Besides many organic supplements were added to the PGM to enhance smooth pollen tube growth such as casein hydrolysate, vitamins etc.

## **4. A protocol to standardize** *in vitro* **pollen germination medium**

To standardize pollen germination medium for any crop initially a set of media are used. Based on the response, one of the media is picked up and modified. Here a procedure to develop a PGM for pigeonpea and its wild species is discussed. Initially two sets of key media(PGM) were used Set I consisted of Brewbaker and Kwack (BK) medium [7] with 1% agar at different levels of sucrose *viz*., 10, 20, 30 and 40% (A–D). With the addition of polyethylene glycol (PEG) 4000 at 15% concentration to each of A, B, C and D; the media E, F, G and H, respectively, were obtained. These were designated as Set II.

Composition of Brewbaker and Kwack's (BK) medium

Key media

**Polyethylene glycol (PEG)**: It is known to be a non-penetrating osmotic agent that decreases water potential of culture medium [21]. In pollen grains, PEG is considered to regulate the permeability of plasma membrane and to give stability to the pollen tube membrane [6] and to give stability to the pollen tube membrane. PEG of different molecular weight has been in pollen cultures of different species and in all cases it has a promontory effect [4, 6, 22]. Generally 10–15% of PEG was added to PGM. PEG 4000 was preferred for pigeonpea and brinjal whereas wheat and rye pollen germination satisfactorily with

**Peptone**: Inclusion of peptone arrested pollen tip bursting in wheat at concentration

tissue culture for various effects such as induction of shoot regeneration in avocado [23], promotion of hairy root formation in ginseng [24] etc. Addition of peptone, contained pollen tube burst to a great extent, it was less than 20% as compared to >85% in the medium which is

**ε‐Amino caproic acid (EACA)**: It is an amino acid derivative, a saturated six carbon fatty acid

used to overcome self-incompatibility and incongruity in *Phaseolus* [25] and *Vigna* [26, 27]. It has been used as a component of PGM of pigeonpea for the first time [1] and subsequently in

**Tryptone**: It was good growth-stimulating nitrogen sources used in the cultivation of *Trichoderma hamatum* and *T. harzianum* [28]. It was found to be the best organic nitrogen source for kefiran production by *Lactobacillus kefiranofaciens* (Dainiel et al., 2015), asparaginase production from *Enterobacter cloacae* [29], production of biosurfactant by *Bacillus subtilis* [30]. In rye pollen germination, tryptone at 50–75 mg concentration gave satisfactory

Besides many organic supplements were added to the PGM to enhance smooth pollen tube

To standardize pollen germination medium for any crop initially a set of media are used. Based on the response, one of the media is picked up and modified. Here a procedure to develop a PGM for pigeonpea and its wild species is discussed. Initially two sets of key media(PGM) were used Set I consisted of Brewbaker and Kwack (BK) medium [7] with 1% agar at different levels of sucrose *viz*., 10, 20, 30 and 40% (A–D). With the addition of polyethylene glycol (PEG) 4000 at 15% concentration to each of A, B, C and D; the media E, F, G and

**4. A protocol to standardize** *in vitro* **pollen germination medium**

H, respectively, were obtained. These were designated as Set II.

Composition of Brewbaker and Kwack's (BK) medium

devoid of peptone however, the level of peptone varied with genotypes.

−1which was used in PGM for the first time [2]. Peptone has been used in plant

addition of PEG 6000.

, CH3

(CH2 ) 4

the medium of wheat and rye [2, 10].

and reproducible level of pollen germination [2].

growth such as casein hydrolysate, vitamins etc.

80–120 mg l

86 Pollination in Plants

(C6 H12O2

Based on the germination/initiation of germination and bursting of pollen, one of the key media was first selected (**Figure 3**). The complete medium for each species was then standardized by altering the concentration of sucrose, boric acid and/or calcium nitrate one by one to obtain maximum pollen germination and good pollen tube growth. First sucrose concentration and temperature were varied to prevent bursting of pollen before germination. Secondly, varying concentrations of boric acid was tried *viz*., 50, 100, … 300 ppm keeping 300 ppm of calcium nitrate (as in BK medium) and other components unchanged. Lastly, the optimum concentration of calcium nitrate was determined (100, 200, … 600 ppm). The observations were recoded for three best concentrations of each boric acid and calcium nitrate.

#### **4.1. Standardization of** *in vitro* **pollen germination medium for** *C. platycarpus*

In the medium (B) containing 20% sucrose, 1% agar and standard Brewbaker and Kwack medium's salts, pollen of *C. platycarpus* showed 99.15% germination and tube length over 172 μm in 1 h duration at 20.5°C. Lowering the concentration of sucrose to 10% (medium A) and leaving other constituents unchanged gave 61.60% germination with a mean tube length of over 83.88 μm. Though the medium (C) containing 30% sucrose gave 94.25% germination, the tube length was reduced considerably (33 μm), also pollen tube burst at the tips.

The best medium which gave 99.15% germination for *in vitro* pollen germination and pollen tube growth of *C. platycarpus* was

**4.2. Standardization of** *in vitro* **pollen germination medium for** *C. volubilis*

On key media G (from Set II) the pollen of *C. volubilis* showed initiation of germination (budding) but almost all pollen grain showed bursting before or after budding. To contain bursting of pollen, sucrose level was increased to 35% and also the temperature was reduced to 18°C.

In order to improve germination, first boric acid concentration was altered keeping Ca(NO3

250 ppm to further improve the germination of *C. volubilis* pollen. At 100 ppm Ca(NO3

Sucrose 35%

Polyethylene glycol 15% Agar 1%

Temperature 18 ± 1°C

Calcium nitrate 100 mg l−1 Potassium nitrate 100 mg l−1 Magnesium sulfate 200 mg l−1

**4.3. Standardization of** *in vitro* **pollen germination media for** *C. cajan* **vars. Pusa 33**

nitrate, it was observed that the medium with 40% sucrose, 250 ppm H3

In the medium D (Set I) only initiation of germination was observed whereas no germination was observed in media A, B and C. On varying the concentrations of boric acid and calcium

, 1% agar and BK salts gave 43.80% germination with abnormal pollen tubes.

From Set II, medium G showed over 45% germination (**Figure 2**) but there was a lot of bursting of pollen grains; medium F also showed little germination. To control the bursting of pollen grains sucrose level was raised to 37.5% and further, different combinations of boric acid and calcium nitrate were tried. Even though over 87% germination was seen, almost all the pollen grains showed bursting after germination. Among the permutations

has the following constituents:

This gives 94.26% pollen germination.

**and IMS‐1**

(NO3 )2

at 300 ppm. At 250 ppm, maximum pollen germination was obtained (approximately 63%) and increasing or decreasing boric acid concentration over 250 ppm did not improve germination. Then, the concentration of calcium nitrate was modified keeping boric acid concentration at

len of *C. volubilis* showed 94.6% germination and pollen tube growth of 27.47 μm. *C. volubilis* showed 94.26% germination and pollen tube growth of 27.47 μm. Increasing the Ca(NO3

concentration to 200 ppm showed. The complete pollen germination medium for *C. volubilis*

)2

89

)2

) 2 , pol-

Pollen Germination *in vitro*

http://dx.doi.org/10.5772/intechopen.75360

BO3

, 300 ppm Ca

**Figure 3.** Procedure for standardization of *in vitro* pollen germination medium of *Cajanus* spp.

## **4.2. Standardization of** *in vitro* **pollen germination medium for** *C. volubilis*

On key media G (from Set II) the pollen of *C. volubilis* showed initiation of germination (budding) but almost all pollen grain showed bursting before or after budding. To contain bursting of pollen, sucrose level was increased to 35% and also the temperature was reduced to 18°C.

