- Email: [email protected]
Simulation of simple models of animal behavior with a digital computer
J. Theoret. Biol. (1969) 23, 400-424
Simulation of Simple Models of Animal Behavior with a Digital Computer F. JAMES ROHLF Department
of Entomology, The University of Kansas, Lawrence,
Kansas,
U.S.A.
DEMOREST DAVENPORT Department
of Biological Sciences, The University of California, Santa Barbara,
Calif,
U.S.A.
(Received 18 September 1968, and in revised form 5 December 1968) The effects of kinetic, klinokinetic, orthokinetic and tropotaxix behavior were simulated on a digital computer in order to study the interactions between these simple behaviors and the effect of sensory adaptation under completely controlled experimental conditions. The conclusions of Ulyott that klinokinetic behavior in the presence of sensory adaptation can cause a directional displacement of an organism along a gradient of stimulus intensity were confirmed. In addition it was found that inverse klinokinetic and direct orthokinetic behavior (with sensory adaptation) result in a marked directional displacement up a gradient. However, direct klinokinetic behavior cancels out the effect of direct orthokinetic behavior.
1. Introduction In the field of invertebrate behavior, undirected locomotory reactions, in which the speed of movement or the frequency of turning depend upon the intensity of stimulation, are defined as kineses (Fraenkel & Gunn, 1961). A klinokinetic response is defined as a change in the rate of change of direction, in response to a change in stimulation level, while an orthokinesis is a change in linear velocity in response to such an intensity change (Gunn, Kennedy & Pielou, 1937; Ewer & Bursell, 1950). In klinokineses, direction of turning is always random. Klinokineses are, therefore, quite distinct from taxes, which by definition always involve non-random “choice” of direction in a gradient. There is ample experimental evidence that the phenomenon of klinokinesis (as distinct from the taxis) exists and is a function of stimulus intensity 400
SIMULATION
0~ K~JNOI~NESIS
401
(Ullyott, 1936; Davenport, Camougis & Hickok, 1960; etc.). Some animals may turn more frequently at high levels of stimulation and some at low levels of stimulation, but in any case the direction of their turns is random. However, the work of Ullyott (1936) and others notwithstanding, there is still no convincing evidence (from experiments including controls to eliminate the possible presence of taxes) that klinokinesis alone, with or without sensory adaptation, can effect the aggregation of organisms at one end or the other of a stimulus gradient (Gunn, 1966). The purpose of the present paper is to find out whether an organism showing only klinokinetic behaviour and sensory adaption can move up and down a smooth gradient. Whether such organisms exist is a separate question. However, even in the most recent literature, the efficacy of klinokinesis in bringing about aggregation or directional displacement in a gradient is often assumed. Furthermore, all too often discussions of the phenomenon are tinged with purposiveness, in such a way that animals are said to approach a “preferred” environment until conditions “improve”. We submit that such words, even if used by authors as convenient shorthand, imply value judgments on the part of the organism in space or time which clearly can have no part in any machinery of displacement in which, by definition, the direction of turns is random relative to the gradient. J. S. Kennedy (1945) has advised that the sign-terms “positive” and “negative” as applied to klinokineses be abandoned, since these terms have directional implications. We recommend the use of his adjectives “direct” and “inverse” for both klinokinesis and orthokinesis, i.e. for situations in which the rate of random turning (or linear velocity) is directly and inversely proportional, respectively, to stimulus intensity. There is enough evidence (uide Fraenkel BE Gunn, 1961) of the adaptive significance of directional “choice” (taxis) to establish its importance Cmly in the behavioral economy of invertebrates. In addition, the adaptive significance of orthokinesis, in which linear velocity is a function of stimulus intensity, is clear (Gunn, 1937, 1966); all other behavior being equal, an organism which moves more slowly under stimulus level I than under stimulus level II will “aggregate” where stimulus level I pertains. However, the only evidence we have of the efficacy of pure klinokinesis in bringing about aggregation, “target-finding”, or displacement’ up or down a gradient is circumstantial and depends upon such observations as those of Ullyott (1936) and demonstrations that in certain animal associations a highly speciesspecific klinokinetic response may be observed .when an animal is in a medium containing a chemical signal from its host (Davenport et al., 1960). Such response specificity clearly suggests that the response may have adaptive significance but controlled experiments to demonstrate this are not at hand.
402
F.
J.
ROHLF
AND
D.
DAVENPORT
It would naturally be desirable to design experiments, using animals, in which one could determine whether an observable klinokinesis, working alone without directional cues, can effect aggregation or displacement in a gradient. But, even if light is used as a stimulus, how are all directional cues and their effects to be indubitably eliminated? If an animal with bilaterally symmetrical sense organs is used, then the possibility of unequal stimulation of the organs in the gradient and a consequent directed response is always at hand. If a single sense organ is involved, then in a gradient there is always the possibility of successive testing of the gradient in time. We have therefore been continuously faced with a problem, unsolved to date, in designing experimental protocol. We may, nevertheless, ask some questions which may give us a possible insight into whether or not an organism exhibiting klinokinetic behavior has a selective advantage. We may ask what the properties are of a theoretical idealized animal having this type of behavior. The klinokinetic response must, of course, be considered alone and then in conjunction with other types of responses (orthokineses and taxes). Clearly it is likely that there may be an interaction between these responses. The displacement of an organism, resulting from the interaction of these responses, is more complex than might first be thought. An intuitive analysis of the results of such interaction is not only extremely difficult but dangerous, although it must be admitted that a long-lasting intuitive distrust by one of us of Ullyott’s interpretation of his results, has led to this effort. An alternative, until such a time as we have an adequate experimental protocol dealing with actual organisms, is to use a mathematical analysis of the behavior of “ideal” organisms. However, just as an experiment to demonstrate the efficacy of klinokinesis in living animals has the disadvantage that it is almost impossible to eliminate directional cues, so our alternative of mathematically analyzing the behavior of an ideal organism has the disadvantage that in order to be able to obtain a solution one may have to simplify the properties of the organism to a point at which one’s assumptions become so unrealistic as to render the results of little real biological significance. Nevertheless, it seems of value to undertake such an analysis. The approach taken in the present study was an analysis of the behavior of an ideal hypothetical organism, using techniques of simulation with a digital computer. The approach has the advantage that it is relatively simple to add complexity to the system. A large number of “experiments” with hypothetical organisms are performed and then a statistical analysis is performed on the results. The mode of attack is very much like the experimental approach, except that here the stimuli being applied are known; only their effect is unknown. We refer the reader to Evans, Wallace $ Sutherland (1967)
SIMULATION
OF
403
KLINOKINESIS
and Naylor el al. (1966) for general accounts on digital simulation techniques. There have been a number of previous efforts to study the motions of organisms. Pearson (1906) investigated the use of the random walk method for the case of random migration (no stimuli). In the classical random walk model an organism moves in a series of steps, where the length of the step, time between steps, and the direction of the step are independent of one another and of preceding steps. If we allow the length of the step to be continuous, we may consider the random walk as a Markov process and describe the system in terms of the Chapman-Kolmogorov equations, which yield the probability of an organism being at any point at any time. One may also solve for the steady-state condition (if it exists) using the Fokker-Planck equation which yields the probability of an organism being at any point after a sufficient period of time has elapsed so that these probabilities no longer depend upon time. General accounts and applications piiq $q +TkTl 4 Set initial state for ith organism r=o, P:=(xt,Y:)=(O,O), M,=O
Compute efkt of previous stimulation Mt=9~Mt-l +qrxt-1. Compute efkt of prcaent stimulation &“.l=qC+qDxt-1
Emu =qEfqF%-1 Em.x,ifXt-1=-M-l
Et- I- q&3-1 - &ml,
otherwise
Construct random munber R, @ 5 R 5 1*O). 1
404
F.
J.
ROHLF
AND
D.
DAVENPORT
P
No
R,(O_
Compute new position of organism
No --a Yes STOP FIG.
1. Flow chart of simulation program. See text for explanation.
of these equations to biology are given in Bailey (1964) and Parzen (1962). Davenport, Kramer & Thorson (1960) investigated klino- and orthokinetic models using both the above approach and that of simulation. They found that a simple klinokinetic model resulted in a uniform distribution of the organisms, whereas an orthokinetic model resulted in a very skewed distribution (the long tail of the distribution was in the direction of the region in which the organisms had the highest velocity). Patlak (1953~) further developed the mathematical methods needed to incorporate the effects of “external forces” (stimulus which causes a taxic component or an avoidance
SIMULATION
OF KLINOKINESIS
405
reaction in the organism’s motion) and “persistence of direction” (which implies that when an organism turns, it will not turn with equal probability in all possible directions). Patlak (1953b) described the application of these methods to the study of the movement of organisms. He found the distinction between klino- and orthokinetic behavior not to be a useful one in terms of his model, since the steady-state distribution was a function of (1) average time between turns, (2) average distance between turns, and (3) average squared distance between turns, “rather than a simple function of average velocity and time between turns alone”. He also described methods for estimating the various parameters in his model and investigated the degree of fit between results previously obtained by others and his model which he fitted to their data. He concluded that the major reason for lack of agreement was probably due to the organism’s accommodation (sensory adaptation) which was not taken into account in his model. However, if the rate of accommodation was very slow it could be ignored. If, on the other hand, it adapts very rapidly, then its behavior will largely be a function of only its previous state and the model may still be used. If the time of accommodation is between these extremes (the organism’s behavior is a function of several of its past states, which seems likely) then the model cannot be used since we no longer have a Markov process. One of the purposes of the present study is to investigate the effect of such a type of adaptation and especially its interaction with other behavioral patterns. The extension of Patlak’s results seemed to pose problems of formidable mathematical complexity. Thus, the present work is entirely in terms of simulation experiments and follows the general approach used by Saila dc Shappy (1963) in their study of random movement and orientation in salmon migration. A disadvantage of the simulation approach is that one does not obtain a general mathematical solution but rather a great quantity of particular solutions. A flow chart of the computer program (coded in FORTRAN IV for the GE 625 computer) is given in Fig. 1. The variables qA through qN are the parameters which specify the behavioral pattern being simulated. The effects of these parameters are discussed separately for each behavioral pattern. The choice of the particular set of values of these parameters to be used in the simulations was fairly arbitrary. Few levels of each parameter could be used since we could not afford to run all possible combinations of a very wide variety of values. In the initial set of simulation models (sections 2.1 to 2.5 below) no sensory adaptation was allowed for (qA = O-0, qB = 1.0, qc = O-0, q,, = l-0, qE = O-0, qF = l-0, q. = O-0, thus Et = X,-J. The multiplicative congruential pseudo random number generator R,+l = (513R&d 2j6 was used
406
F.
