Simulation of the Coevolution of Insects and Flowers

Simulation of the Coevolution of Insects and Flowers Alexander Bisler Fraunhofer Institute for Computer Graphics, Darmstadt, Germany abisler@igd.fhg.de, Abstract. Flowers need insects for their pollination and insects rely on the nectar and the pollen as a food resource. But instead of visiting all flowers, the insects limit their visits to a small number. This paper presents a simulation of the behavior of the insects which results in a specialized perception of blossom colors and fragrances by the insects. A coevolution can be observed only based on simple interactions modified over generations by a genetic algorithm. The results show a mutual benefit expressed in more rewards for the insects and more efficient pollination of the flowers. 1 Introduction Pollination is an important step in the reproduction of seed plants. The process of pollination requires pollinators as agents that carry or move the pollen grains from the anther to the receptive part of the carpel [1 3]. In order to be pollinated flowers developed different ways. A few flowers are able to self-pollinate (autogamy). Others rely upon water (hydrophily) or wind pollination (anemophily) which is only useful in situations where large populations of a very limited number of species are present. Most flowers are pollinated by animals; for example insects (entomophily), birds or bats (zoophily). In order to attract animals the flowers use visual and olfactory cues. Some species use color patterns to stand out against a background of green foliage. The colors used are adapted to the sensors of the animals. Showy petals or sepals with obvious shape, size, and color for the animal s visual sensor are important. Flowers have also evolved a wide array of odors to assist in attracting animals. After the pollinator landed on the blossom, a means of putting pollen onto the pollinator had to be developed, so that the pollen would be transferred effectively to the next flower visited. The animal s reward for its visit is usually nectar or pollen. During the evolution, certain flower species have specialized to attract only specific pollinators. On the other hand, certain insect species have specialized to collect nectar and pollen only from specific flowers. At some point certain species became totally dependent on others and vice versa. This paper reports on the results of a simulation which explains how this coevolution takes place, to show the benefit for both flowers and insects, and

to demonstrate that this mutual evolutionary influence is based only on selforganization. Only insect pollination will be considered for this paper. 2 Computational Model Self-Organization can be found in many complex systems [4] which include biological systems [5]. Computer simulations are used to understand the collective behavior of (artificial) societies [6]. After a model has been established, it can be used to examine the significance of each single parameter and therefore provide some insight. In order to model this example of self-organization in living systems insects and flowers are modeled as agents. While the flowers remain immobile on a meadow, the insects show a typical behavior by flying around searching for flowers to collect nectar. A genetic algorithm is used to make an adaptation of the attributes of insects and flowers possible [7,8]. The attributes of insects and flowers are encoded in genomes. As each species can only use their own genotypes for the production of offspring, each species will run its own genetic algorithm. This implies that there is no competition between flower species or between insect species, since the population size of each genetic algorithm remains unmodified. The modification of the population sizes remains to be solved in future versions. In the following two subsections the genetic information is presented which is used to describe the genotypes of flowers and insects. Additionally, the construction of their phenotypes is explained. Since the genetic algorithm runs offline, an explicit graphical modeling of the agents was not necessary. A visualization has been implemented only for debugging and representational purposes. 2.1 Modeling Flowers Prusinkiewicz and Lindenmayer wrote a standard book on the modeling of plants [9]. The stem of a flower is not explicitly modeled as only the blossom is crucial for the insect s awareness. A manually created model has been used for graphical representations. The blossom is composed of petals which are circularly arranged. Other shapes are not supported. Lu et al. proposed a construction of petals based on Bezier patches with 16 control points [10]. For our purposes, an alpha texture of a petal is mapped on a quad strip which is bent along a Bezier curve. The internal organs of the blossom are not modeled. Genotype of a Flower The genome contains the following information: size of the blossom which is defined by a bounding box number of petals color of petals

shape of petals control points of the Bezier curve along which a petal is bent fragrance The colors are encoded in the HSV system 1 which provides a better interpolation between colors than the RGB system. This is, of course, a simplification. The visual sensors of insects are also known to use different bands of the spectrum where flowers can be detected more easily. For example bee vision does not include red, but extends into the ultraviolet. This simplification can be justified to show the principles, rather than the details, of this coevolution. The shape of a petal is given by a limited number of prepared alpha textures. The way, the quad strip is bent, is controlled by two Bezier curves which are defined by seven control points 2. The number of fragrances is limited to 10 types. They are circularly organized like the hue value in the HSV system to avoid border effects while mutating the value. Phenotype of a Flower The alpha textures are gray-scaled bitmaps. Thus, they can be easily modulated by the flower s color. A variation of blossoms is shown in Fig. 1. Fig.1. Variety of blossoms. 2.2 Modeling Insects The body of an insect is not explicitly modeled. The simulation focuses more on the insect s behavior. The size of an insect is not modifiable. Only a radius is used to approximate the insect s body size. The radius is constant for all insects of the same species 1 Hue Saturation Value 2 They share one.