In order to improve germination, first boric acid concentration was altered keeping Ca(NO3 )2 at 300 ppm. At 250 ppm, maximum pollen germination was obtained (approximately 63%) and increasing or decreasing boric acid concentration over 250 ppm did not improve germination.

Then, the concentration of calcium nitrate was modified keeping boric acid concentration at 250 ppm to further improve the germination of *C. volubilis* pollen. At 100 ppm Ca(NO3 )2 , pollen of *C. volubilis* showed 94.6% germination and pollen tube growth of 27.47 μm. *C. volubilis* showed 94.26% germination and pollen tube growth of 27.47 μm. Increasing the Ca(NO3 )2 concentration to 200 ppm showed. The complete pollen germination medium for *C. volubilis* has the following constituents:

This gives 94.26% pollen germination.

Sucrose 20%

88 Pollination in Plants

Agar 1%

Temperature 20.5 ± 2°C

**Figure 3.** Procedure for standardization of *in vitro* pollen germination medium of *Cajanus* spp.

Boric acid 100 mg l−1 Calcium nitrate 300 mg l−1 Magnesium sulfate 200 mg l−1 Potassium nitrate 100 mg l−1

## **4.3. Standardization of** *in vitro* **pollen germination media for** *C. cajan* **vars. Pusa 33 and IMS‐1**

In the medium D (Set I) only initiation of germination was observed whereas no germination was observed in media A, B and C. On varying the concentrations of boric acid and calcium nitrate, it was observed that the medium with 40% sucrose, 250 ppm H3 BO3 , 300 ppm Ca (NO3 )2 , 1% agar and BK salts gave 43.80% germination with abnormal pollen tubes.

From Set II, medium G showed over 45% germination (**Figure 2**) but there was a lot of bursting of pollen grains; medium F also showed little germination. To control the bursting of pollen grains sucrose level was raised to 37.5% and further, different combinations of boric acid and calcium nitrate were tried. Even though over 87% germination was seen, almost all the pollen grains showed bursting after germination. Among the permutations and combinations of media tried by changing the concentration of boric acid and calcium nitrate, six combinations showed better germination (70–80%) but accompanied with bursting of pollen tubes.

#### *4.3.1. Pretreatments*

In order to standardize optimal media for pollen germination different methods of preconditioning the pollen were attempted. Heat treatment of pollen at 40°C for 30 min or at 60°C for 20 min did not improve germination of *C. cajan* pollen. Pre-hydration for 30 min in Petri dish lined with moist filter paper also did not improve germination. The stamen of pigeonpea is diadelphous (five short and five long stamens). Pollen extracted separately from short and long stamens showed approximately 70% germination. Pollen extracted from single anthers also showed about 70% germination. Thus, germination percentage did not vary between different anthers. Out of six combinations of media tried, the PGM with 250 ppm boric acid and 300 ppm calcium nitrate showed maximum normal pollen tubes.

Another pre-treatment of pollen was tried by incubating young buds (12 h or 36 h before anthesis) at 20.5°C in

To the above medium EACA at a concentration of 250–750 mg l−1 was added to arrest pollen

Pollen Germination *in vitro*

91

http://dx.doi.org/10.5772/intechopen.75360

The results of studies on *in vitro* pollen germination of *Cajanus cajan* and its wild species are

Based on the initial tests on *C. lineatus* it was found that modification of medium "G" could give optimum germination but due to non-availability of pollen, PGM could not be optimized.

**Optimal media composition** *C. platycarpus C. volubilis C. cajan* **vars. Pusa‐33 and IMS‐1**

Sucrose (%) 20 35 27.5 Boric acid (ppm) 100 250 250 Calcium nitrate (ppm) 300 100 300 Potassium nitrate 100 100 100 Magnesium sulfate 200 200 200 PEG 4000 (%) — 15 15 Agar (%) 1 1 1 Incubation period (h) 1 3 3

Sucrose 37.5%

PEG 4000 15% Agar 1%

Temperature 20.5 ± 20°C

Boric acid 250 mg l−1 Calcium nitrate 300 mg l−1 Potassium nitrate 100 mg l−1 Magnesium sulfate 300 mg l−1

Incubation temperature (°C) 20.5 18 20.5 % Pollen germination 99.15 94.26 92.52

**Table 1.** Results of *in vitro* pollen germination of *C. cajan* and its two wild relatives.

Pollen tube length (μm) 172.56 ± 0.315 27.47 ± 0.933 16.34 ± 0.264

tip bursting.

summarized in **Table 1**.

Then upon anther dehiscence, pollen was collected and germination was tested on six different media. The pollen from treatments (a), (b) and (c) showed good germination on medium G5 but burst soon after.

For treatments (d) and (e) media G1 –G6 were used both for pre-treatments and germination tests. The pollen from the treatment (d) showed much variation for germination and bursting. Pollen extracted from the treatment (e) showed improved germination for *C. cajan* var. Pusa 33 and IMS-1. Among the media tested (G1 to G6 ) the medium (G5 ) with 250 ppm H3 BO3 and 300 ppm Ca(NO3 )2 showed over 92% germination for both varieties. Besides media G3 , G4 and G6 also showed over 90% germination but there was pollen tube bursting. The result indicated pre-treatment of young buds (36 h before anthesis) in PGM (G5 ) for 36 h and germinating pollen in the same PGM gave over 92% germination Ca(NO3 )2 some other media (G3 , G4 and G6 ) also showed over 90% pollen germination but there was bursting of pollen tubes at tips and variation in pollen tube length.

Thus, the complete media for both Pusa-33 and IMS-1 should have the following composition:

and combinations of media tried by changing the concentration of boric acid and calcium nitrate, six combinations showed better germination (70–80%) but accompanied with

In order to standardize optimal media for pollen germination different methods of preconditioning the pollen were attempted. Heat treatment of pollen at 40°C for 30 min or at 60°C for 20 min did not improve germination of *C. cajan* pollen. Pre-hydration for 30 min in Petri dish lined with moist filter paper also did not improve germination. The stamen of pigeonpea is diadelphous (five short and five long stamens). Pollen extracted separately from short and long stamens showed approximately 70% germination. Pollen extracted from single anthers also showed about 70% germination. Thus, germination percentage did not vary between different anthers. Out of six combinations of media tried, the PGM with 250 ppm boric acid and 300 ppm calcium nitrate showed maximum

Another pre-treatment of pollen was tried by incubating young buds (12 h or 36 h before

Then upon anther dehiscence, pollen was collected and germination was tested on six different media. The pollen from treatments (a), (b) and (c) showed good germination on medium

tests. The pollen from the treatment (d) showed much variation for germination and bursting. Pollen extracted from the treatment (e) showed improved germination for *C. cajan* var. Pusa

to G6

also showed over 90% germination but there was pollen tube bursting. The result indicated

also showed over 90% pollen germination but there was bursting of pollen tubes at tips and

Thus, the complete media for both Pusa-33 and IMS-1 should have the following composition:

were used both for pre-treatments and germination

some other media (G3

) with 250 ppm H3

) for 36 h and germinating pol-

BO3 and

, G4 and

 and G6 )

, G4

) the medium (G5

)2

showed over 92% germination for both varieties. Besides media G3

bursting of pollen tubes.

*4.3.1. Pretreatments*

90 Pollination in Plants

normal pollen tubes.

anthesis) at 20.5°C in

G5

G6

(a) Petri dish lined with moist filter paper for 36 h.

(d) Pollen germination media for 12 h

(e) Pollen germination media for 36 h.

For treatments (d) and (e) media G1

)2

variation in pollen tube length.

33 and IMS-1. Among the media tested (G1

but burst soon after.