J.
ROHLF
AND
D.
DAVENPORT
2. Methods and Results 2.1. A SIMPLE KINETIC MODEL As a simplest model we may visualize a hypothetical organism placed in the center of a grid. The position, P,, of the organism on this grid can be denoted by a pair of (X,, Y,) co-ordinates (X, denotes position along gradient at time t). When the organism is initially placed in the center of the grid (0, 0), the initial direction in which it faces is determined at random (with equal probabilities). Four possibilities are allowed: it may face up (+ Y), down (- Y), left (- X), or right (+ X). Next, a decision is made as to whether or not the organism will continue in the direction in which it is currently facing or will change its direction by turning clockwise or counter-clockwise (qH = 0.5, qr = O-0). After this is determined, the organism is then allowed to take a step of length l-0 (qJ = 1.0, qx = 0.0) in that the new direction. The turning decision and stepping is then repeated 100 times. Figure 2 shows
(-j 5... .....*....*.*...
0.:... ... i.d 0
~ 0
FIG. 2. Trace of paths taken by simulated organisms based upon a simple kinetic model. The “organism” started at the point X = 0, Y = 0 (shown as the intersection of the broken lines) and during each of the 100 time intervals in which it was followed, it could move one unit clockwise, counterclockwise, or continue in its present direction with probabilities of l/4, l/4 and l/2, respectively. On this and all of the subsequent figures each tick mark along the axes represents 10 units on a grid laid over a hypothetical area.
an example of the path by four such individuals. The final position of the organism after 100 steps, Pioo, is recorded and the entire process repeated until the results of 500 such trials have been accumulated. When this experiment was carried out, we found that the mean position of the 500 organisms along the X-axis was X = O-058 (Table 1) with a standard deviation of 12,602 (Table 2). The resultant distribution of organisms across
SIMULATION
407
OF KLINOKINESIS
TABLE 1
Mean position, x, of 500 independent artificial organisms after 100 unit time intervals Pattern of tropotaxis Decreasing accuracy as Constant +X end is approached
None Orthokinesis direct (faster at + end)
- 1.620 0 1.654 + 0.338
- 23.166 0 21.454 + 18.729
- 12.296 0 13.247 + 13,290
- 27.368 0 23.717 + 23.392
- 0.512 0 0.058 + 0.262
- 16540 0 17.492 + 16.552
- 14.780 0 14608 + 16.364
- 19.052 0 17906 + 16.858
I -076 0 -0.893 + -2.074
- 14.054 0 12.913 + 12.956
- 17.347 0 18.397 + 20.197
- 14.568 0 14.548 + 13.884
none
inverse (faster at -end)
Increasing accuracy as +X end is approached
Note: Within the cells -, absence, respectively.
f,
0, correspond
to inverse, direct klinokinesis
and its
TABLE 2
Standard deviations of the position of 500 independent artificial organisms after 100 unit time intervals
None Orthokinesis direct (faster at +end)
Pattern of tropotaxis Decreasing accuracy as constant +Xendis approached
Increasing accuracy as +Xend is approached
- 12.879 0 12.883 + 11.684
- 19462 0 16.223 + 14062
0 +
8,646 8.042 8224
- 23.678 0 19221 + 17.697
none
- 12.500 0 12.602 + 12.087
- 13.022 0 11.018 + 10460
- 10.577 0 10.379 + 11.252
- 13.687 0 12.919 + 12.401
inverse (faster at -end)
- 11.810 0 12.079 + 14.132
0 +
- 13.269 0 14.645 + 17.411
0 +
Note: Within the cells -, absence, respectively.
8.340 8.424 8.181
+, 0, correspond to itwerse, direct klinokiiesis
9.455 8.813 9-626 and its
408
F.
I
J.
ROHLF
AND
D.
DAVENPORT
I
-30
-20
-10
b
IO
20
30
X
FIG. 3. Summary plot for the simple kinetic model showing the final position of 500 independent organisms after 100 unit time intervals. T?= 0.058.
the grid (Fig. 3) is reminiscent of correlation equal to zero and mean allowed to take fewer steps, or more the same but the variance increases 2.2. (WITHOUT
A SIMPLE A MODEL
a bivariate normal distribution with a at x = 0, Y = 0. If the organisms were steps, the mean position would be about as a function of time.
KLINOKINETIC OF SENSORY
MODEL ADAPTATION)
We may now complicate the behavior of our hypothetical organisms slightly by changing the decision rule for deciding whether the organism continues in the direction in which it was facing or changes its direction, by making the probability of a change of direction a simple function of its position along the X-axis. In the present experiments, we let the probability of a turn (qH+qIl?,) be 0~5+0@05X,-, for direct and 0~5-0~005X,-, for inverse klinokinesis (probability of clockwise turn = 0.5, qJ = O-5, qK = qL = O-0). Figure 4 is an example. This would mean that if we imagined our grid going from plus to minus 100 (representing a gradient going from a low level of stimuIation at the left to a high level at the right side of the grid), our “organism” would have a turning probability ranging from 0 to 1. When such an “experiment” was run (see Fig. 5) we found mean values of
SIMULATION
OF KLINOKINESIS
0.......” .......... M 0
* ~ : .. ... .. ... ... . 0
....... : : y i
........
0
.. . .,........~.........
0
F?!d OX
m
6
FIG. 4. Trace of pathstaken by simulatedorganismsbasedupon a modelof direct klinokineticbehavior.A gradient of stimulusintensityis assumed to exist(for this andall subsequent figures)alongthe X-axis.Probability of a turn equals0.5 + OG05X1-I.
. .
30 -
I
. y
0
-_-__-__________; . t
-10
'. ..
t
-20 -3o-
, -40
.
..
.,. __-_-_ .
.:
:
: I
. ’
.
:,
--,
+,,
. I. .: ;:. . . : : ., : . 1.: '_._ ..;I. . :::.:* ..:::: 1.1 1 ; . . 1.. *'.: . :. . ..j' . . . ~1 : :: .:: :.. :
.
, I
.. .
. .
; ‘.i. . :.’ . . . . .. ‘. ‘.’ ... I,-. . I I I , . I . -30 -20 -10 b IO 20 30 X
FIG. 5. Summaryplot for the direct klinolcineticmodelshowingthe tial positionsof 500independent organisms after 100unit timeintervals.x = O-262.
X = O-262 for direct klinokinesis and a value of X = O-512 for inverse klinokinesis. These values are not significantly different from zero. This model appears to bear out Ullyott’s conclusion that klinokinesis without sensory adaptation can have no aggregating effect.
410
F.
J.
ROHLF
AND
D.
DAVENPORT
2.3. AN ORTHOKINETIC MODEL Independent of the above modification to the program, one can make the length of the “step” taken in each unit time interval a function of the position of the organism along the X-axis. For example, at the right it might take larger steps and at the left take smaller ones (Fig. 6 shows examples). In
0m ...0...
0 .. .. ... ... .. .. . ... ...,
LIYB
0.....i.... Eld
d
Y 0 lY!I?cJ .. ... .. . . . . ... ... . .. .. ... .. x
.. .. 0
FIG. 6. Trace of paths taken by simulated organisms based upon a model of direct orthokinetic behavior. Length of step equals 1 .O + 0.02X, 1.
, I I I I I I
40-
30’ i
“‘I’..
201
,:
.,
.’
:
:.
.I
I
-30
I
t
I :..
!
j
, -20
I -10
0
‘.
:’
. t IO
. I 20
I 30
I 40
x FIG.
7. Summary plot for
the
direct orthokinetic
model. R = 1 e654.