but varies between species. The simulation is meant to model only the adaptation of the insects sensors and behavior and not their body shape or size. The radius of the visual sensor and the capability of the olfactory sensor to smell flowers in a specific distance are also constant. Genotype of an Insect The genome contains the following information: favorite blossom color favorite fragrance consistency The consistency value is a parameter of the behavior routines as the other two values are. It decides whether an insect after it decided to fly towards a specific flower changes its decision again. The higher the value the less likely a new flower is chosen. Phenotype of an Insect The phenotype consists of an autonomous agent which flies over a meadow full of flowers and shows a behavior which depends on the genetically encoded parameters. The radius of the body is used to test whether the insect can crawl into a blossom to collect nectar. Since the behavioral model of the insects represents the most interesting part of the simulation, it is described in its own section (see section 3). 2.3 Fitness Functions Each flower is given a fitness value according to the number of successful pollinations. For the insects the number of visits, during which nectar could be collected, is used as the fitness value. 2.4 Mutation Basically, a mutation of a specific parameter is done by adding a positive or negative offset. If a hue value of the HSV colors or a fragrance value leaves the defined range, they are corrected by a modulo operation. The offset for fragrances is limited to±1. Some constraints are applied when mutating the control points of the two Bezier curves which define the bending of a petal. They ensure that the mutation would only result in curve types which are acceptable as outlines of blossoms. The Bezier curves are also restricted by the bounding box which defines the overall size of a blossom. The larger a flower grows the more its visibility improves. But in order to stop infinite growth, a limit is set which can be interpreted as a maximal stability. If this limit were passed, the stem could no longer hold the blossom due to its heavy weight.

3 Insect Behavior The insect s behavior is modeled with a simple finite state machine. The main states are: searching, flying towards a flower, and landing & collecting nectar. The insect starts in the state searching. It flies over the meadow. If one or more flowers come within the range of its sensors, it can decide whether or not to fly towards one of these flowers. The radius of the olfactory sensor is always larger than that of the visual sensor. First the insect tries to find a flower which matches its favorite color and/or fragrance. Other flowers are more likely ignored. The favorite color expresses the part of the color spectrum which can be perceived better than other parts. This also applies to the fragrance. 3 The longer an insect searches for a specific type of flower the more likely it decides to fly towards other types. Additionally, the closer a flower is to the insect s position the more likely it is chosen. Fig.2. Visibility of different colors. 3.1 Perception of Flowers How well a flower is perceived by an insect depends on the flower s size, the area covered by all petals, its distance to the insect, its color, and its fragrance. 3 Putting the perception this way was easier than modeling a complex, physically correct sensor.

The area covered by all petals is calculated using an overhead camera view of the blossom. Then the ratio is calculated of pixels belonging to the blossom (with a color differing from the background) to all pixels of the maximal area which is a circle around the center of the blossom (shown in Fig. 1). The visibility of the color is determined by calculating the distance to green. Since green blossoms would not stand out against the background of green foliage, they receive the worst value. The greater the difference of the color s hue value to that of green, the better the visibility. All colors are rated the same, as shown in Fig. 2, relative to a specific limit. If the flower s fragrance matches the insect s favorite, it is better perceived than other flowers in that vicinity. Otherwise the distance of the fragrance value to the value of the favorite fragrance is calculated and weighted. 3.2 Pollination After reaching a flower, the insect lands on it and tries to crawl into the blossom. Depending on the shape of the blossom this might not be possible. If the petals form a closed calyx, the radius of the insect s body is tested against the inner radius (Fig. 3). If the insect is too big, it has to leave without any reward. Fig.3. Inner radius of a calyx. Pollen is attached to the insect when it successfully collects its reward. If one of the next visited flowers is of the same species it is successfully pollinated. The amount of pollen an insect can carry is limited and, after several visits, older pollen is shaken off. 3.3 Reinforcement Learning Some insects use a simple reinforcement mechanism to achieve better results and to be able to adapt to what kind of blossoms are actually available in their vicinity (see for example [11]). For this purpose, a simple memory has been implemented which associates colors and fragrances with a success value. This value is increased each time the insect gains a reward for visiting a flower and decreases otherwise. This success value influences the flower preference. The

capacity of the memory is limited to distinguish 10 types of fragrances and also 10 types of colors 4. 4 Implementation The program used the GAlib for the genetic algorithms [12]. Each species runs its own genetic algorithm. The algorithms of the flowers have to be divided into two steps which frame the algorithms of the insect species. For example after constructing all flower populations, their phenotypes are randomly placed on a meadow (a central data structure). Then, for each insect population, each insect flies over the meadow; starting from a random location. After a specific number of simulation steps, the number of successful collections of nectar is used as the insect s fitness value. But the successful pollination of the flowers still have to be added together. After the last insect has been handled, the flowers on the meadow contain the necessary data for their fitness value. 5 Test Scenarios Several test scenarios have been evaluated. After finding a parameter set for which the simulation resulted in a specialization of flowers and insects, single parameters were modified to see their influence on the system. There were 4 flower species and 4 insect species. The genetic algorithms ran over 50 generations. The population size for each species was set to 50. The probability of mutation was set to 30% and the probability of crossover to 90%. The offsets used by the mutations had to be limited, otherwise the system did not converge. Fig.4. These diagrams show for each flower species (differently shaded bars) the percentage of pollination by each insect species. 4 All possible colors are reduced to 10 types.

