## Abstract

From the agent-based, correlated random walk model presented, we observe the effects of varying the maximum turning angle, δmax, tree density, ω, and pollen carryover, κmax, on the distribution of pollen within a tree population by examining pollination graphs. Varying maximum turning angle and pollen carryover alters the dispersal of pollen, which affects many measures of connectivity of the pollination graph. Among these measures the clustering coefficient of fathers is largest when δmax is between 60 and 90∘. The greatest effect of varying ω is not on the clustering coefficient of fathers, but on the other measures of genetic diversity. In particular when comparing simulations with randomly placed trees with that of actual tree placement of C. florida at the VCU Rice Center, it is clear that having specific tree locations is crucial in determining the properties of a pollination graph.

### Keywords

- pollination network
- correlated random walk
- agent-based model
- pollen carryover
- tree density

## 1. Introduction

While the movement of genes from one generation to the next ensures the cohesiveness of plant species through time and space [1, 2, 3], the extent to which individual sites and populations are functionally connected is mitigated by both biotic and abiotic factors [4]. For wind dispersed pollen, features such as the direction and speed of the wind and physical properties of individual pollen grains [5] play prominent roles in how genes are carried across the landscape. In addition to intrinsic factors, site-specific features, such as the structural complexity of the landscape and co-occurring species [6], also influence connectivity to an extent that it is easy to discern.

Genes that are dispersed via active agents—animal mediated pollination—add increasing layers of involvement for at least two reasons. First, the way in which an animal disperser identifies, perceives, and interacts with features in the environment directly impact realized genetic connectivity. Over the last decade, enough work has been focused on this topic to denote a new sub-discipline of population genetic research, dubbed landscape genetics [7, 8], has been is devoted to developing methods for this task. From the perspective of the plant population, the characteristics of the intervening landscape determine the overall *porosity* of the landscape. Second, while the determination of which subset of features is important as it specifies the matrix through which pollinators move, the consequences to plant population genetic structure from alternative modes of pollinator movement is largely unexplored—even for the same plant species, alternative pollinating species leave discernibly distinct genetic structure [4, 9].

In this manuscript, we examine how the way in which pollinating individuals move across the landscape may influences population genetic structure. Here we develop an agent-based model (ABM) to simulate pollinator movement across a spatially explicit landscape. Individual pollinators are tracked as they pick up and disperse pollen among a set of individual plants. We adopt an underlying model of a correlated random walk (CRW) [10], where the direction and rate of movement is both temporally autocorrelated, though constrained. Within this framework, we explore the extent to which the spatial arrangement of trees interacts with variation in model parameters in producing variation in pollination statistics. We then apply this model to a data set from an natural population of the understory tree, *Cornus florida* L. (Cornaceae) [11]. This data set consists of both spatial and genetic information upon which previous landscape genetic studies have been conducted [11].

## 2. Background and methods

The agent based model developed herein uses two different categories of actors; trees act as the source and destination of pollen, and pollinating agents move individual pollen grains across the landscape. While the *trees* are spatially fixed on the landscape, the movement characteristics of the pollinating agents determine the ability of the plant population to maintain population genetic structure and determine relative reproductive output for individual trees. The movement of pollinating agents across the landscape is defined as a correlated random walk parameterized by inertia and speed. Across model runs, the aggregate movements of pollinating agents define a *de facto* pollination network whose characteristics are used to infer the robustness of the overall mating network and provide insights into population genetic stability. Across replicate runs, we extract parameters describing pollinator-movement dynamics parameters (average and maximum dispersal distances), pollen network robustness (pollen donor connectance and spatial clustering), and future population genetic structure (pollen donor density and diversity).

### 2.1. Field characteristics

The field size for model runs is set as a square

where

Previous simulation and empirical work has shown that density of pollen donors can have significant impacts on the genetic structure and diversity of offspring [6, 12], and as such should be a parameter across which we evaluate the other features of this model. The simulation field has rigid boundaries, and is considered impermeable. As such pollinators cannot leave the field nor are new pollinators allowed to enter the field during a model run. When an pollinator comes into contact with the edge of the field, its subsequent heading is set such that it ‘bounces’ off of the barrier at the opposite angle from which it approached.

