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Holling's “hungry mantid” model for the invertebrate functional response considered as a Markov process. Part II: Negligible handling time

Abstract

In this paper we analyse a stochastic model for invertebrate predation taking account of the predator's satiation. This model approximates Holling's “hungry mantid” model when handling time is negligible (see Part I). For this model we derive equations from which we can calculate the functional response and the variance of the total catch. Moreover we study a number of approximations which can be used to calculate these quantities in practical cases in a relatively simple manner.

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Abbreviations

a :

rate constant of digestion

b :

maximum of rate “constant” of prey encounter in the mantid

c :

satiation threshold for search

c′ :

satiation threshold for pursuit in the mantid

c i :

ℰ(w1/2(N- ℰN)ϱi)

:

expectation operator

f :

rate of change of satiation during search

F :

functional response: mean number of prey eaten per unit of time

g :

rate “constant” of prey capture

h :

probability generating function of N conditional on S = s times p

H:

probability generating function of N

mi :

ℰϱ1

n, N :

number of prey caught

p :

probability density of S

pn :

simultaneous probability (density) of N and S

q :

probability of strike success

r :

dummy variable in generating function

s, S :

satiation

T s :

search time

T d :

digestion time

v :

asymptotic rate of increase of var v

V :

asymptotic rate of increase of var N

w :

weight of edible part of prey

W :

standard Wiener process

x :

prey density

z:

(ℰN{S = s}-ℰN)p

β :

rate constant of prey escape time maximum pursuit time

ζ :

(ℰv{S = φ + w 1/2ϱ}-ℰv)π

θ :

present time as a fraction of the time from the start to the end of the experiment

λ :

hazard rate of T s

μ :

mean time between (downward) passages of S through c

v :

w−1/2(N-φ)

ξ :

edible prey biomass density

π :

probability density of σ, number pi

ϱ :

parameter of Weibull distribution of T s ∶ ϱ = (1/2acx(-g′(c)))1/2

σ :

w−1/2(S )

φ :

satiation in the guzzler approximation: solution to dφ/dt = f(φ) + ξg(φ), φ(0)=ℰS(0).

Φ :

biomass functional response: wF

ψ:

total biomass catch in the guzzler approximation: solution to dψ/dt = ξg(φ), ψ(0) = 0

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Metz, J.A.J., van Batenburg, F.H.D. Holling's “hungry mantid” model for the invertebrate functional response considered as a Markov process. Part II: Negligible handling time. J. Math. Biology 22, 239–257 (1985). https://doi.org/10.1007/BF00275717

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Key words

  • Stochastic models
  • Invertebrate predation
  • Functional response