Asymptotic Behavior of the Bifurcation Diagrams for Semilinear Problems with Application to Inverse Bifurcation Problems

Shibata, Tetsutaro

doi:10.1155/2015/138629

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Asymptotic Behavior Of λ(α) and λ±(α)1

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2016

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“…Then by the mean value theorem, for

α - ε < θ < α

, we have

\begin{matrix} true \\ right F (α) - F (θ) & = & left f (α) (α - θ) - \frac{1}{2} f^{'} (α_{θ}) {(α - θ)}^{2}, \end{matrix}

where

α - ε < α_{θ} < α

. By (2.6) and (2.9), we obtain

\begin{matrix} true \\ right \sqrt{λ} & = & left \frac{1}{\sqrt{2}} \int_{0}^{α} \frac{1}{\sqrt{f (α) (α - θ) - f^{'} (α_{θ}) {(α - θ)}^{2} / 2}} d θ \\ = & left \frac{1}{\sqrt{2}} ({, \int_{α - ε}^{α}, \frac{1}{\sqrt{f (α) (α - θ) - f^{'} (α_{θ}) {(α - θ)}^{2} / 2}}, d, θ) \\ left (} + \int_{0}^{α - ε} \frac{1}{\sqrt{f (α) (α - θ) - f^{'} (α_{θ}) {(α - θ)}^{2} / 2}} d θ) \\ : = & left \frac{1}{\sqrt{2}} (Z_{1} + Z_{2}) . \end{matrix}

To prove the following Lemma , we use the arguments recently developed in . Lemma Let

m, k

be the constants which satisfy one of (1.5)–(1.7).…”

Section: Asymptotic Behavior Of λ(α) and λ±(α)mentioning

confidence: 95%

Inverse bifurcation problems for diffusive Holling–Tanner population model

Shibata

2016

Mathematische Nachrichten

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Key words Asymptotic behavior of bifurcation curve, Holling-Tanner population model, inverse problems MSC(2010) 34C23, 34F10We consider the equation which is called Holling-Tanner population modelwhere λ > 0 is a bifurcation parameter and m, k > 0 are unknown constants. In this paper, we determine the unknown constants m, k from the asymptotic behavior of the bifurcation curve λ(α), where α = u λ ∞ > 0.

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“…Then by the mean value theorem, for