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The optimal search path in a patchy environment
J. theor. BioL (1990) 145, 177-182
The Optimal Search Path in a Patchy Environment RICHARD A. STILLMANt AND WILLIAM J. SUTHERLAND
School of Biological Sciences, University of East Anglia, Norwich N R 4 7TJ, U.K. (Received on 11 August 1989, Accepted in revised form on 9 March 1990) The optimal search path in a patchy environment is considered assuming an animal is able to alter walking speed, turning angle or turn alternation according to the quality of the patch. The optimal search path was determined for both an environment in which the patches were depleting and one in which they were non-depleting. Walking speed had a considerable effect on the time spent in different quality patches. Turning angle only had an effect when the animal was able to detect when patch boundaries were crossed, and degree of turn alternation had a negligible effect. Turning angle and degree of turn alternation had greater effects when depletion occurred. An immediate change in search path when encountering a patch boundary could markedly affect time in the better sites. Processes such as memory and detection of patch quality may be more important than simple modifications of the search path in response to quality. Introduction
Animals exploiting patchy environments are known to modify their search path according to the quality of each patch (Gunn, 1937; Fraenkel & Gunn, 1961; Smith, 1974) but there is surprisingly little theoretical work to show the expected behaviour. In this paper we show the consequences o f modifying walking speed, turning angle and frequency of turn alternation on the rate at which animals acquire resources. The Model
The model considers an animal in a habitat consisting o f a square grid of 20 x 20 patches with each patch being 100 units square. The habitat has wrap-around margins, so that an animal moving off one side reappears on the opposite side. This assumption has the advantage that it removes edge effects. Half the patches are allocated at random to be o f good quality (1 unit of resource per patch) and the remaining patches allocated as poor quality (no resources per patch), giving 200 rich and 200 poor patches. The resources may be considered as being light, humidity or food. At the beginning of each simulation the animal starts at a random location within the grid and moves in a straight line in a random direction at a given speed for a time interval. At the end o f this time interval the animal turns in a set direction and at a set angle. The animal can potentially remember the state o f patch occupied during it's previous step and so detect when patch boundaries are crossed. The t Present addresses: Ornithology Branch, Nature Conservancy Council, Northminster House, Peterborough PEI 1UA, U.K. 177
0022-5193/90/014177 + 06 $03.00/0
© 1990 Academic Press Limited
178
R. A. STILLMAN
AND
W. J. SUTHERLAND
simulation continues for 100 time intervals after which the total resource capture is calculated. The simulation was repeated 2000 times for each combination of values and the results averaged. The effect of modifying walking speed, turning angle and degree of turn alternation according to the quality of patch occupied by the animal was investigated. Each of these parameters was considered separately, keeping the other two constant. Values of turn alternation varied between 1 (perfect turn alternation) and 0 (complete preference for one side) with a value of 0.5 signifying random direction change. In an initial set of simulations foraging behaviour depended solely on the quality of patch occupied, whereas in a second set foraging behaviour was only altered when the animal crossed a boundary between two patch types. In simulations assuming no depletion the rate of resource intake per time interval was expressed as a percentage (10%) of the value for that patch. In simulations incorporating depletion the animal consumed 10% of the available food in the patch in each time interval. All depleted patches were replenished each time interval such that 5% of the total reduction in resources were replaced, leading to an asymptotic increase to the maximum level. This replenishment might represent nectar replacement, the growth or reproduction of prey, or the movement of prey making a different subset visible to predators. Although the grid is finite, once the animal has crossed the wrap-round margins and moved across the grid back to the once exploited patches, they will be restored to a maximum level. The environment can thus been considered as infinitely large. Results
Figure 1 shows the consequences of modifying foraging behaviour within high and low quality patches for resource capture, and Fig. 2 shows the effect of modifying behaviour when crossing a patch boundary. As is intuitively obvious, intake is higher if the animal walks slowly in the good patch and quickly in the poor [Fig. l(a)]. If depletion is included [Fig. l(b)] then the result is very similar but low speeds in the good patch result in depletion and reduced intake. Increasing the rate of depletion resulted in higher optimal speed in better sites. Adjusting walking speed on crossing a patch boundary [Fig. 2(a) and (b)] shows a similar response. Intake is maximized by increased speed on crossing from a rich to poor patch, and by decreased speed on crossing from a poor to rich patch. Figure 1(c) shows that in the absence of depletion there is no benefit from altering turning angle. A greater turning angle, associated with turn alternation, tends to reduce the rate at which patches are traversed. However, because the angle at which boundaries are crossed varies, a greater turning angle will also tend to increase the probability of leaving a newly" entered patch. If depletion occurs, then angle does have an effect on intake but the effect is small, even with massive changes in angle [Fig. l(d)]. There is a response if turning angle is altered on crossing a patch boundary. Increased turning angle on entering a poor patch greatly increases resource capture, but decreases resource capture on entry to a rich patch [Fig. 2(c)].
