Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics

Kunquan Lan; Wei Lin

doi:10.3934/dcdsb.2018256

Article Contents

2019, Volume 24, Issue 4: 1943-1960. Doi: 10.3934/dcdsb.2018256

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Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics

Kunquan Lan^1, , and
Wei Lin^2,

1.
Department of Mathematics, Ryerson University, Toronto, Ontario, Canada M5B 2K3
2.
School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai 200433, China

* Corresponding author: Kunquan Lan
* Corresponding author: Kunquan Lan

Received: January 31, 2017

Revised: May 31, 2018

Early access: October 2018

Published: January 2019

The first author was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada under grant no. 250187-2013 and 135752-2018, and the Shanghai Key Laboratory of Contemporary Applied Mathematics, and the second author was supported in part by the NNSF of China under grants no. 11322111 and no. 61773125.

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Abstract

Lyapunov type inequalities for (linear or nonlinear) Hammerstein integral equations are established and applied to second order differential equations (ODEs) with general separated boundary conditions. These new inequalities provide necessary conditions for the Hammerstein integral equations and these boundary value problems to have nonzero nonnegative solutions. As applications of these inequalities for nonlinear ODEs, we obtain extinction criteria and optimal locations of favorable habitats for populations inhabiting one dimensional heterogeneous environments governed by reaction-diffusion equations with spatially varying growth rates and external forcing.

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Mathematics Subject Classification: Primary: 47H30; Secondary: 35P30, 45A05, 47H10, 92D25.

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Figure 1. (a) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{1}$, where $k = 3$, $a = 0$, $\beta = 4$, $\delta = 1$, and $T = 0.8$. (b) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{2}$, where $k = 3$, $a = 0$, $\beta = 2.5$, $\delta = 2$, and $T = 0.8$. (c) The monotonically decreasing curve of $\eta_{1}$, as defined in (4.12), with respect to the variable $a\in[0, 1-T] = [0, 0.2]$ for $(\beta, \delta, T)$ in different area. Here, the curve, plotted by squares, corresponds to the parameters in (a) and the curve, plotted by dots, to the parameters in (b)

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Figure 2. (a) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{3}$, where $k = 3$, $a = 1-T$, $\beta = 2.5$, $\delta = 2$, $T = 0.3$, and $a_{1}(T) = 0.1$. (b) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{4}$, where $k = 3$, $a = 1-T$, $\beta = \delta = 3$, $T = 0.3$, and $a_{1}(T) = \frac{1-T}{2} = 0.35$. (c) The optimal position with $m_{a, k}$ when $(\beta, \delta, T)\in \mathcal{D}_{5}$, where $k = 3$, $a = 1-T$, $\beta = 3$, $\delta = 3.5$, $T = 0.3$, and $a_{1}(T) = 0.6$. (d) The unimodal curve of $\eta_{1}$, as defined in (4.12), with respect to the variable $a\in[0, 1-T] = [0, 0.7]$ for $(\beta, \delta, T)$ in different area. Here, the curve, plotted by the dash line, corresponds to the parameters in (a), the curve, plotted by the solid line, to the parameters in (b), and the curve, plotted by the dotted line, to the parameters in (c)

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