SIMULATION
OF KLIIWKINEbIS
411
these experiments, a step size function (qM+q&) was computed as 1 .O+O.O2X,-i. Thus, in a given experiment, the size of the step could range from a minimum of 0.0 at X = -5Oupto3~0atX=+lOO.Themost dramatic effect of orthokinetic behavior (see Fig. 7) is that it strongly skews the distribution of individuals along the X-axis. The long tail of the distribution is in the direction of the region in which the organisms have a higher velocity. A mean of X = 1.654 and a median of 0417 were obtained for direct orthokinesis. The standard deviation (13.386) of the position of the organisms along the X-axis was also slightly increased over that found for the previous runs. Clearly, this model bears out results obtained in experiments (Gunn, 1937) indicating that, other behaviors being equal, the region of highest density of organisms occurs in regions where activity (speed of movement) is reduced. 2.4. TROPOTAXIC BEHAVIOR Also, independent of the above, we may endow our organism with paired sense organs so that it is capable of detecting differences in intensity of a stimulus along a gradient when it is facing at right angles to this gradient. Since taxic behavior has been experimentally demonstrated to have such a strong effect, we felt it would be of interest to consider only models in which the ability of the organism to make the “correct” decision was something less C
FIO. 8. Trace of paths taken by simulated organisms based upon a model of positive tropotaxis.Probabilityofaturntowardsth6+XendoftbeOridis2/3~~olganinn is facing acraas the gradient (up or down). When facing along the gradient the probabilie of a turn is l/2. 27 T.B.
412
P.
J.
ROHLF
AND
D.
DAVENPORT
than perfect. Several models were considered. In the simplest case, the organism has a constant probability (equal to 2/3) of making a “correct” turn when it is facing across the X-axis (qJ = l/2, qK = l/6, qL = O-0; see Fig. 8 for examples). Another possibility is to have the probability of a correct turn be a function of the position along the X-axis so that an organism may have either an increasing or a decreasing ability to make a “correct” turn as it moves towards a stimulus source. The results obtained from these runs were as expected. With a constant turning preference of two-thirds towards the + X end of the gradient (stimulus source) a mean of x = 17.492 was obtained. An organism in which accuracy increased as it approached the stimulus (probability of a turn towards +X equalled 2/3+O-OO17X,-1; qJ = l/2, qK = l/6, qL = 04017) gave a mean of 1 = 17.906. An organism which approached the stimulus with a decreasing accuracy gave a mean of only 1 = 14.608.
30 20
:-- :.-:I‘, .’ ,,_.:
_. .
IO-
_‘.
.:.: :
0
------.
:
,.
;
:
..‘.:‘::‘i .;.
~ I:
:
++-:~+;;
;---‘----
.., _
..
,.
.,
. . ..
a.-. I.
:. -10
.’
i.,
..,..:. y
-.
: 1 I
‘;;
,,
,
-------_
,
-_-_-_-
4
‘.
y‘ ,,.; 1.’ ,
‘.
FIG. 9. Summary plot for the combined direct klinokinetic 2 = 0.338. 2.5. SIMULATED
and direct orthokinetic
model.
INTERACTIONS
When orthokinetic and taxic behavior are combined it is noted (see Table 1) that direct orthokinetic behavior complements a taxic response and shifts the mean further along the X-axis (faster locomation near a stimulus is more effective unless the accuracy of the tropotaxic response decreases as one approaches the stimulus source).
snfuLATIoN
OF
413
KLIN~IcINESIS
Klinokinetic and orthokinetic behavior can interact in such a way that a direct klinokinetic behavior and a direct orthokinetic behavior results in a less skewed distribution (see Fig. 9) than would result from the presence of orthokinetic behavior by itself (Fig. 7). This simulation would indicate that in an organism which moves more swiftly at high levels of stimulation than at low (direct orthokinesis), the demonstrated tendency to aggregate at low levels of stimulation would be reduced if at the same time the animal made more frequent turns at the high Ievel of stimulation. Inverse klinokinesis interacts with direct orthokinesis to cause an extremely skewed distribution (Fig. IO). This simulation would appear to indicate that, in an organism which moves more swiftly at high levels of stimulation than low (direct orthokinesis), the demonstrated tendency to aggregate at low levels becomes markedly greater if the animal makes more frequent turns at low levels of stimulation (i.e. shows an inverse klinokinesis). Figure 11 shows examples of 50 40
I------
. . .:
FIG. model.
10. Summary x= 1*62O.
plot
for
the combined
: ‘.
. . .
inverse
klinokinetic
and
direct
orthokinetic
414
F.
J.
ROHLF
AND
D.
DAVENPORT
the trace of single individuals. This bears out one’s intuitive impression that slow moving, tightly turning animals are likely to be “trapped” where they are.
0m ....,....0
Fm. 11. Trace of paths taken by simulated organisms based upon the combined inverse kIinolcmetic and direct orthokinetic model,
The most effective combination of factors for moving individuals (on the average) towards a stimulus was found to be direct orthokinetic behavior and inverse klinokinetic behavior coupled with a taxic response (Fig. 12). This simulation appears to be the most efficient of all our models in bringing about aggregation nearer the stimulus source. It says that organisms which have an ability to select the side of strongest stimulus (positive taxis) are rendered more efficient as target-finders if at the same time they move faster and straighter as they approach the target (Fig. 13 shows an example). This too bears out intuitive conclusions and reassures us that the simdation program is working satisfactorily. But to return to our original question concerning the adaptive significance of klinokinesis, we see that so far our simulations indicate that, far from increasing the probability of target-finding in organisms which have a positive taxis, a tendency to make more frequent turns at the higher levels of stimulation decreases the efficiency of target-finding. How, then, can we claim that the demonstrated phenomenon of the marked increase in the rate of turning (random in direction), which occurs when certain commensals are placed in a series of increasing concentration (each without a gradient) of a highly specific host-factor, renders greater their chances of encountering the host?
SIMULATION
OF
415
KLINOKCINESIS
I
50
-
40’-
. .
30 2010-
.* . . . . .. . .I 0 .,“-,-~r: . : .
y
-IO-
**..
-2o-
. :: . * ..
-30
-
.
.
-40’” -5o:-
! 1 -10
IO
20
I
I
I
30
40
50
I
60
70
80
1
I
I
I
90
100
110
120
X FIG. 12. Summary plot for the combined inverse klinokinetic, positive tropotaxic model. R= 23-166.
"..-.......ll-.
0
direct orthokinetic
and
-"..*...."..".........
Y
11 0
X
FIG. 13. Trace of path taken by simulated orga&n based upon the combii klinokiaetic, dimct orthokinetic aad positive tropotaxic modei.
inverse
416
F.
2.6.
J.
ROHLF
SIMULATION
AND
D.
DAVENPORT
OF SENSORY
ADAPTATION
The procedure used for the simulation of the effect of sensory adaptation was more complicated than those described above. First a basal level of stimulation (Ebasal) was defined. In the present experiments this was set at the level of stimulation equivalent in position to 50 units to the left of the starting point (low end of the gradient). Next, a maximal level of stimulation (E,,) was defined as being equivalent to the level of stimulation encountered when an organism was 50 units to the right of the starting point (high end of the gradient). Then a “memory” (M,) was simulated for the organism to “remember” the levels of stimulation it had encountered previously. At any given point in time the value for the memory was computed as qA = O-2 times the previous value for the memory plus qB = 0.8 times the current level of stimulation encountered by the organism (determined by the position of the organism along the X-axis). Thus, if an organism was steadily moving up or down the X-axis, the value for the memory would lag slightly behind the actual co-ordinate position of the organism along the X-axis. Once the organism stopped, the value of the memory would rather quickly catch up. Finally the “construction” of the organism was changed so that its behavior was not a direct function of either the present level of stimulation or the “memory” defined above but was determined by the following rules. If, at a given point in time, the organism experienced a level of stimulation higher “than it was accustomed to” (i.e. it was at a position on the X-axis higher than the value of its memory), then the organism would behave as if it were receiving the maximum level of stimulation (50 in the present case). This value of 50 was then saved in a second “memory” (EJ. If at a given point in time the position of the organism along the X-axis is equal to or less than the value of its memory, then the value in the second memory is decreased by certain proportions (qG) of the difference between its present value and the basal level (several values for this proportion were used; Table 3 and Figs 14 t0 21 were based on a Vahe qG = 0.2). The behavior of the organism was then made a direct function of this second memory (E,, see Fig. 1). The net effect of this somewhat arbitrary system of rules is that if an organism is heading along the X-axis in a positive direction, it will always react as if it were receiving the maximum level of stimulation (since at each step it would encounter a level of stimulation higher than that to which it was previously exposed). If, however, an organism is on the average heading down or across the gradient or is quiescent, then its response would rapidly become that which would result from its experiencing the basal level of stimulation. In summary, the results of our introduction of sensory adaptation into our models were as follows (see Table 3): first, a control model indicated
SIMULATION
OF
417
KLINOKINESIS
that randomly moving organisms (without orthokinetic, klinokinetic or tropotaxic behavior) with sensory adaptation showed no significant directional displacement (mean position along the gradient was X = 0,698). However, the introduction of sensory adaptation clearly facilitated displacement by orthokinesis, klinokinesis or tropotaxis, acting alone or in concert. Moreover, it is particularly interesting to note that when in our initial models klinokinesis without sensory adaptation resulted in no significant average displacement, now, with sensory adaptation, this behavior resulted in marked displacement. TABLE 3
Mean positions, X, of 500 independent artifcial organisms after I00 unit time intervals with the presence of sensory adaptation Pattern of tropotaxis
constant Orthokinesis ffa”gr at
-0
+end)
+
I1OllC
-
inverse (faster at -end) N&e: Within the cells -, absence, respectively.