SEARCH
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IN A PATCHY
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FIG. 1. The consequences of modifying foraging behaviour on the rate of resource capture. In each simulation only one variable was adjusted and these are (a) and (b) speed; (c) and (d) turning angle; (e) and (f) probability of turn alternation. The default values are speed = 10; turning angle = 450; P = 0.5. The figures on the left (a), (c) and (e) are derived from simulations without depletion while those on the right (b), (d) and (f) incorporate depletion.
180
R. A. S T I L L M A N
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W. J. S U T H E R L A N D
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Probability FUG. 2. The consequences o f modifying foraging behaviour on crossing a patch boundary. In each simulation only one variable was adjusted and these are (a) and (b) speed; (c) and (d) turning angle; (e) a n d (f) probability of turn alternation. The default values are speed = 10; turning angle = 45°; P = 0-5. The closed circles represent m o v e m e n t from a poor to a rich patch, and the open circles represent the reverse case. T h e figures on the left (a), (c) a n d (e) show the results without depletion, and those on the right (b), (d) a n d (g) incorporate depletion.
Increased turning angle on entering a patch, increases the chance that an animal will immediately leave the patch. The benefit of increased turning on leaving a good patch is greatly decreased when depletion occurs [Fig. 2(d)], because this behaviour tends to maintain the animal within one patch where resources are rapidly depleted. Thus increased turning angle will only increase intake when animals can detect patch boundaries and in particular when depletion is limited or absent. It has been suggested that animals should alternate between fight and left turns in poor sites (and thus maintain directionality) and turn consistently to one side in good sites (and thus stay within these patches) (Smith, 1974). However, Fig. l(e)
SEARCH
PATHS
1N A P A T C H Y
ENVIRONMENT
181
and (f) shows that the consequence of this behaviour is small. In the absence of depletion the amount of turn alternation had no affect on intake rate. Circular movement reduces the probability of leaving the initial patch, whilst turn alternation increases the probability. However, on crossing a boundary between two patch types, circular movement will increase the probability of leaving the new patch, whilst turn alternation will decrease the probability. When depletion is included, there is again a small response to a large change in behaviour [Fig. l(f)]. The probability of turn alternation on crossing a patch boundary [Fig. 2(e) and (f)] similarly has no effect on intake regardless of whether depletion occurs. Discussion
The model assumes a random distribution of good and poor patches. It is possible that the response would be different for a gradient of habitat qualities. However, modifying the simulations so that patch quality formed a gradient across the habitat produced the same results: without depletion only walking speed affected the time spent in the better quality areas and with depletion the other two made only a trivial contribution. The results presented use extreme differences in patch quality; using less extreme differences between good and poor patches only reduces the magnitude of the response. The response resulting from altering walking speed varied with patch size. With very small patches the animal leaves a patch every time interval and there is no opportunity for responding. At the other extreme of very large patches animals are unlikely to move out of patches and again cannot respond to the patches. Running the simulations with various patch sizes showed that there were no conditions where turning angle or turning probability had a large affect on intake rate. It appears counter-intuitive that turning angle only shows a response when altered as an animal crosses a boundary, and that turn alternation has no affect. However, computer models of the morphology of rhizomatous plants have similarly shown that branching angle and the probability of branching have little affect on aggregation (Sutherland & Stillman, 1988). These results differ from those of Pyke (1978) and Cody (1971) who stated that a degree of directionality results in a higher food intake. Their models determine the search paths which results in harvesting the highest fraction of food within a grid. Our model differs in considering an equilibrium solution in an infinite environment. Furthermore, as Pyke (1978) points out, the conclusions of Cody (1971) are strongly dependent upon the nature of the boundary of the grid. Thc actual search paths of animals differ markedly between studies. Many studies have shown that animals slow down in better patches, for example, woodlice Porcellio scaber walk slower in areas of high humidity (Gunn, 1937) and plovers move more slowly in better patches (Pienkowski, 1983). Many insects and mites show increased turning behaviour after finding a food item (Hassell & Southwood, 1978), but two detailed studies of bumblebees showed no correlation between turning angle and patch quality (Soltz, 1986; Heinrich, 1979). Many studies show turn alternation (see review by Pyke, 1978) and in detailed studies of the search paths
182
R.A.