300 ppm Ca(NO3

(b) Agarified medium containing 37.5% sucrose in (a) for 36 h.

pre-treatment of young buds (36 h before anthesis) in PGM (G5

len in the same PGM gave over 92% germination Ca(NO3

(c) Agarified medium containing 37.5% sucrose +15% PEG in (a) for 36 h

–G6

To the above medium EACA at a concentration of 250–750 mg l−1 was added to arrest pollen tip bursting.

The results of studies on *in vitro* pollen germination of *Cajanus cajan* and its wild species are summarized in **Table 1**.

Based on the initial tests on *C. lineatus* it was found that modification of medium "G" could give optimum germination but due to non-availability of pollen, PGM could not be optimized.

**Table 1.** Results of *in vitro* pollen germination of *C. cajan* and its two wild relatives.

#### **4.4. Varietal response to PGMs**

It was noticed that there are genotypic differences for pollen germination. In pigeonpea, among the genotypes tested, IDTSP51 alone did not require EACA, but for others the requirement varied(250–750 mg l −1). Similar response was seen with wheat pollen germination. The medium supporting more than 90% pollen germination were selected for each genotype though some of them showed pollen tube bursting which indicated the need for fine tuning of the medium. The genotypes HW 971 and HW 741 responded well in M19 giving >95% pollen germination and pollen tube length of >400 μm. This is adjudged as the best medium since it did not require supplementing either EACA or peptone. It is noted that some genotypes (HD 2833, HW 971, HW 1095 etc.) require EACA alone in PGM whose response were at par with medium having EACA + peptone. Some varieties such as MACS 6195, HW 1095 etc. showed better responses (>92% germination) in PGM with a combination of EACA (500 or 750 mg) and peptone water (100 mg). These varieties responded poorly when peptone water was increased beyond 100 mg which reduced the pollen tube length (**Table 2**).

Among the varieties tested, two of them *viz*., HD 2833 and HW 1095 showed more that 94% in all the media tested. The genotype HD 2833 showed the maximum pollen tube length of

**Pollen germination**

**(%)**

M19 + 750 E + 100 P 92.91 ± 4.53 371.87 ± 16.36 M19 + 500E + 120 P 90.82 ± 4.76 278.125 ± 18.13 M19 + 750 E + 120 P 96.37 ± 3.75 190 ± 3.65

achieved a mean pollen tube length of 190 μm in the same medium The variety HD 2833 showed a mean pollen tube length in the range of 334.37 to 713 μm followed by MACS6145 with 360.58 to 603.13 μm and HW671 with 462.50 to 551.25 μm for mean pollen length. Similar kind of genotypic differences for pollen germination requirement have also been reported [1].

A simple or a complex medium may be developed to suit the germination requirement of pollen (binucleate or trinucleate pollen). The protocol suggested would give a better guidance in development of PGM. The author has developed pollen germination medium (PGM) for many crops *viz.* pigeonpea and its wild species, wheat, rye and brinjal by using the base constituents of PGM of Brewbaker and Kwack [7]. These media were supplemented with polyethylene glycol, e-amino caproic acid, peptone etc. besides the addition of BK salts. Grasses pollen so far considered as recalcitrant can also be germinated in the artificial medium e.g. wheat and rye. The grass pollen is released at high moisture content (30–40%) as compared to 1–5% in the case of orthodox species. This trait makes it unsuitable for *in vitro* germination. With combination of maltose, PEG and tryptone/peptone contained the initial pollen bursting and resulted in successful pollen germination of wheat and rye. The grasses pollen was so far classified as recalcitrant [31]. It is observed that monocots require low salt content than the dicots in the pollen germination medium. Also wild species require minimal medium than the domesticate ones.

−1 peptone and the pollen of HW1095

**Pollen tube length (μm)**

http://dx.doi.org/10.5772/intechopen.75360

**Range for pollen tube length (μm)** 93

Pollen Germination *in vitro*

−1 calcium nitrate + BK

**(mean ± S.E)**

−1 boric acid + 30 mg l

−1 EACA and 120 mg l

713 μm in PGM with 750 mg l

**Variety Pollen germination**

**medium**

P = peptone; E = EACA.\*M19 = 19% Maltose + 13% PEG 6000 + 50 mg l

**Table 2.** Response of wheat genotypes in pollen germination medium.

**5. Conclusion**

salts + 1% agar.

**Author details**

Address all correspondence to<EMAIL_ADDRESS>
ICAR-Indian Agricultural Research Institute, The Nilgiris, Tamil Nadu, India

Jayaprakash P

**Table 2.** Response of wheat genotypes in pollen germination medium.

Among the varieties tested, two of them *viz*., HD 2833 and HW 1095 showed more that 94% in all the media tested. The genotype HD 2833 showed the maximum pollen tube length of 713 μm in PGM with 750 mg l −1 EACA and 120 mg l −1 peptone and the pollen of HW1095 achieved a mean pollen tube length of 190 μm in the same medium The variety HD 2833 showed a mean pollen tube length in the range of 334.37 to 713 μm followed by MACS6145 with 360.58 to 603.13 μm and HW671 with 462.50 to 551.25 μm for mean pollen length. Similar kind of genotypic differences for pollen germination requirement have also been reported [1].

## **5. Conclusion**

**4.4. Varietal response to PGMs**

92 Pollination in Plants

requirement varied(250–750 mg l

**Variety Pollen germination**

**medium**

length (**Table 2**).

It was noticed that there are genotypic differences for pollen germination. In pigeonpea, among the genotypes tested, IDTSP51 alone did not require EACA, but for others the

tion. The medium supporting more than 90% pollen germination were selected for each genotype though some of them showed pollen tube bursting which indicated the need for fine tuning of the medium. The genotypes HW 971 and HW 741 responded well in M19 giving >95% pollen germination and pollen tube length of >400 μm. This is adjudged as the best medium since it did not require supplementing either EACA or peptone. It is noted that some genotypes (HD 2833, HW 971, HW 1095 etc.) require EACA alone in PGM whose response were at par with medium having EACA + peptone. Some varieties such as MACS 6195, HW 1095 etc. showed better responses (>92% germination) in PGM with a combination of EACA (500 or 750 mg) and peptone water (100 mg). These varieties responded poorly when peptone water was increased beyond 100 mg which reduced the pollen tube

**Pollen germination**

M19 + 500 E + 100 P 97.51 ± 2.98 553.75 ± 23.14 334.37–713.00

**(%)**

M19 + 750E 94.32 ± 3.67 667.7142 ± 32.18

M19 + 750 E + 100 P 95.38 ± 3.09 334.37 ± 18.39 M19 + 500 E + 120 P 97.12 ± 1.87 680.00 ± 24.255 M19 + 750 E + 120 P 97.22 ± 1.74 713.0 ± 21.55 HW2044 M19 + 500 E 98.02 ± 1.03 265.60 ± 9.88 265.60–467.50

M19 + 750E 98.01 ± 0.076 467.5 ± 9.55 HW5207 M19 + 750E 98.0 ± 0.043 313.529 ± 12.45 246–350.62 M19 + 500 E + 120 P 98 ± 0.054 350.625 ± 14.91 MACS 6145 M19 98.11 ± 0.87 360.58 ± 12.05 360.58–603.13 M19 + 750 E + 100 P 97.21 ± 1.54 603.13 ± 22.48 M19 + 750 E + 120 P 97 2.63 ± 2.11 546.25 ± 26.89 HW 971 M19 + 500 E 98 .13 ± 1.42 462.50 ± 18.76 462.50–551.25 M19 + 500 E + 100 P 98.0 ± 0.6 551.25 ± 25.85 HW 741 M19 98.23 ± 0.042 403.75 ± 21.13 241.87–403.75 M19 + 500 E 97.23 ± 1.99 241.87 ± 8.55

