Fransz, H.G.

Rabbinge, R.

English

Name not available

Hierarchical modelling of predator-prey relationships in an acarine system

NARCIS 

Wageningen Yield 

HIERARCHICAL MODELLING OF PREDATOR PREY RELATIONSHIPS IN A.\ ACARINE SYSTEM 
Introduction 
The aim of this study i~ the construction of predator-prey models for field 
use. These models describe, as such, a sub process in the population models for 
the fruit tree red spider mite (Panonychus ulmi) - predacious mite system 
(Rabbinge and y.d. Vriec, 1974). 
in these models only population densities of the fruit tree red spider mite 
below the economic damage level are considered. This regulation below this level 
seens to be possible with predacious mites (Rabbinge, 1975). Regulation means 
-then that the predacious mites cause a sufficiently increasing mortality rate when 
the prey density (P-ulmi) increases. In the fruit tree red spider mite - predacious 
mite system, (Panonychus ulmi - Amblyseius potentillae) three predator "species" 
(protonymphe, deutonymphe and adult females) and four prey "species" (larvae, 
protonymphe, deutonymphe and adult females) can be distinguished, the preference 
for the different stages depending on the state of the predator. The complexity 
of these interrelationships requires a detailed analysis of the predation process 
and in this introduction 1s shown how such a detailed analysis may lay the basis 
for a simple formulation of the predator-prey rela.tionships in the higher ordered 
population models. 
Modelling of predator-prey relations 
Detailed analysis of predator-prey systems are given by several authors. 
Holling (1966) gave a general description of the predation process and developed 
a mathematical model which provides an explanation of the three fundamental types 
of functional response curves he distinguished. In these models searching periods 
are calculated as a function of the state variables at the beginning of searching, 
such as the hunger level, the gut content~ of the predator. However these variables 
change during ~earching and so do the searching periods. Therefore computations 
of the values of the state variables at the beginning of the periods are not 
suffici-ent so that the generai applicabir'ity of Holling's analytical models is 
questionable. Fransz in his simulation models, in which more flexible numerical 
(CSHP), a simulation language developed by IBM ( 1968), too.k this i~consideration 
and paid also attention .to the stochastic character of the predation_process with 
a s 
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ts of 
disks of 5 cm2 of the leaves of the lima bean, placed upside down on wet cotton 
wool in Petri-dishes, with a number of fresh eggs of Tetranychus urticae and/or 
adult males and one to three ten days old Typhlodromus occidentalis female. In 
figure 1 a conceptual model_ of this_predator-prey system is given. The system 
consists of one Typhlodrornus occidentalis female on a leaf disk with constant numbers 
of Tetranychus urticae males and/or eggs. The elements of the predator-prey system 
are represented by blocks and the relationships by arrows. The number of captures 
per t1me unit (predation rate) depends on the number of encounters per time unit 
(encountering rate) and the success ratio (number of successful encounters divided 
by the number of encounters). 
The success ratio is influenced by the hunger level of the predator, the gut 
content, and the frequency of encounters which may induce response waning, termed 
inhibition by prey by Holling (1966). The gut content of the predator is affected 
by the encountering rate,. the feeding time and digestion. The only state variables 
in this system are the density of the prey i.e. the Actual Number of Eggs (ACNE), 
the Actual Number of Males (ACNH), the GUTCONtent of the predator adult female, 
the total distance covered by the prey males on the leaf disk (DIST} .which determines 
the webbing density, and the indicator of the ~ngagement of the predator (H) which 
is one when it is handling prey and zero otherwise. The other factors are rate 
variables or auxiliary variables. 
Simulation 
The conceptual model is used to build mathematical models of the system with 
the purpose to understand the predation process. The state variables considered 
e. g. the_ gut content and the webbing density vary with time because they depend on 
processes like ingestion, digestion and locomotion. As a consequence other system 
elements change because they are determined by the stat;e variables. 
The number of prey killed per time unit is also not constant, and the models 
are used to simuiate the dynamics of the system at different prey densities and to 
compute the. expectation values of the numbers of prey killed per predator and 
other output variables as a function of time. 
In fig. 2 a time film of the predation with a single predator is given. A 