48.320 14.381
Increasing accuracy as +X end is approached
-0
67.106 38.264
-0
72.151 45.182
+
19.686
+
29.090
19.882
-
32.620
-
35530
+” -14.070 0.698
0+
15.854 4.918
0+
20.896 9.096
- -6+44 0 -14909 + -24.528
2.580 0 -1.699 + -8460
-1,542
+, 0, correspond
1.702 0 -2.848 + -9.332
to inverse, direct klinokinesis
and its
3. Discwsion
Let us, then, conclude by attempting to evaluate these behaviors in accordance with their ability to effect displacement as shown by the models. Klinokinesis alone, with no sensory adaptation, results in no significant displacement. With sensory adaptation displacement appears. Inverse klinokinetic behavior results in displacement toward the ‘higher’ (+x> end of the gradient (mean position X = 19.882 on the X-axis). Direct klinokinetic behavior (more frequent turns at higher levels of stimulation) results in displacement toward the low (- X’) end of the gradient (see Figs 14 and 1s.)
418
F.
J.
ROHLF
AND
D.
DAVENPORT
0
X
FIG. 14. Trace of paths taken by simulated organisms based upon the combined direct klinokinetic and sensory adaptation model. See text for an explanation of the method of simulating the effect of sensory adaptation.
. :. . .
:. _‘.
. .
:
:
”
.
X 15. Summary plot for the combined direct klinokinetic model. R= -12.940. FIG.
and sensory adaptation
SIMULATION
OF ICLINOICINESIS
419
This bears out the theoretical conclusion of Ullyott (1936). Inverse klinokinetic behavior with sensory adaptation appears to have a slightly stronger displacement effect than does direct. Orthokinesis alone without sensory adaptation results in a slight displacement of the mean, which moves to the “fast” end of the gradient, but the highest density (mode) is in the region of low speed. Orthokinesis with sensory adaptation drastically shifts the mean of distribution to the “fast”‘end; displacement of the organisms will then be toward the region -where they are travelling faster (see Figs 16 and 17). This
X Fro. 16. Trace of paths taken by simulated organisms based upon the combined direct orthokinetic and sensory adaptation model.
implies that Gunn’s conclusion (1937) regarding aggregation in Porcellio can only be reached for organisms without or with only very slow sensory adaptation. Although in our models of this situation the average displacement of the organisms is up the gradient, the distribution will be skewed so that the greatest concentration of the organisms will be closer to the region of low speed. From the point of view of natural behavior, under these conditions, some organisms will travel a great distance in the direction of the “fast” end of the gradient, but most will be stuck closer to the region of low speed. Positive tropotaxis without sensory adaptation results in a marked displacement toward the plus end, while negative results in similar displacement toward the minus end. Sensory adaptation appears to have little effect on this behavior in our models. Table 3 also shows that the effect of combining orthokinetic and klinokinetic behavior with sensory adaptation can be predicted by the effects of each taken separately. As exampks, an inverse klinokinesis (straight runs at high levels of stimulation) plus direct orthokinesis (faster at higher levels of
F. J. ROHLF
AND D. DAVENPORT
401
-2o-30;
t
-401 I
-20
-iO
-IL-...
IO
L
20
30
1
40
-I-
50
60
70
x FIG. 17. Summary plot for the combined direct orthokinetic model. 8= 14.381.
and sensory adaptation
stimulation) results in a great displacement toward the high end of stimulation, the two behaviors in combination resulting in a greater displacement than either alone (mean position X = 48.320, also see Figs 18 and 19). However, direct klinokinesis (straighter at low levels of stimulation) plus direct orthokinesis result in essentially no displacement; the two effects cancel each other out (mean position X = - 1542, see also Figs 20 and 21). The above are the most dramatic effects of combinations of behavior. The reader can interpret other interrelationships of these ways of behaving from the numerical data in the table. If we introduce tropotaxic behavior it can be seen that with sensory adaptation adding tropotaxis to direct orthokinesis and inverse klinokinesis effects the greatest displacement of all (mean position along the gradient w = 72.151). This means that an animal that senses asymmetries in stimulation and runs fast and straight as it moves into higher levels of stimulation
SIMULATION
OF
KLINOKINBSIS
421
6050 4030 20 * IO f
0 - - - - - - _ -.
-to-
.
-20
-
-30
-
-40
-
-50
-10
Fxo. 18. Summary plot for the combined inverse kIin&inetic, direct ortbokinetic, and sensory adaptation model. 2 = 48.320.
will have the greatest probability of climbing the gradient toward the stimuhrs source. At the opposite extreme we seethe organism with a positive taxis and an inverse orthokinesis. In such an animal which sensesand turns towards higher levels of stimulation but which moves slower as it climbs the gradient, displacementwill not be great. Finally, even though the simulations with sensory adaptation were more complicated than those without, they are still obviously oversimplifications of the behavior of real organisms.Many sorts of refinements could be made on the models which we have used, such as, for instance, the introduction of
422
F. J. ROHLF
AND D. DAVENPORT
FIG. 19. Trace of paths taken by simulated organisms based upon the combined inverse klinokinetic, direct orthokinetic and sensory adaptation model.
different rates of sensory adaptation or by making stimulus intensity a nonlinear function of the position of an organism along the X-axis. However, the study is sufficient to demonstrate that the very simple type of sensory adaptation used in these models is enough to effect displacement in organisms showing orthokinetic and klinokinetic responses without tropotaxis. These results confirm the general conclusions reached by Fraenkel & Gunn (1961) and suggest that the term “klinokinesis” should not yet be dropped, as suggested by Gunn (1966). Clearly the next phase of research in this area should be to study the degree of agreement between the behavior of an actual organism with the behavior predicted by our models, using estimates of behavioral parameters (probability of turning, velocity, etc.) determined from studies of particular organisms. Perhaps it will be possible to resolve a number of dilemmas, the most important being to assign some raison d’btre or selective advantage to the direct klinokinesis exhibited by certain associated animals to a speciesspecific factor emenating from their host (Davenport et al., 1960). Our models indicate that such behavior gives such an animal no advantage in hostfinding, even when working in concert with other behaviors. Obviously, much remains to be done to relate our models to real animals.
SIMULATION
OF
423
KLINOKINESIS
30 20
.
IO y
o-1 -10,
-20
:..y ‘.;.- *:: . . . . :;‘,i.Jy .*** .* -1. . ..f’.:. : *- . .I . .
I
.i
-30 -20
-30
-10
0
IO
20’
30
40
50
60
X
FIO. 20. Summary plot for the combined direct klinokinetic, direct orthokinetic and sensory adaptation model. x = -1.542.
Y 0 .. ... .. .. ... : El 0
... ...
0lTYE!d .... . .. 0
FIG. 21. Trace of paths taken by simulated organisms based upon the combined direct klinokinetic, direct orthokinetic and sensory adaptation model.
This is Contribution No. 1387 from the department of Entomology, the University of Kansas, Lawrence, Kansas. Computer time for this project was made available by Western Data Processing Center, Los Angeles, California (IBM 7040-7094 DCS) and the University of Kansas Computation Center (IBM 7040 and GE 625). The figures were prepared using the B/L plotter at the University of Kansas Computation Center. The computer programs were prepared with the
424
F.
J.
ROHLF
AND
D.
DAVENPORT
assistance of Mr Ron Bartcher. The authors would like to thank Dr C. D. Michener who read and criticized the manuscript and Patricia Rohlf who assisted in the preparation of the illustrations. This work was partially supported by grants from the Office of Naval Research (Contract NONR 4-222(03)) and the National Science Foundation (GB 5137) to Dr Davenport and from the National Institutes of Health (GM 1193) to Dr R. R. Sokal. REFERENCES BAILEY, N. T. J. (1964).“The Elements of StochasticProcesses.” New York: Wiley. DAVENPORT, D., CAMOUGIS, G. & HICKOK, J. F. (1960). Anim. Behav. 8, 3, 209. DAVENPORT, D., KRAMER, H. & THORSON, J. (1960).TEMPO, Center for Advanced
Studies,GeneralElectricCo., SantaBarbara,Calif., ReportSP-95.
EVANS, G. W., WALLACE, G. F. & SUTHERLAND, G. (1967). “Simulation Using Digital Computers.” New York: Prentice Hall. EWER, D. W. & BURSELL, E. (1950). Be/z&our, 3,40. FRAENKEL, GO~TFRIED S. & GUNN, DAVID L. (1961). “The Orientation of AnimalsKineses, Taxes and Compass Reactions”. New York: Dover. GUNN, D. L. (1937). J. exp. Biol. 14, 178. GUNN, D. L. (1966). In “Insect Behaviour”, Symposium of the Royal Entomological Society No. 3, pp. 109-110. P. T. Haskell, ed. GUNN, D. L., KENNEDY, J. S. & PIELOU, D. P. (1937). Nature, Lord. 140, 1064. KENNEDY, J. S. (1945). Nature, Lord. 154, 754. NAYLOR, T. H., BALINT~Y, J. L., BURDICK, D. S. & CHU, K. (1966). “Computer Simulation Techniques.” New York: Wiley. PATLAK, CLIFFORD L. (1953~). Bull. math. Biophys., 15, 311. PATLAK, CLIFFORDL. (19536). Bull. math. Biophys. 15,431. PARZEN, E. (1962). “Stochastic Processes.” San Francisco: Holden-Day. PEARSON.K. (1906). “Mathematical contributions to theorv of Evolution. XV. A mathematical theory of random migration”. Draper’s Co. Research Memoirs, Biometric Series III. 54 pp. SAILA, S. B. & SHAPPY, R. A. (1963). J. Cons. perm. int. Explor. Mer, 28, 153. ULLYO~~, P. (1936). J. exp. Biol. 13, 253. ,
.