HD2833 \*M19 + 500 E 97.9 418.8 ± 15.39

HW1085 M19 + 500 E 98.18 ± 1.11 244.375 ± 11.44

HW1095 M19 + 500 E 98.01 ± 0.43 268.75 ± 11.83 190–371.87

−1). Similar response was seen with wheat pollen germina-

**Pollen tube length (μm)**

**Range for pollen tube length (μm)**

**(mean ± S.E)**

A simple or a complex medium may be developed to suit the germination requirement of pollen (binucleate or trinucleate pollen). The protocol suggested would give a better guidance in development of PGM. The author has developed pollen germination medium (PGM) for many crops *viz.* pigeonpea and its wild species, wheat, rye and brinjal by using the base constituents of PGM of Brewbaker and Kwack [7]. These media were supplemented with polyethylene glycol, e-amino caproic acid, peptone etc. besides the addition of BK salts. Grasses pollen so far considered as recalcitrant can also be germinated in the artificial medium e.g. wheat and rye. The grass pollen is released at high moisture content (30–40%) as compared to 1–5% in the case of orthodox species. This trait makes it unsuitable for *in vitro* germination. With combination of maltose, PEG and tryptone/peptone contained the initial pollen bursting and resulted in successful pollen germination of wheat and rye. The grasses pollen was so far classified as recalcitrant [31]. It is observed that monocots require low salt content than the dicots in the pollen germination medium. Also wild species require minimal medium than the domesticate ones.

## **Author details**

Jayaprakash P

Address all correspondence to<EMAIL_ADDRESS>
ICAR-Indian Agricultural Research Institute, The Nilgiris, Tamil Nadu, India

## **References**

[1] Jayaprakash P, Sarla N. Development of an improved medium for germination of *Cajanus cajan* (L.) Millsp. pollen *in vitro*. Journal of Experimental Botany. 2001;**52**:851-855

[15] Jayaprakash P, Sabesan T. *In vitro* pollen germination of some wild species of pigeonpea (*Cajanus cajan*) using PGM droplet technique. Indian Journal of Genetics and Plant

Pollen Germination *in vitro*

95

http://dx.doi.org/10.5772/intechopen.75360

[17] Vasil K. Effect of boron on pollen germination and pollen growth. In: Linskens JF, editor. Pollen Physiology and Fertilization. Amsterdam, The Netherlands: North Holland

[18] Visser T. Germination and storage of pollen. Meded Landbouwhogeschool Wageningen.

[19] Acar I, Ak BE, Sarpkaya K. Effect of boron and gibberellic acid on *in vitro* pollen germination of pistachio (*Pistacia vera* L.). African Journal of Biotechnology. 2010;**32**:5126-5130

[20] Steinhorst L, Kudla J. Calcium—A central regulator of pollen germination and tube

[21] Powell GM III. Polyethylene glycol. In: Davidson RE, editor. Handbook of Water Soluble

[22] Ferrari TE, Wallace DH. Germination of *Brassica* pollen and expression of incompatibil-

[23] Nhut DT, Thi NN, Khiet BLT, Luan VQ. Peptone stimulates *in vitro* shoot and root regeneration of avocado (*Persia Americana Mil*). Science Horticulture. 2008;**115**:124-128

[24] Sivakumar G, Yu KW, Hahn EJ, Pack YY. Optimization of organic nutrients for ginseng hairy roots production in large scale bioreactors. Current Science. 2005;**89**:641-649

[25] Nagarajan P, Walton PD. The effect of ε-amino caproic acid on self-incompatibility and

[26] Baker LR, Chen NC, Park HG. Effect of an immunosuppressant on an interspecific cross

[27] Chen NC, Parrot HF, Jacob T, Baker LR, Carlson PS. Interspecific hybridization of food legumes by unconventional methods of breeding. In:Proceedings of the First International

Mungbean Symposium.Aug 16-19, Univ. Philipines, Los Bonas. 1978. pp. 247-252

[28] Bianca Caroline RR, Brochetto-Braga MR, Tauk-Tornisielo SM, Carmona EC, Arruda VM, Netto JC. Comparative growth of *Trichoderma* strains in different nutritional sources, using bioscreening C automated system. Brazilian Journal of Microbiology. 2009;**40**:

[29] Anjana S, Islam H. Optimization of medium components for extracellular glutaminase free asparaginase from *enterobacter cloacae*. International Journla of Current

incongruity in *Phaseolus*. Journal of Horticultural Science. 1988;**63**:433-450

growth. Biochimica et Biophysica Acta. 2013;**1833**:1573-1581

ity *in vitro*. Euphytica. 1975;**24**:751-765

of genus *Vigna*. Horticultural Science. 1975;**10**:313

Microbiological Applied Science. 2015;**4**(1):296-309

Gums and Resins. New York: McGraw-Hill; 1980. pp. 1801-1831

[16] Johri BM, Vasil IK. Physiology of pollen. The Botanical Review. 1961;**27**:325-381

Breeding. 2013;**73**:211-215

Publishing Co.; 1963

1955;**55**:1-68

404-410

**References**

94 Pollination in Plants

2015;**75**(4):446-452

Neerlandica. 1982;**31**:113-119

Newsletter. 1997;**4**:28-29

Holland; 1964. pp. 145-151

Phytologist. 1992;**120**:459-462

Phytologist. 2011;**197**:668-679

Manual. New Delhi: Narosa; 1992. p. 33

2017;**77**(1):25-31

[1] Jayaprakash P, Sarla N. Development of an improved medium for germination of *Cajanus cajan* (L.) Millsp. pollen *in vitro*. Journal of Experimental Botany. 2001;**52**:851-855

[2] Jayaprakash P, Annapoorani S, Vikas VK, Sivasamy M, Kumar1 J, Saravannan K, Punniakotti, Sheeba D. An improved *in vitro* germination medium for recalcitrant bread wheat pollen (*Triticum aestivum* L.). Indian Journal of Genetics and Plant Breeding.

[3] Linskens HF. Pollen. Handbuch der Pflanzenphysiologie. Vol. XVIII1967. pp. 368-406

[4] Zhang HQ, Croes AF. A new medium for pollen germination *in vitro*. Acta Botanica

[5] Shivanna KR, Saxena NP, Seetharama N. An improvised medium for *in vitro* pollen germination and pollen tube growth of chickpea. International Chickpea and Pigeonpea

[6] Read SM, Clarke A, Bacic A. Stimulation of growth of cultured *Nicotiana tabacum* W-38 pollen tubes by polyethylene glycol and Cu(II) salts. Protoplasma. 1993;**177**:1-14

[7] Brewbaker JL, Kwack BH. The calcium ion and substances influencing pollen growth. In: Linskens HF, editor. Pollen Physiology and Fertilization. Amsterdam: Elsevier North

[8] Roberts IN, Gaunde TC, Harrod G, Dickinson HG. Pollen stigma interaction in *Bracssica oleracea*: A new pollen germination medium and its use in elucidating the mechanism of

[9] Cheng C, McComb JA. *In vitro* germination of wheat pollen on raffinose medium. New

[10] Jayaprakash P, Sheeba D, Vikas VK, Sivasamy M, Kumar J, Annapoorani S, Krishan N. A reproducible *in vitro* pollen germination medium for recalcitrant cereal rye pollen (*Secale cereale* L.): Effect of tryptone. Indian Journal of Genetics and Plant Breeding.