a prey (catch=l.) to start a period of handling prey (H=l, S=O). The capture of 
a prey may take some time (CATIM). To account for this a state variable RCTIM will 
be introduced. RCTIM is set equal to CATIM at the begin~ing of a handling period, 
. ... 
and decreases with time until it is· zero at the end of -CATn·!. At the end of CATIM 
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at t 2 the predator starts feeding. This increases the gut. content (GUTCON), but 
dEc teases the content of the prey egg (EGG). At t 3 the prey is empty ant:r-up till 
to the next feeding period, GUTCON decreases by digestion. At t 4 the predator 
abandons the prey (ABAND=l.) to start a new searching period. At t 5 a second prey 
is captured, which may be abandoned before it is emptied. 
Ingestion and digestion 
The rate of decrease of the gut content is assumed to be exponential that is 
equal the gut content multiplied with the relative ~igestion rate DIGEST, a parameter 
that only depends on the temperature, vlhich is here kept constant. 
The ingestion rate (INGRT) of the predator depends on 
I 0\ 0 
whether the predator is handling or searching prey (whether H equals l$D50) 1 • 
2. whether it actually feeds on the prey handled (RCTIM equals 0) 
3. on the content of the prey. (EGG or Male greater than 0) 
4. onthe gut content of the predator. 
lt is assumed that the ingestion rate dA/dt is proportional to the difference between 
the actual gut content (A) and the maximum gut content (M): dA/dt = c * (M-A), 
where c is a positive constant. Calculations with experimentally determined values 
of A and t showed that M is 1.08 egg equivalents and cis 1.32 min-l •. 
Stochasticy of the predation process 
The integer H is initially zero and becomes when the predator catches a prey 
(CATC=l.). H becomes zero again when t~e prey is abandoned (ABAND=l.). CATCH and 
ABAND are random variables which may become 1 or zero during a time interval. The 
process of catching has been observed to be a Poisson process. Irrespective of 
handl~ng time, the number of prey caught in ~t has a Poisson.p~~b?bility · 
distrib~ti_op_, ... w~.t;,h __ aver~ge values.-~ Ji~!:,_ ... ,~- .P~_i,ng ~he success ratio, E the average 
number of encog~ters.per t;me unit. This number depends on the actual number of 
eggs (ACNE) on the leaf disk, the coincidence in.space (COINAl, the locomotion 
velocity of the predator (VELPRE). and the activity of the predator (ACTPRE): 
E = ACNE x VELPRE * COINAE * ACTPRE. 
Also ~he abandonment of the prey is a Poisson process; duri.ng each time step 
there exists a small probability that the predator leaves its prey, This 
probability seems to.be determined by hunger in the first place, since hungry· 
redators continue feedin while satiated redators abandon their 
prey even when it is not empty (fig. 3). 
Deterministic simulation of the process is the variables 
are considered as proportions on a continuous scale of individuals in a pop~lation, 
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which in fact are subjected to these events or conditions. Also the probability 
of abc:trrdormlt~nt PR:Auaird the probability of catching PRC are considered to be 
proportions and all other variables represent population means. The stochastic models 
give good results, but requires an excessive amount of computing time. The 
deterministic model requires much less computing time, but its results are wrong, 
because of the many curvilinear relationships that are involved (fig. 4). A new 
method, compound simulation, has the advantages and disadvantages of both buth the 
latter on a permissable level. It is a deterministic model which provides the 
expectation values of the stochastic model, but with much less computing time. 
Basically, this is achieved by applying every time step the deterministic model to 
homogeneous classes of individuals which are then de-homogenized, but reclassed 
again and again. Some results of compound simulation are given in fig. 5 in 
comparison with the results of independent verification experiments. In this way 
an explanation for the functional response of the predator to prey density is given 
from the underlying physiological and ethological processes. 
Prey utilizing and numerical response 
The utilization of~the prey is computed with the compound model. It depends to 
a large extent on the gut content and because of the observed relation between the 
gut content of.the predator and its fecundity (Fransz, 1974) the numerical response 
to prey density may also be derived from the computed gut content of the predator. 
Predation process in popul~tion models of the fruit tree red 
spider mite system 
The models on the predation process explain this process and provide the basis 
for its incorporation in the population models. In the fruit tree red spider mite -
predacious mite system three predator "species" and four prey "species" are 