STILLMAN
AND
W. J. S U T H E R L A N D
of thrushes, Smith (1974) showed a reduction in turn alternation following prey capture. These models show that responding to patch quality has little affect on the time spent in better sites but that recognizing and responding to patch boundaries can markedly affect the time in better sites. Our models assume animals only adjust walking speed, probabilities of turning and turn alternation. However animals may remember and return to rich sites (as do starlings Sturnus vulgaris Tinbergen, 1981) or restrain their searching to the host scent surrounding the prey (Waage, 1979). Such processes may be more important than simple modifications of the search path. We t h a n k D i a n a Bell a n d a referee for useful c o m m e n t s a n d N.E.R.C. for f u n d i n g R.A.S.
REFERENCES COPY, M. L. (1971). Finches flocks in Mohave desert. Theor. pop. Biol. 2, 142-158. FRANKEL, G. & GUNN, D. L. (1961). The Orientation of Animals. New York: Dover. GUNN, D. L. (1937). The humidity reactions of the woodlouse, Porcelio scaber. J. expl. Biol. 14, 178-186. HASSELL, M. P. ~¢. SOUTHWOOD, T. R. E. (1978). Foraging strategies of insects. Ann. Rev. Ecol. System. 9, 75-98. HEINRICH, B. (1979). Resource heterogeneity and patterns of movement in foraging bumblebees. Oecologia 140, 235-245. PIENKOWSK1, M. W. (1983). Changes in the foraging pattern of plovers in relation to environmental factors. Anita. Behav. 31, 244-264. PYKE, G. H. (1978). Are animals efficient harvesters? Anita. Behav. 26, 241-256. SMITH, J. N. M. (1974). The food searching behaviour of two European thrushes. 1I The adaptiveness of the search patterns. Behaoiour 49, 1-61. SOLTZ, R. L. (1986). Foraging plant selection in bumblebees: hindsight and foresight. Behaviour 99, 1-21. SUTHERLAND , W. J. 8£ STILLMAN, R. A. (1988). The foraging tactics of plants. Oikos 52, 239-244. TINBERGEN, J. (1981). Foraging decisions in starlings (Sturnus vulgaris L.) Ardea 69, 1-67. WAAGE, J. K. (1979). Foraging for patchily-distributed hosts by the parasitoid Nemeritus canescens. J. Anim. Ecol. 48, 353-371.
The Optimal Search Path in a Patchy Environment RICHARD A. STILLMANt AND WILLIAM J. SUTHERLAND
School of Biological Sciences, University of East Anglia, Norwich N R 4 7TJ, U.K. (Received on 11 August 1989, Accepted in revised form on 9 March 1990) The optimal search path in a patchy environment is considered assuming an animal is able to alter walking speed, turning angle or turn alternation according to the quality of the patch. The optimal search path was determined for both an environment in which the patches were depleting and one in which they were non-depleting. Walking speed had a considerable effect on the time spent in different quality patches. Turning angle only had an effect when the animal was able to detect when patch boundaries were crossed, and degree of turn alternation had a negligible effect. Turning angle and degree of turn alternation had greater effects when depletion occurred. An immediate change in search path when encountering a patch boundary could markedly affect time in the better sites. Processes such as memory and detection of patch quality may be more important than simple modifications of the search path in response to quality. Introduction
Animals exploiting patchy environments are known to modify their search path according to the quality of each patch (Gunn, 1937; Fraenkel & Gunn, 1961; Smith, 1974) but there is surprisingly little theoretical work to show the expected behaviour. In this paper we show the consequences o f modifying walking speed, turning angle and frequency of turn alternation on the rate at which animals acquire resources. The Model
The model considers an animal in a habitat consisting o f a square grid of 20 x 20 patches with each patch being 100 units square. The habitat has wrap-around margins, so that an animal moving off one side reappears on the opposite side. This assumption has the advantage that it removes edge effects. Half the patches are allocated at random to be o f good quality (1 unit of resource per patch) and the remaining patches allocated as poor quality (no resources per patch), giving 200 rich and 200 poor patches. The resources may be considered as being light, humidity or food. At the beginning of each simulation the animal starts at a random location within the grid and moves in a straight line in a random direction at a given speed for a time interval. At the end o f this time interval the animal turns in a set direction and at a set angle. The animal can potentially remember the state o f patch occupied during it's previous step and so detect when patch boundaries are crossed. The t Present addresses: Ornithology Branch, Nature Conservancy Council, Northminster House, Peterborough PEI 1UA, U.K. 177