[11] Rodriguez-Enriquez MJ, Mehdi S, Dickinson HG, Grant-Downton RT. A novel method for efficient *in vitro* germination and tube growth of *Arabidopsis thaliana* pollen. New

[12] Shivanna KR, Rangaswamy NS. Test for pollen viability. In: Pollen Biology: A Laboratory

[14] Singh IS, Bharti IS, Nandwal AS, Goswami C, Verma SK. Effect of temperature on *in vitro*

[13] Shivanna KR and Rangaswamy NS. Pollen Biology: Laboratory Manual. 1992

pollen germination in pigeonpea. Biologia Plantarum. 1992;**34**

self-incompatibility. Theoretical and Applied Genetics. 1983;**65**:231-238

[30] Gudina JE, Elisabete CF, Rodrigues AI, José AT, Lígia RR. Biosurfactant production by *Bacillus subtilis* using corn steep liquor as culture medium. Frontiers Microbiology. 2015:6-59

**Chapter 7**

Provisional chapter

. The greatest effect of

**Comparison of Pollination Graphs**

Comparison of Pollination Graphs

James H. Lee, David M. Chan and Rodney J. Dyer

James H. Lee, David M. Chan and Rodney J. Dyer

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 cluster-

DOI: 10.5772/intechopen.74553

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

Keywords: pollination network, correlated random walk, agent-based model, pollen

While the movement of genes from one generation to the next ensures the cohesiveness of plant species through time and space [1–3], 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

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

ing coefficient of fathers is largest when δmax is between 60 and 90∘

locations is crucial in determining the properties of a pollination graph.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74553

Abstract

carryover, tree density

1. Introduction

[31] Franchi GG, Piotto B, Nepi M, Baskin CC, Baskin J, Pacini E. Pollen and seed desiccation tolerance in relation to degree of developmental arrest, dispersal, and survival. Journal of Experimental Botany. 2011:1-15

## **Chapter 7**

Provisional chapter

## **Comparison of Pollination Graphs**

Comparison of Pollination Graphs

James H. Lee, David M. Chan and Rodney J. Dyer

James H. Lee, David M. Chan and Rodney J. Dyer

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74553

#### Abstract

[30] Gudina JE, Elisabete CF, Rodrigues AI, José AT, Lígia RR. Biosurfactant production by *Bacillus subtilis* using corn steep liquor as culture medium. Frontiers Microbiology.

[31] Franchi GG, Piotto B, Nepi M, Baskin CC, Baskin J, Pacini E. Pollen and seed desiccation tolerance in relation to degree of developmental arrest, dispersal, and survival. Journal

2015:6-59

96 Pollination in Plants

of Experimental Botany. 2011:1-15

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.

DOI: 10.5772/intechopen.74553

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 [1–3], 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

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

complexity of the landscape and co-occurring species [6], also influence connectivity to an extent that it is easy to discern.

2.1. Field characteristics

2.2. Tree characteristics

simulated on this landscape, ω, is determined as

Simulations were also run for a field size of 3100 ffiffiffiffiffi

ω ¼ τ

trees per square unit, of ω ∈f g 0:0250; 0:0500; 0:0750; 0:1000; 0:1500 .

it 'bounces' off of the barrier at the opposite angle from which it approached.

For tracking purposes, each tree, T, is numbered such that 1 ≤ T ≤ τ. Let Yð Þ T ¼ y

sired those seeds, and the frequency each tree fathered seeds on other trees.

heterogeneity influences broad trends in pollen connectivity.

The field size for model runs is set as a square 100 � 100 units grid. The density of the trees

where τ is the number of trees. In our simulation runs, we used tree densities, measured in

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

541 p � 1100 ffiffiffiffiffi

used based on data for trees sampled from the experimental natural population at the Virginia Commonwealth University, Rice Rivers Center (http://ricerivers.vcu.edu). This population was used as it has been the focus of previous work on pollen-mediated gene flow [11]. The tree density that results from this data set was ω ¼ 0:071552 trees per square unit, which was also simulated as a uniformly distributed scenario along with others previously mentioned.

the location of tree T, which is static. For the initial model runs, trees are randomly placed using a uniform random distribution within the allotted field size. For both simplicity and comparison to temperate tree species, we assume that all trees are self-incompatible and that all successful pollination and fertilization produces non-inbred individuals. This means that all pollination distances will be strictly greater than zero since at most one tree can occupy any location on the landscape. We also applied the model to a spatial arrangement of trees mimicking a natural population for which we have already conducted extensive empirical studies of insect-mediated pollination and gene flow. To create the pollination graph, data was collected to determine the number of seeds fertilized on each tree, which pollen donor trees are

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

10; 000 ¼ 0:0001τ, (1)

Comparison of Pollination Graphs

99

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541 p ≈ 133:2794 � 47:2927 units. This was

ð Þ T 1 ; y ð Þ T 2 � � be

100 � 100 ¼ τ

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 subdiscipline 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

complexity of the landscape and co-occurring species [6], also influence connectivity to an

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 subdiscipline 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, alterna-

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

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

tive pollinating species leave discernibly distinct genetic structure [4, 9].

previous landscape genetic studies have been conducted [11].

2. Background and methods

donor density and diversity).

extent that it is easy to discern.

98 Pollination in Plants

The field size for model runs is set as a square 100 � 100 units grid. The density of the trees simulated on this landscape, ω, is determined as

$$
\omega = \frac{\tau}{100 \times 100} = \frac{\tau}{10,000} = 0.0001\tau,\tag{1}
$$

where τ is the number of trees. In our simulation runs, we used tree densities, measured in trees per square unit, of ω ∈f g 0:0250; 0:0500; 0:0750; 0:1000; 0:1500 .

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 3100 ffiffiffiffiffi 541 p � 1100 ffiffiffiffiffi 541 p ≈ 133:2794 � 47:2927 units. This was used based on data for trees sampled from the experimental natural population at the Virginia Commonwealth University, Rice Rivers Center (http://ricerivers.vcu.edu). This population was used as it has been the focus of previous work on pollen-mediated gene flow [11]. The tree density that results from this data set was ω ¼ 0:071552 trees per square unit, which was also simulated as a uniformly distributed scenario along with others previously mentioned.

#### 2.2. Tree characteristics

For tracking purposes, each tree, T, is numbered such that 1 ≤ T ≤ τ. Let Yð Þ T ¼ y ð Þ T 1 ; y ð Þ T 2 � � be the location of tree T, which is static. For the initial model runs, trees are randomly placed using a uniform random distribution within the allotted field size. For both simplicity and comparison to temperate tree species, we assume that all trees are self-incompatible and that all successful pollination and fertilization produces non-inbred individuals. This means that all pollination distances will be strictly greater than zero since at most one tree can occupy any location on the landscape. We also applied the model to a spatial arrangement of trees mimicking a natural population for which we have already conducted extensive empirical studies of insect-mediated pollination and gene flow. To create the pollination graph, data was collected to determine the number of seeds fertilized on each tree, which pollen donor trees are sired those seeds, and the frequency each tree fathered seeds on other trees.

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 [14–18].

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 nmax time steps.

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 β be the total number of pollinators in a simulation, and let Xð Þi n ¼ x ð Þi 1,n; x ð Þi 2,n be the location of the i th pollinator, 1 ≤ i ≤ β, at time step n, 0 ≤ n ≤ nmax. For all simulations, we assume β ¼ 1000 and nmax ¼ 600.

The initial position of each pollinator, Xð Þi 0 is uniformly distributed throughout the field. Each pollinator's initial heading, �180∘ ≤ θð Þi 1 ≤ 180∘ is chosen from a uniform random distribution. At each subsequent time step, the pollinator's new heading θð Þi nþ1 is dependent upon its current heading θð Þi n and a random number δ ð Þi nþ1. That is

$$
\Theta\_{n+1}^{(i)} = \Theta\_n^{(i)} + \delta\_{n+1}^{(i)} \tag{2}
$$

Sample paths based on different values of δmax are shown in Figure 1. As δmax increases the paths do not generally travel away as far from the starting point and loop around more often.

Comparison of Pollination Graphs

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If a pollinator visits a flower, it will collect pollen from that individual. Pollen will be deposited with a probability of Pκ, where κ is the number of previously visited flowers, to account for pollen carryover. Carryover probability Pκ ¼ 0 if κ > κmax where κmax is the maximum pollen

carryover. In the simulations the pollination carryovers used were κmax ∈ f g 1; 3; 5; 7; ∞ .

