Integral equations—a reference text

Zabreyko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, Solomon G.; Rakovshchik, L. S.; Stetsenko, V. Ya.

doi:10.1007/978-94-010-1909-5

Cited by 178 publications

(118 citation statements)

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“…From the spectral theory of compact, linear, positive self-adjoint operators, we know that a positive dominant eigenvalue corresponding to a unique (up to constant multiple), nonnegative eigenfunction exists (Lax, 2002;Zabresȋko et al, 1975).…”

Section: Critical Patch Size (ω Bounded)mentioning

confidence: 99%

Integrodifference equations in patchy landscapes

Musgrave

Lutscher

2013

J. Math. Biol.

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In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with populationlevel dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation.First, we formulate a random-walk model in a patchy landscape with patchdependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters.Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior ii

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Section: Critical Patch Size (ω Bounded)mentioning

confidence: 99%

Integrodifference equations in patchy landscapes

Musgrave

Lutscher

2013

J. Math. Biol.

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“…These equations have been studied in several papers and monographs (see for example Krasnosel'skii et al [10], Zabrejko et al [14], Appell [1,2] and references therein).…”

Section: Jomentioning

confidence: 99%

“…Existence theorems for (1) and (2) proved with the help of the above mentioned tools require rather strong hypotheses, and so the result are not entirely satisfactory. A survey of such existence theorems is given in the book of Zabrejko et al [14] and in the paper of Appell [2]. The goal of this paper is to prove theorems on existence of solutions of the equations (1) and (2) under weaker hypotheses.…”

Section: Jomentioning

confidence: 99%

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Integrable solutions of Hammerstein and Urysohn integral equations

Banaś

1989

J Aust Math Soc A

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“…(10]. p. 384, see also (11], (12]. Actually, the functional equation (3.13) differs slightly from that discussed in [10] because the line of discontinuity is not a closed contour bounding a finite domain.…”

Section: :Yj = 'Y;(p)mentioning

confidence: 99%

On Periodic Pollaczek Waiting Time Processes

Cohen

1996

Athens Conference on Applied Probability and Time Series Analysis

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Integral equations—a reference text

Cited by 178 publications

References 0 publications

Integrodifference equations in patchy landscapes

Integrodifference equations in patchy landscapes

Integrable solutions of Hammerstein and Urysohn integral equations

On Periodic Pollaczek Waiting Time Processes

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