6 Results The simulation s output consisted of huge amounts of data for which many diagrams had to be displayed. Here, only the most interesting diagrams are shown to describe the results 5. Fig.5. These diagrams show for each insect species (differently shaded bars) the percentage of rewarded visits on each flower species. Instead of starting with one specific blossom color per flower species, they were randomly distributed. The same applied for the flower s fragrances and for the insect s favorite colors and fragrances. As a result, not all flowers of a certain species had the same blossom color, but the variety decreased over time and was very limited at the conclusion. This can be explained with a feedback loop: flowers which happened to be chosen more often in the beginning received a higher fitness value and therefore had a greater chance to produce offspring. On the other hand, insects with the corresponding favorite color or fragrance could find these flowers more often and receive more rewards. In Fig. 4, the distribution of visits (in percentage) from each insect species is shown for each flower species. Flower species 1 (black bars) started with a quite even distribution and ended with a distribution which favored insect species 1, 3, and 4, while 2 contributed only 13%. Flower species 2 (gray bars) was mostly visited by insect species 1 right from the beginning and specialized even more. Finally, its pollination depended over 90% on insect species 1. Flower species 3 (bars with horizontal stripes) specialized in insect species 2 followed by 3 and 4 and flower species 4 (bars with diagonal stripes) specialized more in insect species 3 followed by 2 and 4. The corresponding specialization can also be observed for the insects (Fig. 5) where species 1 (black bars) specialized in flower species 2, and 2 (bars with 5 In all diagrams the y-axis is a percentage ranging from 0 to 1. I.e. the value 1 corresponds to 100%.

Fig.6. Percentage of rewarded visits for each insect species. horizontal stripes) in 3 and 4. Insect species 3 (bars with diagonal stripes) favored flower species 4 closely followed by 1 and 3, while 4 (gray bars) favored 1 followed by 4 and 3. The number of visits successful plus unsuccessful ones was quite constant while the ratio of successful visits rose (Fig. 6). The number of successful pollinations through the insects also rose (Fig. 7). This might be the most important result. The specialization process leads to an increased productivity for the insects (more rewards per visits) and to an increased efficiency of pollination which is a benefit for the flowers. 7 Conclusion and Future Work The simulation of the coevolution of colors and fragrances used by flowers and the colors and fragrances best perceived by insects could be based on a simple behavioral model for the insects. Driven by reinforcing feedback loops, the system moved to specialization due to its mutual benefits for insects and flowers. In future versions, the competition between species could be introduced by calculating an average fitness value for each species which can be compared to others. According to the order of these fitness values, the population sizes could be adjusted. But then flower populations should start with a limited set of genomes to distinguish them at the beginning. Since this scenario can lead to the extinction of entire populations, the system should be open so that new species can be established. The implementation of a more physically based simulation of the insect s visual and olfactory sensors might lead to other interesting results. However, it is not considered to be of sufficient importance to further demonstrate the principles of coevolution for this paper. Further research regarding the influence of insects which only take the reward and do not contribute to the pollination process would be beneficial.

Fig.7. Percentage of pollinations done by each insect species. 8 Acknowledgment I wish to thank Tobias Saul for the implementation of the simulation in the context of his Master thesis. References 1. E. Strasburger, P. Sitte, and H. Ziegler, Lehrbuch der Botanik fuer Hochschulen. Spektrum Akad. Verlag, 1998. 2. S. L. Buchmann and G. P. Nabhan, The Forgotten Pollinators. Island Press, 1996. 3. R. E. Koning, Pollination adaptations, 1994, (5-14-105). 4. S. A. Kauffman, Origins of Order: Self-Organization and Selection in Evolution. Oxford: Oxford University Press, 1989. 5. C. Camazine, J. Deneubourg, N. Franks, J. Sneyd, G. Theraulaz, and E. Bonabeau, Self-Organization in Biological Systems. Princetown University Press, 2001. 6. G. N. Gilbert and K. G. Troitzsch, Simulation for the Social Scientist. Taylor & Francis, Inc., 1999. 7. J. H. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor: University of Michigan Press, 1975. 8. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley Longman Publishing Co., Inc., 1989. 9. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants. Springer-Verlag New York, Inc., 1996. 10. Z. Lu, C. Willis, and D. Paddon, Perceptually realistic flower generation, in Proceedings of the 8th International Conference in Central Europe on Computer Graphics, Visualization and Interactive Digital Media 2000, 2000. 11. A. Balkenius, A. Kelber, and C. Balkenius, Simulations of learning and behaviour in the hawkmoth deilephila elpenor, in ICSAB: Proceedings of the seventh international conference on simulation of adaptive behavior in From animals to animats. Cambridge, MA, USA: MIT Press, 2002, pp. 85 92. 12. M. Wall, GAlib: A C++ library of genetic algorithm components, 1996.