A path with δmax ¼ 0 would be along a straight line, which is not shown here.

2.4. Pollination

Figure 1. Sample paths based on δmax.

where δ ð Þi nþ1 ∈ ð Þ �δmax; δmax for each n ¼ 1, …, nmax. Similarly, the initial step size of each pollinator, r ð Þi 1 ¼ 1 at time n ¼ 1. Each subsequent step size, r ð Þi nþ1 ∈ ð Þ 0; rmax uniformly distributed for each n ¼ 1, … , nmax. In Cartesian coordinates, the position of the i th pollinator at each subsequent time step will be

$$\mathbf{X}\_{n+1}^{(i)}\left(\mathbf{X}\_{n}^{(i)};\ r\_{n+1}^{(i)},\ \ \boldsymbol{\theta}\_{n+1}^{(i)}\right) = \left(\mathbf{x}\_{1,n}^{(i)} + r\_{n+1}^{(i)}\cos\left(\boldsymbol{\theta}\_{n+1}^{(i)}\right),\ \ \mathbf{x}\_{2,n}^{(i)} + r\_{n+1}^{(i)}\sin\left(\boldsymbol{\theta}\_{n+1}^{(i)}\right)\right) \tag{3}$$

for each n ¼ 1, … , nmax. In our simulations we used values of δmax ∈ 0∘ ; 15∘ ; 30∘ ; 45∘ f ; 60∘ ; 75∘ ; 90∘ ; 120∘ ; 150∘ ; 180∘ g.

Sample paths based on different values of δmax are shown in Figure 1. As δmax increases the paths do not generally travel away as far from the starting point and loop around more often. A path with δmax ¼ 0 would be along a straight line, which is not shown here.

#### 2.4. Pollination

2.3. Pollinator movement

100 Pollination in Plants

animal taxa [14–18].

location of the i

heading θð Þi

where δ

; 60∘

pollinator, r

ð Þi

; 75∘ ; 90∘

ð Þi

Xð Þi nþ1 Xð Þi n ; r ð Þi nþ1; θð Þi nþ1

each subsequent time step will be

; 120∘

β ¼ 1000 and nmax ¼ 600.

process continued for nmax time steps.

The initial position of each pollinator, Xð Þi

n and a random number δ

; 150∘ ; 180∘ g.

pollinator's initial heading, �180∘ ≤ θð Þi

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

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

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.

th pollinator, 1 ≤ i ≤ β, at time step n, 0 ≤ n ≤ nmax. For all simulations, we assume

n þ δ ð Þi

nþ1 ∈ ð Þ �δmax; δmax for each n ¼ 1, …, nmax. Similarly, the initial step size of each

n ¼ x ð Þi 1,n; x ð Þi 2,n

nþ1 is dependent upon its current

nþ1 ∈ ð Þ 0; rmax uniformly distrib-

nþ1

th pollinator at

; 15∘ ; 30∘ ; 45∘ f

(3)

nþ1 (2)

0 is uniformly distributed throughout the field. Each

1 ≤ 180∘ is chosen from a uniform random distribution.

ð Þi

; x ð Þi 2,n þ r ð Þi nþ1sin θð Þi

nþ1  be the

Let β be the total number of pollinators in a simulation, and let Xð Þi

ð Þi nþ1. That is

1 ¼ 1 at time n ¼ 1. Each subsequent step size, r

¼ x ð Þi 1,n þ r ð Þi nþ1cos θð Þi

θð Þi nþ1 ¼ θð Þi

uted for each n ¼ 1, … , nmax. In Cartesian coordinates, the position of the i

for each n ¼ 1, … , nmax. In our simulations we used values of δmax ∈ 0∘

At each subsequent time step, the pollinator's new heading θð Þi

If a pollinator visits a flower, it will collect pollen from that individual. Pollen will be deposited with a probability of Pκ, where κ is the number of previously visited flowers, to account for pollen carryover. Carryover probability Pκ ¼ 0 if κ > κmax where κmax is the maximum pollen carryover. In the simulations the pollination carryovers used were κmax ∈ f g 1; 3; 5; 7; ∞ .

Figure 1. Sample paths based on δmax.

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 γð Þ 1 � γ k�1 pollen grains onto the k th flower visited, where γ depends upon the type of pollen as well as the type of pollinator. In this study, the probability that an pollinator distributes pollen from one tree to another tree is given by

$$P\_{\kappa} = \begin{cases} \rho (1 - \rho)^{\kappa - 1} & \text{if } \kappa \le \kappa\_{\text{max}} \\ 0 & \text{otherwise} \end{cases} \tag{4}$$

The connectance, L, of a graph is defined as the proportion of realized pollination events to the

As with the previous parameters, connectance has been used in a number of settings, see [23, 25–28], 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 m and tree f where tree f ∈ ϕm, then denote bf ,m to be the weight of that edge, which is equal to the number of times tree f fathers seeds on tree m.

seeds that tree f fathers on tree m. The weighted diversity of fathers for a mother tree m is a weighted measurement of the number of fathers that contribute pollen to seeds on m, accounting for the various number of seeds fathered by each father tree. The weighted diversity of

> Pj j ϕm f ¼1 bf ,m � �2

Pj j ϕm f ¼1 bf ,m

The average weighted diversity of fathers is the mean average of the weighted diversity of fathers

X μ

m¼1

The average pollination distance for an pollinator i is the average of the distances between any

� � be the location of mother tree mð Þi where pollinator i delivers pollen from tree

y mð Þi ð Þ 1 � y

� � be the location of father tree f

f ð Þi ð Þ 1 � �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ y mð Þi ð Þ 2 � y

f ð Þi ð Þ 2

τ τð Þ � 1 : (6)

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� �, which is a τ � τ matrix such that bf ,m is the number of

� �2 : (7)

bFm, (8)

, achieved by pollinator i for all such

f ð Þi ð Þ 2 � �2

ð Þi and Y mð Þi ð Þ ¼

, (9)

, for each

m are the number of

number of possible pollination events [24]. The connectance of the graph is given by

Pτ f ¼1 Pτ m¼1 af ,m

L ¼

fathers, bFm, is computed for each m in the graph by the formula

over all mother trees, and is given by the formula

two trees mated by i. Let Y f

Dð Þi ¼ 1 μð Þi

y mð Þi ð Þ 1 ; y mð Þi ð Þ 2

pairings is

f

where 0 ≤ μ ≤ τ is the total number individuals in the graph.

ð Þi ð Þ ¼ y

ð Þi to tree mð Þi .  The average pollination distance, Dð Þi

bFm ¼

E ¼ 1 μ

With this we create the matrix B ¼ bi,j

where κ is the number of previously visited flowers, and r is the chance of pollination when κ ¼ 1. For the simulation work we used r ¼ 0:30 based on work by [21].

#### 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, ω, pollination carryover, κ, and pollinator maximum turning angle, δmax. To determine the effect of these parameters on the landscape pattern of connectivity, we estimated the number of fathers per mother, Φm, connectance, L, average weighted diversity of fathers, E, average pollination distance, D, average maximum pollination distance, D~ , and the weak and strong clustering coefficients of fathers, Cweak and Cstrong.

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, f , if that tree contributes pollen to another tree. We will refer to a tree as a mother tree, m, if that tree is accepts pollen from another tree. Let ϕm be the set of trees which father seeds on tree m, then the number of fathers for each tree m in the graph is ϕm � � � �, m ¼ 1, 2, … , τ, where ∣ � ∣ denotes cardinality. The set containing the number of fathers per mother for all trees in the graph is

$$\Phi\_m = \{ |\phi\_1|, \ |\phi\_2|, \ \dots, \ |\phi\_\tau| \}. \tag{5}$$

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 τ � τ adjacency matrix, A ¼ ai,j � �, where af ,m ¼ 1 if tree f fathers at least one seed on tree m, and 0 if not. Thus A is a binary representation of the connectance of the graph. Since the trees do not self-pollinate, the number of possible interactions on this matrix is τ τð Þ � 1 .