I
Simulation of Simple Models of Animal Behavior with a Digital Computer F. JAMES ROHLF Department
of Entomology, The University of Kansas, Lawrence,
Kansas,
U.S.A.
DEMOREST DAVENPORT Department
of Biological Sciences, The University of California, Santa Barbara,
Calif,
U.S.A.
(Received 18 September 1968, and in revised form 5 December 1968) The effects of kinetic, klinokinetic, orthokinetic and tropotaxix behavior were simulated on a digital computer in order to study the interactions between these simple behaviors and the effect of sensory adaptation under completely controlled experimental conditions. The conclusions of Ulyott that klinokinetic behavior in the presence of sensory adaptation can cause a directional displacement of an organism along a gradient of stimulus intensity were confirmed. In addition it was found that inverse klinokinetic and direct orthokinetic behavior (with sensory adaptation) result in a marked directional displacement up a gradient. However, direct klinokinetic behavior cancels out the effect of direct orthokinetic behavior.
1. Introduction In the field of invertebrate behavior, undirected locomotory reactions, in which the speed of movement or the frequency of turning depend upon the intensity of stimulation, are defined as kineses (Fraenkel & Gunn, 1961). A klinokinetic response is defined as a change in the rate of change of direction, in response to a change in stimulation level, while an orthokinesis is a change in linear velocity in response to such an intensity change (Gunn, Kennedy & Pielou, 1937; Ewer & Bursell, 1950). In klinokineses, direction of turning is always random. Klinokineses are, therefore, quite distinct from taxes, which by definition always involve non-random “choice” of direction in a gradient. There is ample experimental evidence that the phenomenon of klinokinesis (as distinct from the taxis) exists and is a function of stimulus intensity 400
SIMULATION
0~ K~JNOI~NESIS
401
(Ullyott, 1936; Davenport, Camougis & Hickok, 1960; etc.). Some animals may turn more frequently at high levels of stimulation and some at low levels of stimulation, but in any case the direction of their turns is random. However, the work of Ullyott (1936) and others notwithstanding, there is still no convincing evidence (from experiments including controls to eliminate the possible presence of taxes) that klinokinesis alone, with or without sensory adaptation, can effect the aggregation of organisms at one end or the other of a stimulus gradient (Gunn, 1966). The purpose of the present paper is to find out whether an organism showing only klinokinetic behaviour and sensory adaption can move up and down a smooth gradient. Whether such organisms exist is a separate question. However, even in the most recent literature, the efficacy of klinokinesis in bringing about aggregation or directional displacement in a gradient is often assumed. Furthermore, all too often discussions of the phenomenon are tinged with purposiveness, in such a way that animals are said to approach a “preferred” environment until conditions “improve”. We submit that such words, even if used by authors as convenient shorthand, imply value judgments on the part of the organism in space or time which clearly can have no part in any machinery of displacement in which, by definition, the direction of turns is random relative to the gradient. J. S. Kennedy (1945) has advised that the sign-terms “positive” and “negative” as applied to klinokineses be abandoned, since these terms have directional implications. We recommend the use of his adjectives “direct” and “inverse” for both klinokinesis and orthokinesis, i.e. for situations in which the rate of random turning (or linear velocity) is directly and inversely proportional, respectively, to stimulus intensity. There is enough evidence (uide Fraenkel BE Gunn, 1961) of the adaptive significance of directional “choice” (taxis) to establish its importance Cmly in the behavioral economy of invertebrates. In addition, the adaptive significance of orthokinesis, in which linear velocity is a function of stimulus intensity, is clear (Gunn, 1937, 1966); all other behavior being equal, an organism which moves more slowly under stimulus level I than under stimulus level II will “aggregate” where stimulus level I pertains. However, the only evidence we have of the efficacy of pure klinokinesis in bringing about aggregation, “target-finding”, or displacement’ up or down a gradient is circumstantial and depends upon such observations as those of Ullyott (1936) and demonstrations that in certain animal associations a highly speciesspecific klinokinetic response may be observed .when an animal is in a medium containing a chemical signal from its host (Davenport et al., 1960). Such response specificity clearly suggests that the response may have adaptive significance but controlled experiments to demonstrate this are not at hand.
402
F.
J.
ROHLF
AND
D.
DAVENPORT
It would naturally be desirable to design experiments, using animals, in which one could determine whether an observable klinokinesis, working alone without directional cues, can effect aggregation or displacement in a gradient. But, even if light is used as a stimulus, how are all directional cues and their effects to be indubitably eliminated? If an animal with bilaterally symmetrical sense organs is used, then the possibility of unequal stimulation of the organs in the gradient and a consequent directed response is always at hand. If a single sense organ is involved, then in a gradient there is always the possibility of successive testing of the gradient in time. We have therefore been continuously faced with a problem, unsolved to date, in designing experimental protocol. We may, nevertheless, ask some questions which may give us a possible insight into whether or not an organism exhibiting klinokinetic behavior has a selective advantage. We may ask what the properties are of a theoretical idealized animal having this type of behavior. The klinokinetic response must, of course, be considered alone and then in conjunction with other types of responses (orthokineses and taxes). Clearly it is likely that there may be an interaction between these responses. The displacement of an organism, resulting from the interaction of these responses, is more complex than might first be thought. An intuitive analysis of the results of such interaction is not only extremely difficult but dangerous, although it must be admitted that a long-lasting intuitive distrust by one of us of Ullyott’s interpretation of his results, has led to this effort. An alternative, until such a time as we have an adequate experimental protocol dealing with actual organisms, is to use a mathematical analysis of the behavior of “ideal” organisms. However, just as an experiment to demonstrate the efficacy of klinokinesis in living animals has the disadvantage that it is almost impossible to eliminate directional cues, so our alternative of mathematically analyzing the behavior of an ideal organism has the disadvantage that in order to be able to obtain a solution one may have to simplify the properties of the organism to a point at which one’s assumptions become so unrealistic as to render the results of little real biological significance. Nevertheless, it seems of value to undertake such an analysis. The approach taken in the present study was an analysis of the behavior of an ideal hypothetical organism, using techniques of simulation with a digital computer. The approach has the advantage that it is relatively simple to add complexity to the system. A large number of “experiments” with hypothetical organisms are performed and then a statistical analysis is performed on the results. The mode of attack is very much like the experimental approach, except that here the stimuli being applied are known; only their effect is unknown. We refer the reader to Evans, Wallace $ Sutherland (1967)
SIMULATION
OF
403
KLINOKINESIS
and Naylor el al. (1966) for general accounts on digital simulation techniques. There have been a number of previous efforts to study the motions of organisms. Pearson (1906) investigated the use of the random walk method for the case of random migration (no stimuli). In the classical random walk model an organism moves in a series of steps, where the length of the step, time between steps, and the direction of the step are independent of one another and of preceding steps. If we allow the length of the step to be continuous, we may consider the random walk as a Markov process and describe the system in terms of the Chapman-Kolmogorov equations, which yield the probability of an organism being at any point at any time. One may also solve for the steady-state condition (if it exists) using the Fokker-Planck equation which yields the probability of an organism being at any point after a sufficient period of time has elapsed so that these probabilities no longer depend upon time. General accounts and applications piiq $q +TkTl 4 Set initial state for ith organism r=o, P:=(xt,Y:)=(O,O), M,=O
Compute efkt of previous stimulation Mt=9~Mt-l +qrxt-1. Compute efkt of prcaent stimulation &“.l=qC+qDxt-1
Emu =qEfqF%-1 Em.x,ifXt-1=-M-l
Et- I- q&3-1 - &ml,
otherwise
Construct random munber R, @ 5 R 5 1*O). 1
404
F.
J.
ROHLF
AND
D.
DAVENPORT
P
No
R,(O_
Compute new position of organism
No --a Yes STOP FIG.
1. Flow chart of simulation program. See text for explanation.
of these equations to biology are given in Bailey (1964) and Parzen (1962). Davenport, Kramer & Thorson (1960) investigated klino- and orthokinetic models using both the above approach and that of simulation. They found that a simple klinokinetic model resulted in a uniform distribution of the organisms, whereas an orthokinetic model resulted in a very skewed distribution (the long tail of the distribution was in the direction of the region in which the organisms had the highest velocity). Patlak (1953~) further developed the mathematical methods needed to incorporate the effects of “external forces” (stimulus which causes a taxic component or an avoidance
SIMULATION
OF KLINOKINESIS
405
reaction in the organism’s motion) and “persistence of direction” (which implies that when an organism turns, it will not turn with equal probability in all possible directions). Patlak (1953b) described the application of these methods to the study of the movement of organisms. He found the distinction between klino- and orthokinetic behavior not to be a useful one in terms of his model, since the steady-state distribution was a function of (1) average time between turns, (2) average distance between turns, and (3) average squared distance between turns, “rather than a simple function of average velocity and time between turns alone”. He also described methods for estimating the various parameters in his model and investigated the degree of fit between results previously obtained by others and his model which he fitted to their data. He concluded that the major reason for lack of agreement was probably due to the organism’s accommodation (sensory adaptation) which was not taken into account in his model. However, if the rate of accommodation was very slow it could be ignored. If, on the other hand, it adapts very rapidly, then its behavior will largely be a function of only its previous state and the model may still be used. If the time of accommodation is between these extremes (the organism’s behavior is a function of several of its past states, which seems likely) then the model cannot be used since we no longer have a Markov process. One of the purposes of the present study is to investigate the effect of such a type of adaptation and especially its interaction with other behavioral patterns. The extension of Patlak’s results seemed to pose problems of formidable mathematical complexity. Thus, the present work is entirely in terms of simulation experiments and follows the general approach used by Saila dc Shappy (1963) in their study of random movement and orientation in salmon migration. A disadvantage of the simulation approach is that one does not obtain a general mathematical solution but rather a great quantity of particular solutions. A flow chart of the computer program (coded in FORTRAN IV for the GE 625 computer) is given in Fig. 1. The variables qA through qN are the parameters which specify the behavioral pattern being simulated. The effects of these parameters are discussed separately for each behavioral pattern. The choice of the particular set of values of these parameters to be used in the simulations was fairly arbitrary. Few levels of each parameter could be used since we could not afford to run all possible combinations of a very wide variety of values. In the initial set of simulation models (sections 2.1 to 2.5 below) no sensory adaptation was allowed for (qA = O-0, qB = 1.0, qc = O-0, q,, = l-0, qE = O-0, qF = l-0, q. = O-0, thus Et = X,-J. The multiplicative congruential pseudo random number generator R,+l = (513R&d 2j6 was used
406
F.