0022-5193/90/014177 + 06 $03.00/0
© 1990 Academic Press Limited
178
R. A. STILLMAN
AND
W. J. SUTHERLAND
simulation continues for 100 time intervals after which the total resource capture is calculated. The simulation was repeated 2000 times for each combination of values and the results averaged. The effect of modifying walking speed, turning angle and degree of turn alternation according to the quality of patch occupied by the animal was investigated. Each of these parameters was considered separately, keeping the other two constant. Values of turn alternation varied between 1 (perfect turn alternation) and 0 (complete preference for one side) with a value of 0.5 signifying random direction change. In an initial set of simulations foraging behaviour depended solely on the quality of patch occupied, whereas in a second set foraging behaviour was only altered when the animal crossed a boundary between two patch types. In simulations assuming no depletion the rate of resource intake per time interval was expressed as a percentage (10%) of the value for that patch. In simulations incorporating depletion the animal consumed 10% of the available food in the patch in each time interval. All depleted patches were replenished each time interval such that 5% of the total reduction in resources were replaced, leading to an asymptotic increase to the maximum level. This replenishment might represent nectar replacement, the growth or reproduction of prey, or the movement of prey making a different subset visible to predators. Although the grid is finite, once the animal has crossed the wrap-round margins and moved across the grid back to the once exploited patches, they will be restored to a maximum level. The environment can thus been considered as infinitely large. Results
Figure 1 shows the consequences of modifying foraging behaviour within high and low quality patches for resource capture, and Fig. 2 shows the effect of modifying behaviour when crossing a patch boundary. As is intuitively obvious, intake is higher if the animal walks slowly in the good patch and quickly in the poor [Fig. l(a)]. If depletion is included [Fig. l(b)] then the result is very similar but low speeds in the good patch result in depletion and reduced intake. Increasing the rate of depletion resulted in higher optimal speed in better sites. Adjusting walking speed on crossing a patch boundary [Fig. 2(a) and (b)] shows a similar response. Intake is maximized by increased speed on crossing from a rich to poor patch, and by decreased speed on crossing from a poor to rich patch. Figure 1(c) shows that in the absence of depletion there is no benefit from altering turning angle. A greater turning angle, associated with turn alternation, tends to reduce the rate at which patches are traversed. However, because the angle at which boundaries are crossed varies, a greater turning angle will also tend to increase the probability of leaving a newly" entered patch. If depletion occurs, then angle does have an effect on intake but the effect is small, even with massive changes in angle [Fig. l(d)]. There is a response if turning angle is altered on crossing a patch boundary. Increased turning angle on entering a poor patch greatly increases resource capture, but decreases resource capture on entry to a rich patch [Fig. 2(c)].
SEARCH
PATHS
IN A PATCHY
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50
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150
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FIG. 1. The consequences of modifying foraging behaviour on the rate of resource capture. In each simulation only one variable was adjusted and these are (a) and (b) speed; (c) and (d) turning angle; (e) and (f) probability of turn alternation. The default values are speed = 10; turning angle = 450; P = 0.5. The figures on the left (a), (c) and (e) are derived from simulations without depletion while those on the right (b), (d) and (f) incorporate depletion.
180
R. A. S T I L L M A N
AND
W. J. S U T H E R L A N D
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Probability FUG. 2. The consequences o f modifying foraging behaviour on crossing a patch boundary. In each simulation only one variable was adjusted and these are (a) and (b) speed; (c) and (d) turning angle; (e) a n d (f) probability of turn alternation. The default values are speed = 10; turning angle = 45°; P = 0-5. The closed circles represent m o v e m e n t from a poor to a rich patch, and the open circles represent the reverse case. T h e figures on the left (a), (c) a n d (e) show the results without depletion, and those on the right (b), (d) a n d (g) incorporate depletion.