The connectance, L, of a graph is defined as the proportion of realized pollination events to the number of possible pollination events [24]. The connectance of the graph is given by

$$L = \frac{\sum\_{f=1}^{\tau} \sum\_{m=1}^{\tau} a\_{f,m}}{\tau(\tau - 1)}. \tag{6}$$

As with the previous parameters, connectance has been used in a number of settings, see [23, 25–28], 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 m and tree f where tree f ∈ ϕm, then denote bf ,m to be the weight of that edge, which is equal to the number of times tree f fathers seeds on tree m. With this we create the matrix B ¼ bi,j � �, which is a τ � τ matrix such that bf ,m is the number of seeds that tree f fathers on tree m. The weighted diversity of fathers for a mother tree m is a weighted measurement of the number of fathers that contribute pollen to seeds on m, accounting for the various number of seeds fathered by each father tree. The weighted diversity of fathers, bFm, is computed for each m in the graph by the formula

$$\widehat{F}\_m = \frac{\left(\sum\_{f=1}^{\left|\phi\_m\right|} b\_{f,m}\right)^2}{\sum\_{f=1}^{\left|\phi\_m\right|} \left(b\_{f,m}\right)^2}.\tag{7}$$

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

$$E = \frac{1}{\mu} \sum\_{m=1}^{\mu} \quad \widehat{F}\_{m\prime} \tag{8}$$

where 0 ≤ μ ≤ τ is the total number individuals in the graph.

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

visited, where γ depends upon the type of pollen as well as the type of pollinator. In this study, the probability that an pollinator distributes pollen from one tree to another tree is given by

> Pκ ¼ rð Þ 1 � r κ�1 if κ ≤ κmax 0 otherwise

where κ is the number of previously visited flowers, and r is the chance of pollination when

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

The parameters we vary in constructing these networks include tree density, ω, pollination carryover, κ, and pollinator maximum turning angle, δmax. To determine the effect of these parameters on the landscape pattern of connectivity, we estimated the number of fathers per mother, Φm, connectance, L, average weighted diversity of fathers, E, average pollination distance, D, average maximum pollination distance, D~ , and the 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, f , if that tree contributes pollen to another tree. We will refer to a tree as a mother tree, m, if that tree is accepts pollen from another tree. Let ϕm be the set of trees which father seeds on tree m, then the number of fathers

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

f fathers at least one seed on tree m, and 0 if not. Thus A is a binary representation of the connectance of the graph. Since the trees do not self-pollinate, the number of possible interac-

�; … ; ϕτ � � �

�, m ¼ 1, 2, … , τ, where ∣ � ∣ denotes cardinality. The set

� � �: (5)

� �, where af ,m ¼ 1 if tree

th flower

(4)

given flower, a pollinator will deposit roughly γð Þ 1 � γ k�1 pollen grains onto the k

(

κ ¼ 1. For the simulation work we used r ¼ 0:30 based on work by [21].

2.5. Statistics

102 Pollination in Plants

directed pollination graph.

for each tree m in the graph is ϕm

� � �

containing the number of fathers per mother for all trees in the graph is

Φm ¼ ϕ1 � � � �; ϕ2 � � �

From this construct, we then create the τ � τ adjacency matrix, A ¼ ai,j

Cweak and Cstrong.

Valdovions et al. [23].

tions on this matrix is τ τð Þ � 1 .

The average pollination distance for an pollinator i is the average of the distances between any two trees mated by i.  Then the average pollination distance, Dð Þi , achieved by pollinator i for all such pairings is

$$\overline{D}^{(i)} = \frac{1}{\mu^{(i)}} \sum\_{m=1}^{\mu^{(i)}} \sum\_{f=1}^{\phi\_m^{(i)}} \frac{1}{\phi\_m^{(i)}} \sqrt{\left(y\_1^{(m^{(i)})} - y\_1^{(f^{(i)})}\right)^2 + \left(y\_2^{(m^{(i)})} - y\_2^{(f^{(i)})}\right)^2},\tag{9}$$

where μð Þi is the number of mother trees pollinated by pollinator i, and ϕi m are the number of father trees pollinating tree m by pollinator i. The average pollination distance, Dð Þi , for each pollinator is averaged over the total number of pollinators, β, to obtain the average pollination distance, D, for the graph

$$
\overline{D} = \frac{1}{\beta} \sum\_{i=1}^{\beta} \ \overline{D}^{(i)}.\tag{10}
$$

D~ ¼ 1 β

other (Figure 2(l)).

densely connected groups.

3. Results and discussion

of parameters. The results follow.

3.1. Number of fathers

pollination events.

X β

D~ ð Þi : (12)

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Comparison of Pollination Graphs

105

number of fathering triplets : (13)

number of fathering triplets : (14)

i¼1

A fathering triplet is the relationship between three trees such that tree f is a father to seeds on both tree m1 and tree m2. These seeds are half-siblings on the paternal side. In particular a weak fathering triangle, as a subset of fathering triplets, is one such that m1 also fathers seeds on m2 (Figure 2(a)–(d)), m2 fathers seeds on m1 (Figure 2(e)–(h)), or both (Figure 2(i)–(l)). A strong fathering triangle is defined as fathering triangle where f , m1, and m2 all father seeds on each

The weak clustering coefficient of fathers, Cweak, is the number of weak fathering triangles in the

Cweak ¼ number of weak fathering triangles

The strong clustering coefficient of fathers, Cstrong is the number of strong fathering triangles in the

Cstrong ¼ number of strong fathering triangles

Cweak and Cstrong are measurements of the tendency of parent trees to be clustered together in

We examine the effect of varying parameters on the graph statistics: the number of fathers per mother, Φm, connectance, L, the average weighted diversity of fathers, E, the clustering coefficient of fathers, C, the average pollination distance, D, and the average maximum pollination distance, D~ . The model was simulated a total of 10 replicate runs for each unique combination

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, Φm. In Figure 3, the number of different fathers per mother in a randomized placement of trees is distributed in a Gaussianlike distribution. This is an expected outcome since gene flow should be directly proportional to the distance traveled by a pollinator. It has been shown, see [29], that the distribution range resulting from a CRW would necessarily result in Gaussian-like behavior for a large number of

pollination graph over the total number of fathering triplets

pollination graph over the total number of fathering triplets

The maximum pollination distance for an pollinator i is

$$\tilde{D}^{(i)} = \max\_{\substack{\boldsymbol{m}^{(i)} \le \boldsymbol{\mu}^{(i)}, \ \boldsymbol{f}^{(i)} \le \boldsymbol{\phi}\_{\text{w}}^{(i)}}} \left( \sqrt{\left( \boldsymbol{y}\_{1}^{(\boldsymbol{m}^{(i)})} - \boldsymbol{y}\_{1}^{(\boldsymbol{f}^{(i)})} \right)^{2} + \left( \boldsymbol{y}\_{2}^{(\boldsymbol{m}^{(i)})} - \boldsymbol{y}\_{2}^{(\boldsymbol{f}^{(i)})} \right)^{2}} \right). \tag{11}$$

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

Figure 2. Fathering triangles. Arrows indicate direction of gene flow.

$$
\tilde{D} = \frac{1}{\beta} \sum\_{i=1}^{\beta} \quad \tilde{D}^{(i)}.\tag{12}
$$

A fathering triplet is the relationship between three trees such that tree f is a father to seeds on both tree m1 and tree m2. These seeds are half-siblings on the paternal side. In particular a weak fathering triangle, as a subset of fathering triplets, is one such that m1 also fathers seeds on m2 (Figure 2(a)–(d)), m2 fathers seeds on m1 (Figure 2(e)–(h)), or both (Figure 2(i)–(l)). A strong fathering triangle is defined as fathering triangle where f , m1, and m2 all father seeds on each other (Figure 2(l)).