J.
ROHLF
AND
D.
DAVENPORT
2. Methods and Results 2.1. A SIMPLE KINETIC MODEL As a simplest model we may visualize a hypothetical organism placed in the center of a grid. The position, P,, of the organism on this grid can be denoted by a pair of (X,, Y,) co-ordinates (X, denotes position along gradient at time t). When the organism is initially placed in the center of the grid (0, 0), the initial direction in which it faces is determined at random (with equal probabilities). Four possibilities are allowed: it may face up (+ Y), down (- Y), left (- X), or right (+ X). Next, a decision is made as to whether or not the organism will continue in the direction in which it is currently facing or will change its direction by turning clockwise or counter-clockwise (qH = 0.5, qr = O-0). After this is determined, the organism is then allowed to take a step of length l-0 (qJ = 1.0, qx = 0.0) in that the new direction. The turning decision and stepping is then repeated 100 times. Figure 2 shows
(-j 5... .....*....*.*...
0.:... ... i.d 0
~ 0
FIG. 2. Trace of paths taken by simulated organisms based upon a simple kinetic model. The “organism” started at the point X = 0, Y = 0 (shown as the intersection of the broken lines) and during each of the 100 time intervals in which it was followed, it could move one unit clockwise, counterclockwise, or continue in its present direction with probabilities of l/4, l/4 and l/2, respectively. On this and all of the subsequent figures each tick mark along the axes represents 10 units on a grid laid over a hypothetical area.
an example of the path by four such individuals. The final position of the organism after 100 steps, Pioo, is recorded and the entire process repeated until the results of 500 such trials have been accumulated. When this experiment was carried out, we found that the mean position of the 500 organisms along the X-axis was X = O-058 (Table 1) with a standard deviation of 12,602 (Table 2). The resultant distribution of organisms across
SIMULATION
407
OF KLINOKINESIS
TABLE 1
Mean position, x, of 500 independent artificial organisms after 100 unit time intervals Pattern of tropotaxis Decreasing accuracy as Constant +X end is approached
None Orthokinesis direct (faster at + end)
- 1.620 0 1.654 + 0.338
- 23.166 0 21.454 + 18.729
- 12.296 0 13.247 + 13,290
- 27.368 0 23.717 + 23.392
- 0.512 0 0.058 + 0.262
- 16540 0 17.492 + 16.552
- 14.780 0 14608 + 16.364
- 19.052 0 17906 + 16.858
I -076 0 -0.893 + -2.074
- 14.054 0 12.913 + 12.956
- 17.347 0 18.397 + 20.197
- 14.568 0 14.548 + 13.884
none
inverse (faster at -end)
Increasing accuracy as +X end is approached
Note: Within the cells -, absence, respectively.
f,
0, correspond
to inverse, direct klinokinesis
and its
TABLE 2
Standard deviations of the position of 500 independent artificial organisms after 100 unit time intervals
None Orthokinesis direct (faster at +end)
Pattern of tropotaxis Decreasing accuracy as constant +Xendis approached
Increasing accuracy as +Xend is approached
- 12.879 0 12.883 + 11.684
- 19462 0 16.223 + 14062
0 +
8,646 8.042 8224
- 23.678 0 19221 + 17.697
none
- 12.500 0 12.602 + 12.087
- 13.022 0 11.018 + 10460
- 10.577 0 10.379 + 11.252
- 13.687 0 12.919 + 12.401
inverse (faster at -end)
- 11.810 0 12.079 + 14.132
0 +
- 13.269 0 14.645 + 17.411
0 +
Note: Within the cells -, absence, respectively.
8.340 8.424 8.181
+, 0, correspond to itwerse, direct klinokiiesis
9.455 8.813 9-626 and its
408
F.
I
J.
ROHLF
AND
D.
DAVENPORT
I
-30
-20
-10
b
IO
20
30
X
FIG. 3. Summary plot for the simple kinetic model showing the final position of 500 independent organisms after 100 unit time intervals. T?= 0.058.
the grid (Fig. 3) is reminiscent of correlation equal to zero and mean allowed to take fewer steps, or more the same but the variance increases 2.2. (WITHOUT
A SIMPLE A MODEL
a bivariate normal distribution with a at x = 0, Y = 0. If the organisms were steps, the mean position would be about as a function of time.
KLINOKINETIC OF SENSORY
MODEL ADAPTATION)
We may now complicate the behavior of our hypothetical organisms slightly by changing the decision rule for deciding whether the organism continues in the direction in which it was facing or changes its direction, by making the probability of a change of direction a simple function of its position along the X-axis. In the present experiments, we let the probability of a turn (qH+qIl?,) be 0~5+0@05X,-, for direct and 0~5-0~005X,-, for inverse klinokinesis (probability of clockwise turn = 0.5, qJ = O-5, qK = qL = O-0). Figure 4 is an example. This would mean that if we imagined our grid going from plus to minus 100 (representing a gradient going from a low level of stimuIation at the left to a high level at the right side of the grid), our “organism” would have a turning probability ranging from 0 to 1. When such an “experiment” was run (see Fig. 5) we found mean values of
SIMULATION
OF KLINOKINESIS
0.......” .......... M 0
* ~ : .. ... .. ... ... . 0
....... : : y i
........
0
.. . .,........~.........
0
F?!d OX
m
6
FIG. 4. Trace of pathstaken by simulatedorganismsbasedupon a modelof direct klinokineticbehavior.A gradient of stimulusintensityis assumed to exist(for this andall subsequent figures)alongthe X-axis.Probability of a turn equals0.5 + OG05X1-I.
. .
30 -
I
. y
0
-_-__-__________; . t
-10
'. ..
t
-20 -3o-
, -40
.
..
.,. __-_-_ .
.:
:
: I
. ’
.
:,
--,
+,,
. I. .: ;:. . . : : ., : . 1.: '_._ ..;I. . :::.:* ..:::: 1.1 1 ; . . 1.. *'.: . :. . ..j' . . . ~1 : :: .:: :.. :
.
, I
.. .
. .
; ‘.i. . :.’ . . . . .. ‘. ‘.’ ... I,-. . I I I , . I . -30 -20 -10 b IO 20 30 X
FIG. 5. Summaryplot for the direct klinolcineticmodelshowingthe tial positionsof 500independent organisms after 100unit timeintervals.x = O-262.
X = O-262 for direct klinokinesis and a value of X = O-512 for inverse klinokinesis. These values are not significantly different from zero. This model appears to bear out Ullyott’s conclusion that klinokinesis without sensory adaptation can have no aggregating effect.
410
F.
J.
ROHLF
AND
D.
DAVENPORT
2.3. AN ORTHOKINETIC MODEL Independent of the above modification to the program, one can make the length of the “step” taken in each unit time interval a function of the position of the organism along the X-axis. For example, at the right it might take larger steps and at the left take smaller ones (Fig. 6 shows examples). In
0m ...0...
0 .. .. ... ... .. .. . ... ...,
LIYB
0.....i.... Eld
d
Y 0 lY!I?cJ .. ... .. . . . . ... ... . .. .. ... .. x
.. .. 0
FIG. 6. Trace of paths taken by simulated organisms based upon a model of direct orthokinetic behavior. Length of step equals 1 .O + 0.02X, 1.
, I I I I I I
40-
30’ i
“‘I’..
201
,:
.,
.’
:
:.
.I
I
-30
I
t
I :..
!
j
, -20
I -10
0
‘.
:’
. t IO
. I 20
I 30
I 40
x FIG.
7. Summary plot for
the
direct orthokinetic
model. R = 1 e654.
