Holling's ?hungry mantid? model for the invertebrate functional response considered as a Markov process. III. Stable satiation distribution

Heijmans, H.J.A.M.

doi:10.1007/bf00277665

Cited by 16 publications

(7 citation statements)

References 14 publications

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“…When the transition rates are constant, p(n) will converge to a steady state distribution, as proven by Heijmans (1984). Under this condition i0(n) can be found by selecting an arbitrary starting value for p(0), solving p(1) from p'(0)=0, p(2) from p'(1)=0 and so on, and finally by rescaling the p(n) values such that they sum to unity.…”

Section: A Finite State Markov Model Of Predation In Prey Mixturesmentioning

confidence: 93%

How to analyse prey preference when prey density varies? A new method to discriminate between effects of gut fullness and prey type composition

Sabelis

1990

Oecologia

100

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How to analyse prey preference when prey density varies? a new method to discriminate between effects of gut fullness and prey type composition Sabelis, M.W. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.

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Section: A Finite State Markov Model Of Predation In Prey Mixturesmentioning

confidence: 93%

How to analyse prey preference when prey density varies? A new method to discriminate between effects of gut fullness and prey type composition

Sabelis

1990

Oecologia

100

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“…Both {T0(t)},;;:; 0 and {T.(t)},il!i 0 are positive semigroups, which is intuitively clear from the biological interpretation, but can also be shown rigorously (see Heijmans (1986) (see Nagel (1986)). Choose A> s(A.).…”

Section: {3(x )mentioning

confidence: 74%

“…Whether the limit argument extends to such properties has to be ascertained in a separate manner. As an example we may refer to Heijmans (1984) who considers both the transient behavior and some properties of the stable i-state distribution (the dominant eigenfunction of the forward equation), as well as the eventual convergence of the p-state towards this distribution, for a model of satiation dependent predatory behavior.…”

Section: Domentioning

confidence: 99%

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Small Parameters in Structured Population Models and the Trotter–Kato Theorem

Heijmans¹,

Metz²

1989

SIAM J. Math. Anal.

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Abstract. The justification of some (often implicit) limit arguments used in the development of structured population models is discussed via two examples. The first example shows how a pair of sink-source terms may transform into a side condition relating the appearance of individuals in the interior of the individual state space to the outflow of individuals at its boundary. The second example considers the usual equation for size-dependent population growth in which it is implicitly assumed that discrete finitely-sized young are produced from infinitesimal contributions by all potential parents. The main mathematical tool for dealing with these examples is the Trotter-Kato theorem for one-parameter semigroups of bounded linear operators.Key words. structured population, limit transition, C0-semigroup, Trotter-Kato theorem AMS(MOS) subject classifications. 92A15, 35A35, 47005 l. Introduction. 1.1. Biological motivation: structured populations, semigroups of operators, and the need for model simplifications. The tenet of the physiologically structured approach to the modeling of the dynamics of populations as set out in Metz and Diekmann (1986) is that, provided all individuals experience the same environmental inputs such as food availability or chance of running into a predator, we may (and should) represent a population as a frequency distribution over a space D of potential states of the individuals comprising the population. (As we frequently need corresponding concepts on the individual and population levels we will, where necessary, use the prefixes iand p-to distinguish the corresponding terms, for example i-state versus p-state, where the latter refers to the frequency distribution.) The main effort in model construction is the determination of an appropriate state representation of i-behavior, where the i-behavior consists of (i) any contributions to population change such as giving birth or dying, and (ii) any quantities relevant to the calculation of the output from the population model, such as the rate at which the individual consumes food. If we make the assumption that the number of individuals is sufficiently large, then for any given course of the environment the present p-state should determine the future p-states in a deterministic and linear fashion. For a constant environment the maps relating subsequent p-states should form a linear semigroup.The transition from i-model to p-model is made through their differential generators. It is here that we leave biology and start doing mathematics: did we really write down a genuine differential generator, and what can be said about the properties of the semigroup so generated?Until now the attention has been mostly restricted to models where the i-state space n is a subset of !Rk, and where the individuals move through n according to the solution of an ordinary differential equation (ODE), possibly alternating with (usually randomly occurring) state jumps, for example, due to an individual losing weight when it splits off a daughter. The reasons ...

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“…to t5 for constant x was proven by Heijmans (1984)). Given the solution to (4) the functional response can be calculated from (.…”

Section: Calculation Proceduresmentioning

confidence: 85%

Sources of variation in predation rates at high prey densities: an analytic model and a mite example

Metz

Sabelis

Kuchlein³

1988

Exp Appl Acarol

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Holling's ?hungry mantid? model for the invertebrate functional response considered as a Markov process. III. Stable satiation distribution

Cited by 16 publications

References 14 publications

How to analyse prey preference when prey density varies? A new method to discriminate between effects of gut fullness and prey type composition

How to analyse prey preference when prey density varies? A new method to discriminate between effects of gut fullness and prey type composition

Small Parameters in Structured Population Models and the Trotter–Kato Theorem

Sources of variation in predation rates at high prey densities: an analytic model and a mite example

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