Increased turning angle on entering a patch, increases the chance that an animal will immediately leave the patch. The benefit of increased turning on leaving a good patch is greatly decreased when depletion occurs [Fig. 2(d)], because this behaviour tends to maintain the animal within one patch where resources are rapidly depleted. Thus increased turning angle will only increase intake when animals can detect patch boundaries and in particular when depletion is limited or absent. It has been suggested that animals should alternate between fight and left turns in poor sites (and thus maintain directionality) and turn consistently to one side in good sites (and thus stay within these patches) (Smith, 1974). However, Fig. l(e)
SEARCH
PATHS
1N A P A T C H Y
ENVIRONMENT
181
and (f) shows that the consequence of this behaviour is small. In the absence of depletion the amount of turn alternation had no affect on intake rate. Circular movement reduces the probability of leaving the initial patch, whilst turn alternation increases the probability. However, on crossing a boundary between two patch types, circular movement will increase the probability of leaving the new patch, whilst turn alternation will decrease the probability. When depletion is included, there is again a small response to a large change in behaviour [Fig. l(f)]. The probability of turn alternation on crossing a patch boundary [Fig. 2(e) and (f)] similarly has no effect on intake regardless of whether depletion occurs. Discussion
The model assumes a random distribution of good and poor patches. It is possible that the response would be different for a gradient of habitat qualities. However, modifying the simulations so that patch quality formed a gradient across the habitat produced the same results: without depletion only walking speed affected the time spent in the better quality areas and with depletion the other two made only a trivial contribution. The results presented use extreme differences in patch quality; using less extreme differences between good and poor patches only reduces the magnitude of the response. The response resulting from altering walking speed varied with patch size. With very small patches the animal leaves a patch every time interval and there is no opportunity for responding. At the other extreme of very large patches animals are unlikely to move out of patches and again cannot respond to the patches. Running the simulations with various patch sizes showed that there were no conditions where turning angle or turning probability had a large affect on intake rate. It appears counter-intuitive that turning angle only shows a response when altered as an animal crosses a boundary, and that turn alternation has no affect. However, computer models of the morphology of rhizomatous plants have similarly shown that branching angle and the probability of branching have little affect on aggregation (Sutherland & Stillman, 1988). These results differ from those of Pyke (1978) and Cody (1971) who stated that a degree of directionality results in a higher food intake. Their models determine the search paths which results in harvesting the highest fraction of food within a grid. Our model differs in considering an equilibrium solution in an infinite environment. Furthermore, as Pyke (1978) points out, the conclusions of Cody (1971) are strongly dependent upon the nature of the boundary of the grid. Thc actual search paths of animals differ markedly between studies. Many studies have shown that animals slow down in better patches, for example, woodlice Porcellio scaber walk slower in areas of high humidity (Gunn, 1937) and plovers move more slowly in better patches (Pienkowski, 1983). Many insects and mites show increased turning behaviour after finding a food item (Hassell & Southwood, 1978), but two detailed studies of bumblebees showed no correlation between turning angle and patch quality (Soltz, 1986; Heinrich, 1979). Many studies show turn alternation (see review by Pyke, 1978) and in detailed studies of the search paths
182
R.A.
STILLMAN
AND
W. J. S U T H E R L A N D
of thrushes, Smith (1974) showed a reduction in turn alternation following prey capture. These models show that responding to patch quality has little affect on the time spent in better sites but that recognizing and responding to patch boundaries can markedly affect the time in better sites. Our models assume animals only adjust walking speed, probabilities of turning and turn alternation. However animals may remember and return to rich sites (as do starlings Sturnus vulgaris Tinbergen, 1981) or restrain their searching to the host scent surrounding the prey (Waage, 1979). Such processes may be more important than simple modifications of the search path. We t h a n k D i a n a Bell a n d a referee for useful c o m m e n t s a n d N.E.R.C. for f u n d i n g R.A.S.
REFERENCES COPY, M. L. (1971). Finches flocks in Mohave desert. Theor. pop. Biol. 2, 142-158. FRANKEL, G. & GUNN, D. L. (1961). The Orientation of Animals. New York: Dover. GUNN, D. L. (1937). The humidity reactions of the woodlouse, Porcelio scaber. J. expl. Biol. 14, 178-186. HASSELL, M. P. ~¢. SOUTHWOOD, T. R. E. (1978). Foraging strategies of insects. Ann. Rev. Ecol. System. 9, 75-98. HEINRICH, B. (1979). Resource heterogeneity and patterns of movement in foraging bumblebees. Oecologia 140, 235-245. PIENKOWSK1, M. W. (1983). Changes in the foraging pattern of plovers in relation to environmental factors. Anita. Behav. 31, 244-264. PYKE, G. H. (1978). Are animals efficient harvesters? Anita. Behav. 26, 241-256. SMITH, J. N. M. (1974). The food searching behaviour of two European thrushes. 1I The adaptiveness of the search patterns. Behaoiour 49, 1-61. SOLTZ, R. L. (1986). Foraging plant selection in bumblebees: hindsight and foresight. Behaviour 99, 1-21. SUTHERLAND , W. J. 8£ STILLMAN, R. A. (1988). The foraging tactics of plants. Oikos 52, 239-244. TINBERGEN, J. (1981). Foraging decisions in starlings (Sturnus vulgaris L.) Ardea 69, 1-67. WAAGE, J. K. (1979). Foraging for patchily-distributed hosts by the parasitoid Nemeritus canescens. J. Anim. Ecol. 48, 353-371.