SIMULATION
OF KLIIWKINEbIS
411
these experiments, a step size function (qM+q&) was computed as 1 .O+O.O2X,-i. Thus, in a given experiment, the size of the step could range from a minimum of 0.0 at X = -5Oupto3~0atX=+lOO.Themost dramatic effect of orthokinetic behavior (see Fig. 7) is that it strongly skews the distribution of individuals along the X-axis. The long tail of the distribution is in the direction of the region in which the organisms have a higher velocity. A mean of X = 1.654 and a median of 0417 were obtained for direct orthokinesis. The standard deviation (13.386) of the position of the organisms along the X-axis was also slightly increased over that found for the previous runs. Clearly, this model bears out results obtained in experiments (Gunn, 1937) indicating that, other behaviors being equal, the region of highest density of organisms occurs in regions where activity (speed of movement) is reduced. 2.4. TROPOTAXIC BEHAVIOR Also, independent of the above, we may endow our organism with paired sense organs so that it is capable of detecting differences in intensity of a stimulus along a gradient when it is facing at right angles to this gradient. Since taxic behavior has been experimentally demonstrated to have such a strong effect, we felt it would be of interest to consider only models in which the ability of the organism to make the “correct” decision was something less C
FIO. 8. Trace of paths taken by simulated organisms based upon a model of positive tropotaxis.Probabilityofaturntowardsth6+XendoftbeOridis2/3~~olganinn is facing acraas the gradient (up or down). When facing along the gradient the probabilie of a turn is l/2. 27 T.B.
412
P.
J.
ROHLF
AND
D.
DAVENPORT
than perfect. Several models were considered. In the simplest case, the organism has a constant probability (equal to 2/3) of making a “correct” turn when it is facing across the X-axis (qJ = l/2, qK = l/6, qL = O-0; see Fig. 8 for examples). Another possibility is to have the probability of a correct turn be a function of the position along the X-axis so that an organism may have either an increasing or a decreasing ability to make a “correct” turn as it moves towards a stimulus source. The results obtained from these runs were as expected. With a constant turning preference of two-thirds towards the + X end of the gradient (stimulus source) a mean of x = 17.492 was obtained. An organism in which accuracy increased as it approached the stimulus (probability of a turn towards +X equalled 2/3+O-OO17X,-1; qJ = l/2, qK = l/6, qL = 04017) gave a mean of 1 = 17.906. An organism which approached the stimulus with a decreasing accuracy gave a mean of only 1 = 14.608.
30 20
:-- :.-:I‘, .’ ,,_.:
_. .
IO-
_‘.
.:.: :
0
------.
:
,.
;
:
..‘.:‘::‘i .;.
~ I:
:
++-:~+;;
;---‘----
.., _
..
,.
.,
. . ..
a.-. I.
:. -10
.’
i.,
..,..:. y
-.
: 1 I
‘;;
,,
,
-------_
,
-_-_-_-
4
‘.
y‘ ,,.; 1.’ ,
‘.
FIG. 9. Summary plot for the combined direct klinokinetic 2 = 0.338. 2.5. SIMULATED
and direct orthokinetic
model.
INTERACTIONS
When orthokinetic and taxic behavior are combined it is noted (see Table 1) that direct orthokinetic behavior complements a taxic response and shifts the mean further along the X-axis (faster locomation near a stimulus is more effective unless the accuracy of the tropotaxic response decreases as one approaches the stimulus source).
snfuLATIoN
OF
413
KLIN~IcINESIS
Klinokinetic and orthokinetic behavior can interact in such a way that a direct klinokinetic behavior and a direct orthokinetic behavior results in a less skewed distribution (see Fig. 9) than would result from the presence of orthokinetic behavior by itself (Fig. 7). This simulation would indicate that in an organism which moves more swiftly at high levels of stimulation than at low (direct orthokinesis), the demonstrated tendency to aggregate at low levels of stimulation would be reduced if at the same time the animal made more frequent turns at the high Ievel of stimulation. Inverse klinokinesis interacts with direct orthokinesis to cause an extremely skewed distribution (Fig. IO). This simulation would appear to indicate that, in an organism which moves more swiftly at high levels of stimulation than low (direct orthokinesis), the demonstrated tendency to aggregate at low levels becomes markedly greater if the animal makes more frequent turns at low levels of stimulation (i.e. shows an inverse klinokinesis). Figure 11 shows examples of 50 40
I------
. . .:
FIG. model.
10. Summary x= 1*62O.
plot
for
the combined
: ‘.
. . .
inverse
klinokinetic
and
direct
orthokinetic
414
F.
J.
ROHLF
AND
D.
DAVENPORT
the trace of single individuals. This bears out one’s intuitive impression that slow moving, tightly turning animals are likely to be “trapped” where they are.
0m ....,....0
Fm. 11. Trace of paths taken by simulated organisms based upon the combined inverse kIinolcmetic and direct orthokinetic model,
The most effective combination of factors for moving individuals (on the average) towards a stimulus was found to be direct orthokinetic behavior and inverse klinokinetic behavior coupled with a taxic response (Fig. 12). This simulation appears to be the most efficient of all our models in bringing about aggregation nearer the stimulus source. It says that organisms which have an ability to select the side of strongest stimulus (positive taxis) are rendered more efficient as target-finders if at the same time they move faster and straighter as they approach the target (Fig. 13 shows an example). This too bears out intuitive conclusions and reassures us that the simdation program is working satisfactorily. But to return to our original question concerning the adaptive significance of klinokinesis, we see that so far our simulations indicate that, far from increasing the probability of target-finding in organisms which have a positive taxis, a tendency to make more frequent turns at the higher levels of stimulation decreases the efficiency of target-finding. How, then, can we claim that the demonstrated phenomenon of the marked increase in the rate of turning (random in direction), which occurs when certain commensals are placed in a series of increasing concentration (each without a gradient) of a highly specific host-factor, renders greater their chances of encountering the host?
SIMULATION
OF
415
KLINOKCINESIS
I
50
-
40’-
. .
30 2010-
.* . . . . .. . .I 0 .,“-,-~r: . : .
y
-IO-
**..
-2o-
. :: . * ..
-30
-
.
.
-40’” -5o:-
! 1 -10
IO
20
I
I
I
30
40
50
I
60
70
80
1
I
I
I
90
100
110
120
X FIG. 12. Summary plot for the combined inverse klinokinetic, positive tropotaxic model. R= 23-166.
"..-.......ll-.
0
direct orthokinetic
and
-"..*...."..".........
Y
11 0
X
FIG. 13. Trace of path taken by simulated orga&n based upon the combii klinokiaetic, dimct orthokinetic aad positive tropotaxic modei.
inverse
416
F.
2.6.
J.
ROHLF
SIMULATION
AND
D.
DAVENPORT
OF SENSORY
ADAPTATION
The procedure used for the simulation of the effect of sensory adaptation was more complicated than those described above. First a basal level of stimulation (Ebasal) was defined. In the present experiments this was set at the level of stimulation equivalent in position to 50 units to the left of the starting point (low end of the gradient). Next, a maximal level of stimulation (E,,) was defined as being equivalent to the level of stimulation encountered when an organism was 50 units to the right of the starting point (high end of the gradient). Then a “memory” (M,) was simulated for the organism to “remember” the levels of stimulation it had encountered previously. At any given point in time the value for the memory was computed as qA = O-2 times the previous value for the memory plus qB = 0.8 times the current level of stimulation encountered by the organism (determined by the position of the organism along the X-axis). Thus, if an organism was steadily moving up or down the X-axis, the value for the memory would lag slightly behind the actual co-ordinate position of the organism along the X-axis. Once the organism stopped, the value of the memory would rather quickly catch up. Finally the “construction” of the organism was changed so that its behavior was not a direct function of either the present level of stimulation or the “memory” defined above but was determined by the following rules. If, at a given point in time, the organism experienced a level of stimulation higher “than it was accustomed to” (i.e. it was at a position on the X-axis higher than the value of its memory), then the organism would behave as if it were receiving the maximum level of stimulation (50 in the present case). This value of 50 was then saved in a second “memory” (EJ. If at a given point in time the position of the organism along the X-axis is equal to or less than the value of its memory, then the value in the second memory is decreased by certain proportions (qG) of the difference between its present value and the basal level (several values for this proportion were used; Table 3 and Figs 14 t0 21 were based on a Vahe qG = 0.2). The behavior of the organism was then made a direct function of this second memory (E,, see Fig. 1). The net effect of this somewhat arbitrary system of rules is that if an organism is heading along the X-axis in a positive direction, it will always react as if it were receiving the maximum level of stimulation (since at each step it would encounter a level of stimulation higher than that to which it was previously exposed). If, however, an organism is on the average heading down or across the gradient or is quiescent, then its response would rapidly become that which would result from its experiencing the basal level of stimulation. In summary, the results of our introduction of sensory adaptation into our models were as follows (see Table 3): first, a control model indicated
SIMULATION
OF
417
KLINOKINESIS
that randomly moving organisms (without orthokinetic, klinokinetic or tropotaxic behavior) with sensory adaptation showed no significant directional displacement (mean position along the gradient was X = 0,698). However, the introduction of sensory adaptation clearly facilitated displacement by orthokinesis, klinokinesis or tropotaxis, acting alone or in concert. Moreover, it is particularly interesting to note that when in our initial models klinokinesis without sensory adaptation resulted in no significant average displacement, now, with sensory adaptation, this behavior resulted in marked displacement. TABLE 3
Mean positions, X, of 500 independent artifcial organisms after I00 unit time intervals with the presence of sensory adaptation Pattern of tropotaxis
constant Orthokinesis ffa”gr at
-0
+end)
+
I1OllC
-
inverse (faster at -end) N&e: Within the cells -, absence, respectively.