Simulations were also run for a field size of

### 2.2. Tree characteristics

For tracking purposes, each tree,

For the second part of the study, coordinates of the trees at the VCU Rice Center were provided by the Dyer Laboratory [13]. These coordinates were used to create pollination graphs to compare with the random location pollination graphs to gauge the extent to which spatial heterogeneity influences broad trends in pollen connectivity.

### 2.3. Pollinator movement

Both natural and managed landscapes contain a broad range of species that are commonly distributed with a high degree of spatial heterogeneity. For tree species, reproductive structures may be nestled among several other taxa both below and above the target species in a mixed forest canopy. Under these conditions, a movement model based upon correlated random walk is preferred over alternatives such as Levy walks due to the complexity of the intervening landscape and the lack of long thoroughfares in the forest. Correlated Random Walk (CRW) models have been widely used to describe foraging behavior across a range of animal taxa [14, 15, 16, 17, 18].

In our simulations, we begin with an allotment of 1000 pollinators starting at random location with a random direction of travel on the simulated landscape. At each discrete time step, each pollinating agents will obtain a new heading based upon its previous heading with a specified random deviance. The individual will then move in this new direction 1 distance unit. This process continued for

If a pollinator is within one unit distance of a tree, it will visit flowers on the tree. Each flower on a tree can be pollinated with equal probability. Pollinators visit one tree at a time. If multiple trees are within 1 unit the closest one is chosen. When visiting a tree, the pollinator may both gather pollen and deposit pollen from other trees. Due to the short length of the simulation we assume there are a sufficient number of flowers to gather pollen from and deposit pollen to on each tree.

Let

The initial position of each pollinator,

where

for each

Sample paths based on different values of

### 2.4. Pollination

If a pollinator visits a flower, it will collect pollen from that individual. Pollen will be deposited with a probability of

As a pollinator visits multiple flowers, the chances that it deposits pollen from a previous flower diminishes with each successive flower visited [19]. It was shown by [20] that from a given flower, a pollinator will deposit roughly

where

### 2.5. Statistics

To characterize pollen movement and how it responds to the parameters of the models, pollination graphs were constructed. The connectivity network is based upon the physical location of individual trees and the pattern of spatial pollen movement created by the pollinators. In this network, each tree is represented as a node and the edges designate the movement of pollen from donor (paternal individual) to recipient (maternal individual), creating a directed pollination graph.

The parameters we vary in constructing these networks include tree density, *number of fathers per mother*, *connectance*, *average weighted diversity of fathers*, *average pollination distance*, *average maximum pollination distance*, *weak and strong clustering coefficients of fathers*,

Each tree has the ability to contribute pollen to other trees and to accept pollen from other trees. When applicable, we will refer to a tree as a father tree, *number of fathers per mother* for all trees in the graph is

This value is similar to the degree distribution in the studies by Ramos-Jilberto et al. [22] and Valdovions et al. [23].

From this construct, we then create the

The *connectance*,

As with the previous parameters, connectance has been used in a number of settings, see [23, 25, 26, 27, 28], where the graphs is between species. In this study the focus is on individuals, and so the connectance is the proportion of realized pollination events between individual plants.

Furthermore, if there is an edge between tree

The *average weighted diversity of fathers* is the mean average of the weighted diversity of fathers over all mother trees, and is given by the formula

where

The average pollination distance for an pollinator

where *average pollination distance*,

The *maximum pollination distance* for an pollinator

The maximum pollination distance for each pollinator is averaged over all of the pollinators to obtain the *average maximum pollination distance* for the graph

A fathering triplet is the relationship between three trees such that tree

The *weak clustering coefficient of fathers*,

The *strong clustering coefficient of fathers*,

## 3. Results and discussion

We examine the effect of varying parameters on the graph statistics: the number of fathers per mother,

### 3.1. Number of fathers

One way to analyze the genetic structure and connectivity within a local plant population is to examine the number of different fathers per mother tree,

Of interest here is that as the maximum pollen carryover increases, the mean number of fathers increases. This is due to the increase in the variability in pollen that each pollinator can distribute, which would increase diversity. The variance in the distribution also increases as pollen carryover increases, which is also due to this increase in diversity.

However, when this distribution is compared with the tree placement at the Rice Center, see Figure 4, we observe a bi-modal distribution of the number of fathers per mother. This distribution is attributed to the spatial heterogeneity of the research site. This influences the genetic structure and connectivity in *C. florida* populations [11] at the Rice Center.

