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(8 citation statements)

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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.…”

confidence: 94%

“…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.…”

confidence: 94%

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

confidence: 99%

“…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…”

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…”

confidence: 99%

“…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.…”

confidence: 99%