48.320 14.381
Increasing accuracy as +X end is approached
-0
67.106 38.264
-0
72.151 45.182
+
19.686
+
29.090
19.882
-
32.620
-
35530
+” -14.070 0.698
0+
15.854 4.918
0+
20.896 9.096
- -6+44 0 -14909 + -24.528
2.580 0 -1.699 + -8460
-1,542
+, 0, correspond
1.702 0 -2.848 + -9.332
to inverse, direct klinokinesis
and its
3. Discwsion
Let us, then, conclude by attempting to evaluate these behaviors in accordance with their ability to effect displacement as shown by the models. Klinokinesis alone, with no sensory adaptation, results in no significant displacement. With sensory adaptation displacement appears. Inverse klinokinetic behavior results in displacement toward the ‘higher’ (+x> end of the gradient (mean position X = 19.882 on the X-axis). Direct klinokinetic behavior (more frequent turns at higher levels of stimulation) results in displacement toward the low (- X’) end of the gradient (see Figs 14 and 1s.)
418
F.
J.
ROHLF
AND
D.
DAVENPORT
0
X
FIG. 14. Trace of paths taken by simulated organisms based upon the combined direct klinokinetic and sensory adaptation model. See text for an explanation of the method of simulating the effect of sensory adaptation.
. :. . .
:. _‘.
. .
:
:
”
.
X 15. Summary plot for the combined direct klinokinetic model. R= -12.940. FIG.
and sensory adaptation
SIMULATION
OF ICLINOICINESIS
419
This bears out the theoretical conclusion of Ullyott (1936). Inverse klinokinetic behavior with sensory adaptation appears to have a slightly stronger displacement effect than does direct. Orthokinesis alone without sensory adaptation results in a slight displacement of the mean, which moves to the “fast” end of the gradient, but the highest density (mode) is in the region of low speed. Orthokinesis with sensory adaptation drastically shifts the mean of distribution to the “fast”‘end; displacement of the organisms will then be toward the region -where they are travelling faster (see Figs 16 and 17). This
X Fro. 16. Trace of paths taken by simulated organisms based upon the combined direct orthokinetic and sensory adaptation model.
implies that Gunn’s conclusion (1937) regarding aggregation in Porcellio can only be reached for organisms without or with only very slow sensory adaptation. Although in our models of this situation the average displacement of the organisms is up the gradient, the distribution will be skewed so that the greatest concentration of the organisms will be closer to the region of low speed. From the point of view of natural behavior, under these conditions, some organisms will travel a great distance in the direction of the “fast” end of the gradient, but most will be stuck closer to the region of low speed. Positive tropotaxis without sensory adaptation results in a marked displacement toward the plus end, while negative results in similar displacement toward the minus end. Sensory adaptation appears to have little effect on this behavior in our models. Table 3 also shows that the effect of combining orthokinetic and klinokinetic behavior with sensory adaptation can be predicted by the effects of each taken separately. As exampks, an inverse klinokinesis (straight runs at high levels of stimulation) plus direct orthokinesis (faster at higher levels of
F. J. ROHLF
AND D. DAVENPORT
401
-2o-30;
t
-401 I
-20
-iO
-IL-...
IO
L
20
30
1
40
-I-
50
60
70
x FIG. 17. Summary plot for the combined direct orthokinetic model. 8= 14.381.
and sensory adaptation
stimulation) results in a great displacement toward the high end of stimulation, the two behaviors in combination resulting in a greater displacement than either alone (mean position X = 48.320, also see Figs 18 and 19). However, direct klinokinesis (straighter at low levels of stimulation) plus direct orthokinesis result in essentially no displacement; the two effects cancel each other out (mean position X = - 1542, see also Figs 20 and 21). The above are the most dramatic effects of combinations of behavior. The reader can interpret other interrelationships of these ways of behaving from the numerical data in the table. If we introduce tropotaxic behavior it can be seen that with sensory adaptation adding tropotaxis to direct orthokinesis and inverse klinokinesis effects the greatest displacement of all (mean position along the gradient w = 72.151). This means that an animal that senses asymmetries in stimulation and runs fast and straight as it moves into higher levels of stimulation
SIMULATION
OF
KLINOKINBSIS
421
6050 4030 20 * IO f
0 - - - - - - _ -.
-to-
.
-20
-
-30
-
-40
-
-50
-10
Fxo. 18. Summary plot for the combined inverse kIin&inetic, direct ortbokinetic, and sensory adaptation model. 2 = 48.320.
will have the greatest probability of climbing the gradient toward the stimuhrs source. At the opposite extreme we seethe organism with a positive taxis and an inverse orthokinesis. In such an animal which sensesand turns towards higher levels of stimulation but which moves slower as it climbs the gradient, displacementwill not be great. Finally, even though the simulations with sensory adaptation were more complicated than those without, they are still obviously oversimplifications of the behavior of real organisms.Many sorts of refinements could be made on the models which we have used, such as, for instance, the introduction of
422
F. J. ROHLF
AND D. DAVENPORT
FIG. 19. Trace of paths taken by simulated organisms based upon the combined inverse klinokinetic, direct orthokinetic and sensory adaptation model.
different rates of sensory adaptation or by making stimulus intensity a nonlinear function of the position of an organism along the X-axis. However, the study is sufficient to demonstrate that the very simple type of sensory adaptation used in these models is enough to effect displacement in organisms showing orthokinetic and klinokinetic responses without tropotaxis. These results confirm the general conclusions reached by Fraenkel & Gunn (1961) and suggest that the term “klinokinesis” should not yet be dropped, as suggested by Gunn (1966). Clearly the next phase of research in this area should be to study the degree of agreement between the behavior of an actual organism with the behavior predicted by our models, using estimates of behavioral parameters (probability of turning, velocity, etc.) determined from studies of particular organisms. Perhaps it will be possible to resolve a number of dilemmas, the most important being to assign some raison d’btre or selective advantage to the direct klinokinesis exhibited by certain associated animals to a speciesspecific factor emenating from their host (Davenport et al., 1960). Our models indicate that such behavior gives such an animal no advantage in hostfinding, even when working in concert with other behaviors. Obviously, much remains to be done to relate our models to real animals.
SIMULATION
OF
423
KLINOKINESIS
30 20
.
IO y
o-1 -10,
-20
:..y ‘.;.- *:: . . . . :;‘,i.Jy .*** .* -1. . ..f’.:. : *- . .I . .
I
.i
-30 -20
-30
-10
0
IO
20’
30
40
50
60
X
FIO. 20. Summary plot for the combined direct klinokinetic, direct orthokinetic and sensory adaptation model. x = -1.542.
Y 0 .. ... .. .. ... : El 0
... ...
0lTYE!d .... . .. 0
FIG. 21. Trace of paths taken by simulated organisms based upon the combined direct klinokinetic, direct orthokinetic and sensory adaptation model.
This is Contribution No. 1387 from the department of Entomology, the University of Kansas, Lawrence, Kansas. Computer time for this project was made available by Western Data Processing Center, Los Angeles, California (IBM 7040-7094 DCS) and the University of Kansas Computation Center (IBM 7040 and GE 625). The figures were prepared using the B/L plotter at the University of Kansas Computation Center. The computer programs were prepared with the
424
F.
J.
ROHLF
AND
D.
DAVENPORT
assistance of Mr Ron Bartcher. The authors would like to thank Dr C. D. Michener who read and criticized the manuscript and Patricia Rohlf who assisted in the preparation of the illustrations. This work was partially supported by grants from the Office of Naval Research (Contract NONR 4-222(03)) and the National Science Foundation (GB 5137) to Dr Davenport and from the National Institutes of Health (GM 1193) to Dr R. R. Sokal. REFERENCES BAILEY, N. T. J. (1964).“The Elements of StochasticProcesses.” New York: Wiley. DAVENPORT, D., CAMOUGIS, G. & HICKOK, J. F. (1960). Anim. Behav. 8, 3, 209. DAVENPORT, D., KRAMER, H. & THORSON, J. (1960).TEMPO, Center for Advanced
Studies,GeneralElectricCo., SantaBarbara,Calif., ReportSP-95.
EVANS, G. W., WALLACE, G. F. & SUTHERLAND, G. (1967). “Simulation Using Digital Computers.” New York: Prentice Hall. EWER, D. W. & BURSELL, E. (1950). Be/z&our, 3,40. FRAENKEL, GO~TFRIED S. & GUNN, DAVID L. (1961). “The Orientation of AnimalsKineses, Taxes and Compass Reactions”. New York: Dover. GUNN, D. L. (1937). J. exp. Biol. 14, 178. GUNN, D. L. (1966). In “Insect Behaviour”, Symposium of the Royal Entomological Society No. 3, pp. 109-110. P. T. Haskell, ed. GUNN, D. L., KENNEDY, J. S. & PIELOU, D. P. (1937). Nature, Lord. 140, 1064. KENNEDY, J. S. (1945). Nature, Lord. 154, 754. NAYLOR, T. H., BALINT~Y, J. L., BURDICK, D. S. & CHU, K. (1966). “Computer Simulation Techniques.” New York: Wiley. PATLAK, CLIFFORD L. (1953~). Bull. math. Biophys., 15, 311. PATLAK, CLIFFORDL. (19536). Bull. math. Biophys. 15,431. PARZEN, E. (1962). “Stochastic Processes.” San Francisco: Holden-Day. PEARSON.K. (1906). “Mathematical contributions to theorv of Evolution. XV. A mathematical theory of random migration”. Draper’s Co. Research Memoirs, Biometric Series III. 54 pp. SAILA, S. B. & SHAPPY, R. A. (1963). J. Cons. perm. int. Explor. Mer, 28, 153. ULLYO~~, P. (1936). J. exp. Biol. 13, 253. ,
.
I