive number) as a parameter, the system exhibits a backward bifurcation: the disease-free equilibrium is stable when 0 < 1 and unstable when 0 > 1, whereas the two endemic equilibria appear from * 0 (a specific positive value reached by 0 and less than unity), one being asymptotically stable and the other unstable, but for 0 > 1 values, only the former remains inside the feasible region. On the other hand, considering social and nonsocial contagion and following the same methodology, we found that the dynamic of the model is simpler than that described above: it has a unique endemic equilibrium point that is globally asymptotically stable.

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Section: Equilibria Local Stability and Backward Bifurcationmentioning

confidence: 94%

“…Moreover, if (19) and (21) are substituted in determinant (18) and trace (17), these are simplified as…”

Section: Equilibria Local Stability and Backward Bifurcationmentioning

confidence: 99%

Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

Lozano-Ochoa¹,

Camacho²,

Vargas-De-León³

2017

Discrete Dynamics in Nature and Society

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“…In Vargas-De-León [22], the global stability of the positive equilibria for two-species mutualisms has been investigated by means of Lyapunov's second method. The models of interest in [22] are…”

Section: Previous Work On Mutualistic Models and Motivationmentioning

confidence: 99%

“…However, this is not actually the case. Even a cursory look at the specific forms of the functionals V 1 , V 2 , V 3 given in (2.4) (or of the functionals in [22], which are particular cases) shows that they cannot deal directly with the situation in which one or more of the functions f 1 , f 2 , g 1 , g 2 are null, as either a denominator or one of the integrals are null. By replacing the problematic integral (null or with null denominator) with a logarithmic term, Vargas-De-León and Gómez-Alcaraz [23] have obtained global stability results for the positive equilibria of the models…”

Section: Previous Work On Mutualistic Models and Motivationmentioning

confidence: 99%

Global stability results for models of commensalism

Georgescu¹,

Maxin

Zhang

2017

Int. J. Biomath.

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We analyze the global stability of the coexisting equilibria for several models of commensalism, first by devising a procedure to modify several Lyapunov functionals which were introduced earlier for corresponding models of mutualism, further confirming their usefulness. It is seen that commensalism promotes global stability, in connection with higher-order self-limiting terms which prevent unboundedness. We then use the theory of asymptotically autonomous systems to prove global stability results for models of commensalism which are subject to Allee effects, finding that commensalisms of appropriate strength can overcome the influence of strong Allee effects.

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“…Note that model (5) also includes models (1) and (2) as its special cases. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Maxin

Georgescu²,

Sega

et al. 2017

Journal of Biological Dynamics

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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.

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Lyapunov functions for two-species cooperative systems

Cited by 20 publications

References 10 publications

Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

Qualitative Stability Analysis of an Obesity Epidemic Model with Social Contagion

Global stability results for models of commensalism

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

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