The spatial heterogeneity is evident in Figure 5. The bimodal distribution of the number of fathers per mother is caused by the variation of tree density across the landscape. There is a high density group of trees in the center of the region, and the density decreases toward the boundary of the region. This region is bounded by a river and a lake on two sides, and a major road and a farm on the other two sides, which are not pictured here.

These two different density regions create the two peaks shown in Figure 4. In particular when

### 3.2. Connectance

The connectance is a measure of how complete a pollination graph is in terms of individuals mating with others. In Figures 6 and 7, we see the effect of density and maximum turning angle. With higher density, and thus more individuals, the connectance is reduced due to a much larger number of potential pairs of individuals. As the maximum turning angle increases the connectance also decreases. If an pollinator travels in a straight line, it will cover a greater spatial distance as it visits more different trees than it would if it just spun around in circles locally. Smaller

If an pollinator does not venture very far from its starting location, as would be the case when

Pollen carryover is important in connectance as well. In Figure 7, it is clear that if the maximum pollen carryover is limited, the connectance of the pollination graph is also limited due to the diversity of pollen an individual would have access to via the pollinator. With smaller

When considering the actual tree locations at the Rice Center, the connectance of the graph is close to half of the connectance value with randomly-placed trees. The differences between the graphs is greatest when

### 3.3. Average weighted diversity of fathers

The average weighted diversity of fathers is clearly affected by density and the maximum turning angle, though the effects of maximum turning angle are more pronounced. In Figure 9, as the maximum turning angle increases, the average weighted diversity of fathers decreases. The greatest change occurs between

The differences in average weighted diversity of fathers as pollen carryover increases are exactly as one would expect, see Figure 10. With higher

When comparing the random tree distribution to that of the Rice Center, the random distribution has a larger average weighted diversity of fathers (not shown here) the average weighted diversity of fathers at the Rice Center is about 60% of the randomly placed trees. Trees that are in densely packed groups are going to be greatly influenced by surrounding trees, but trees at greater distances will have less of a comparative impact on the fatherhood of seeds.

### 3.4. Average and maximum average pollination distances

The average and maximum pollination distances behave similar to the connectance. As with the connectance, as the density increases both distances decrease, which is due to a greater number of shorter pollination events lowering the averages, see Figures 11 and 12. The change in density has a smaller effect on the maximum pollination distance. Also as the maximum turning angle increase both distances decreases as well since the pollinators do not travel as far. This decrease is more dramatic at lower densities.

Using Rice Center locations (not shown here) if

We see the effects of pollen carryover in the average pollination distance is shown in Figure 11 and in the maximum pollination distance in Figure 12. As expected, increasing

The results are similar with the Rice Center model. The results at the Rice Center are slightly higher in distances due to the potential longer distances at the edges of the region.

### 3.5. Clustering coefficients of fathers

The clustering coefficient is a measure of the interconnectedness of the graph, as well as how the genes are shared within the graph. For a field of randomly placed trees, there is a maximum value for the weak clustering coefficient of fathers,

At the other extreme, for large

Modeled data on the weak clustering coefficient of fathers,

Unexpectedly, varying the tree density,

Figures 16–18 are corresponding plots for strong clustering coefficient of fathers,

## 4. Conclusions

From the agent-based, correlated random walk model presented, we observe the effects of varying the maximum pollen carryover, *Cornus florida*.

When

Changing the pollinator movement by increasing the maximum turning angle,

When *C. florida* population. While neither of these extremes may be biologically relevant in *C. florida* populations, we note that the clustering coefficient of fathers,

Major changes are observed when comparing simulations using randomly-placed trees with simulations using the tree-placement at the Rice Center. When using the Rice Center data, we see a bimodal distribution in the number of fathers per mother, the connectance values are halved, the average weighted diversity of fathers is lower, the average pollination distance is lower when

## Conflict of Interests

The authors declare that they have no conflicts interests.

## Funding

The collection of data and construction of an initial model were supported by a National Science Foundation grant (DEB-0640803) to Rodney J. Dyer and David M. Chan. James H. Lee was supported by a grant from the VCU Rice Center to complete an advance model. This manuscript is VCU Rice Center Manuscript #69.