distinguish~d (larvae, prot.onymphe, deutonymphe and adult female of. the prey and 
protonymphe, deutonymphe and adult female of the predator}. In the population 
models the expectation va-lues of the numbers of prey and predator are calculated. 
Calculations with the complex compound model made clear that the system reaches 
an equilibrium within a few hours, which means that the gut content of the predator 
oscillates with a small amplitude on a level, depending on predator and prey-density 
and the temperature of the system. 
at a higher hierarchical level. It suffices to work with the expectation values of 
prey risk and prey.valu~ as a function of the state variable that characterizes 
individual prey to be caught, prey value for instance in units of gut content per 
killed prey expresses the utilizing and the preference of a prey. The quantification 
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of the hunger level of the predator is here easy, because well fed predators show 
adark reddishcolour, while hungry predators uare whit~h and transpara.nt. A colour 
scale was developed which relates the behaviour of the predator (success ratio) and 
the quantity of leaf and animal pigments in the predator, that cause the colour. 
The dependence on temperature of the relative digestion rate or Relative Decrease 
of the.Condition Variable (RDCV) is quantified in thermostat experiments. The 
ingestion rate or the rate of increase of the condition variable (INCAF) depends on 
the prey risk and the prey value, both being gut content dependent and on the 
density of the prey and the predator. In case of one predator "species", 
A. potentillae adult female, and one prey "species", P. ulmi adult female, this J.S 
written in Continuous System 2-fodelling Program: 
DECAF RDCV * CTAF 
INCAF = RISAF * PAF * TTAF * PV 
CTAF = INTGRL (5., INCAF- DECAF) 
DECAF rate of decrease of the condition variable 
CTAF = Condition Typhl. Adult Female 
RISAF = RISK Panonychus ulmi Adult Female· . 
PAF = number of Panonychus ulmi Adult Female 
TTAF = Total number of Typhlodromus Adult Female 
PV = Prey Value 
· · · s1."tuation the prey risk is calculated as predation rate h In the equJ.lJ.brJ.um prey density ' w en 
TTAF = 1. per unit of surface. 
Since in the equilibrium state 
DECAF = INCAF, 
the prey value equals 
RDCV x CTAF 
PV = RISAF * PAF x 1. 
The in this way e~perimentally found relations, prey risk in dependence of 
temperature and condition variable, prey value in dependence of condition variable, 
and relative decrease of the condition variable in dependence of temperature are 
input relations for the population model. This way of programming the predation 
process makes preference experiments superfluous as is demonstrated by the agreement 
of simulated and experimental results of a system with P~ ulmi larvae and P .. ulmi 
variable is found by other experiments. In this way the complex predator-prey 
relationships are incorporated in a very simple way in the population models, without 
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loosing any of the explanatory value of the models, because they are based on a 
detauileduuanalysis uof the predatj on process The results of the population modsls and 
their possible application in the field.are discussed by Rabbinge (in prep.) in a 
description of the models and their results. 
~ 
~ 
PREY INTERACTION PREDATOR 
I 
l'ig. ):().I Conceptual model of a predator-prey system. One Typhlodromus orcit/enlalt:\· female on a leaf with N" .. nU-Lf ... n 
numhcrs of Tclranychus urlicae nmlcs and c~gs. 
~ I D [s 
I [ I I l H 0 
~_j CATCH 
I 
I ABA NO 0 
0
CATIM"" 
"' 
RCTIM 
0MAXEC~ D EGG 
MAXGUT 
,<; ~GUTCON o. I 
0 
'• 
t f t t It TIME 2 3 4 5 6 7 
--? 
Fig. 'I 2 I A time film of the pred< 
process with a single predator. 
.. __ .- .... -·· 
1.0 egg eq. 
.). GUTCON 
Fig. l<l I The frequency of abandonmcnts as related to the gut content when 
the predator handle5 eg~ (A) or males (B). 
63 
. . . ... .... -. ~ 
PRE 
hr·1 
.6 
~ 
'/ 
~ • • COMPOmlO 
,to• o STOCHI.ST with ?~ ~. con f. int. 
0 DETERM 
• experimer.tcl ret:.:l!s 
5 10 
PRE 
hr·1 
.6 
.2 
B 
• COMPOUND 
o COMPOUND wil!-:~u1 
prey replenishr;.er.t 
O+------~r------~----------,----------------
0 5 10 15 20 c.-n·2 
OEt:EGG 
!4! The functional res?o:-1se to the egg de.nsity for the last six hours CA) 
~c first six hours (B) of a 24-h period. 
81 
.e 
.. 
0 5 
ti" 
• 16-24 hr 
o- 6hr 
• o- 6 hr Wtthout prey replenishment 
·- o- 6hr with much webbing 
• experimental results 
10 
-2 20 em 
DEN MAL 
Fig. 1-'i I The functional response to male density as computed 1: 
program C 0 M P 0 UN D, and obtained by independent experiments. 
. .J::: u Or '-.J 
j 
I 
~ ! : 
0 1 
• 1 •• 
--:·'~---·---·· 
-· I ------ .... - -·-
---··-;- --~-··· 
I 
--·-··--- _;..._ --
--.-.. i. -·~-


Hierarchical modelling of predator-prey relationships in an acarine system

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Wageningen Yield