The weak clustering coefficient of fathers, Cweak, is the number of weak fathering triangles in the pollination graph over the total number of fathering triplets

$$\mathcal{C}\_{\text{weak}} = \frac{\text{number of weak father triangles}}{\text{number of father triangles}}.\tag{13}$$

The strong clustering coefficient of fathers, Cstrong is the number of strong fathering triangles in the pollination graph over the total number of fathering triplets

$$\mathcal{C}\_{\text{strong}} = \frac{\text{number of strong father triangles}}{\text{number of father triangles}}.\tag{14}$$

Cweak and Cstrong are measurements of the tendency of parent trees to be clustered together in densely connected groups.

## 3. Results and discussion

pollinator is averaged over the total number of pollinators, β, to obtain the average pollination

X β

Dð Þi

f ð Þi ð Þ 1 � �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ y mð Þi ð Þ 2 � y

: (10)

1

A: (11)

f ð Þi ð Þ 2 � �2

i¼1

D ¼ 1 β

> y mð Þi ð Þ 1 � y

The maximum pollination distance for each pollinator is averaged over all of the pollinators to

The maximum pollination distance for an pollinator i is

ð Þi ≤ ϕð Þi m

obtain the average maximum pollination distance for the graph

Figure 2. Fathering triangles. Arrows indicate direction of gene flow.

0s @

D~ ð Þi ¼ max mð Þi ≤ μð Þi , f

distance, D, for the graph

104 Pollination in Plants

We examine the effect of varying parameters on the graph statistics: the number of fathers per mother, Φm, connectance, L, the average weighted diversity of fathers, E, the clustering coefficient of fathers, C, the average pollination distance, D, and the average maximum pollination distance, D~ . The model was simulated a total of 10 replicate runs for each unique combination of parameters. The results follow.

#### 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, Φm. In Figure 3, the number of different fathers per mother in a randomized placement of trees is distributed in a Gaussianlike distribution. This is an expected outcome since gene flow should be directly proportional to the distance traveled by a pollinator. It has been shown, see [29], that the distribution range resulting from a CRW would necessarily result in Gaussian-like behavior for a large number of pollination events.

Figure 3. Number of fathers per mother. Tree density ω ¼ 0:071552 trees per square unit. Maximum pollinator turning radius δmax ¼ 45∘ . Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

is attributed to the spatial heterogeneity of the research site. This influences the genetic structure

Figure 5. Tree coordinates (scaled) at the VCU Rice Center. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552

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

These two different density regions create the two peaks shown in Figure 4. In particular when κmax ¼ ∞, the first peak is approximately 3 fathers per mother on 29 mother trees, and the second peak is 32 fathers per mother on 12 mother trees. This bimodality is present whether an

δmax ¼ 180∘ . Clearly, knowing the locations of tree is critical in understanding the gene flow in

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 δmax increases the potential for mating to occur between trees with greater distance between them. The graph connectance is three to five times greater with small δmax

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

decreases. When δmax ¼ 0, the connectance of the graph is nearly four times greater if the

, the effect of maximum pollen carryover on the connectance of the graph

, or with pure dispersal, i.e.,

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and connectivity in C. florida populations [11] at the Rice Center.

road and a farm on the other two sides, which are not pictured here.

pollinator is restricted to moving in a straight line, i.e., δmax ¼ 0∘

a particular area (Figure 5).

than it is with large δmax.

δmax is close to 180∘

3.2. Connectance

trees per square units.

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

Figure 4. Number of fathers per mother. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Maximum pollinator turning radius δmax ¼ 45∘ . Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

Figure 5. Tree coordinates (scaled) at the VCU Rice Center. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square units.

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 κmax ¼ ∞, the first peak is approximately 3 fathers per mother on 29 mother trees, and the second peak is 32 fathers per mother on 12 mother trees. This bimodality is present whether an pollinator is restricted to moving in a straight line, i.e., δmax ¼ 0∘ , or with pure dispersal, i.e., δmax ¼ 180∘ . Clearly, knowing the locations of tree is critical in understanding the gene flow in a particular area (Figure 5).

#### 3.2. Connectance

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

Figure 3. Number of fathers per mother. Tree density ω ¼ 0:071552 trees per square unit. Maximum pollinator turning radius δmax ¼ 45∘ . Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

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

Figure 4. Number of fathers per mother. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Maximum pollinator turning radius δmax ¼ 45∘ . Pollination chance diminishing with larger carryover. Maximum

pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

106 Pollination in Plants

pollen carryover increases, which is also due to this increase in diversity.

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 δmax increases the potential for mating to occur between trees with greater distance between them. The graph connectance is three to five times greater with small δmax than it is with large δmax.

If an pollinator does not venture very far from its starting location, as would be the case when δmax is close to 180∘ , the effect of maximum pollen carryover on the connectance of the graph decreases. When δmax ¼ 0, the connectance of the graph is nearly four times greater if the

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 κmax the diversity of pollen distributed is greatly decreased and thus the diversity of pollination

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

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 15 and 90∘ and tends to even out at the extremes. As described earlier, with greater potential for a variety of fathers, the average number of fathers contributing pollen to mother trees increases as the maximum turning angle of pollinators decreases. This adds a greater genetic diversity to the tree population. Pollinators that travel in straighter paths not only distribute pollen greater distances, but also with greater diversity. The effects of density are more pronounced at smaller maximum turning angles, where we see

Figure 8. Connectance. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Pollination

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

a reduction of the average weighted diversity of fathers as the density increases.

. Connectance values for simulations run with the

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events is reduced as well.

graphs is greatest when δmax is close to 0∘

3.3. Average weighted diversity of fathers

Rice Center data are shown in Figure 8.

Figure 6. Connectance. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

Figure 7. Connectance. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f0:025; 0:050; 0:075; 0:100; ; 0:125; 0:150g (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly-placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

maximum pollen carryover is unlimited versus the case when pollen carryover is limited to only one flower. However, when δmax ¼ 180∘ , the connectance of the graph is only about twice if the maximum pollen carryover is unlimited.

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 κmax the diversity of pollen distributed is greatly decreased and thus the diversity of pollination events is reduced as well.

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 δmax is close to 0∘ . Connectance values for simulations run with the Rice Center data are shown in Figure 8.

#### 3.3. Average weighted diversity of fathers

maximum pollen carryover is unlimited versus the case when pollen carryover is limited to only one flower. However, when δmax ¼ 180∘ , the connectance of the graph is only about twice

Figure 7. Connectance. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f0:025; 0:050; 0:075; 0:100; ; 0:125; 0:150g (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly-placed trees). Pollination chance diminishing with larger car-

Figure 6. Connectance. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Pollination

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

if the maximum pollen carryover is unlimited.

ryover. Maximum pollen carryover κmax ¼ ∞.

108 Pollination in Plants

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 15 and 90∘ and tends to even out at the extremes. As described earlier, with greater potential for a variety of fathers, the average number of fathers contributing pollen to mother trees increases as the maximum turning angle of pollinators decreases. This adds a greater genetic diversity to the tree population. Pollinators that travel in straighter paths not only distribute pollen greater distances, but also with greater diversity. The effects of density are more pronounced at smaller maximum turning angles, where we see a reduction of the average weighted diversity of fathers as the density increases.

Figure 8. Connectance. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit. Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

The differences in average weighted diversity of fathers as pollen carryover increases are exactly as one would expect, see Figure 10. With higher κmax, the larger the diversity in pollen increases the average weighted diversity in fathers. This increase is small for large δmax due to the increase in the number of multiple visits by pollinators to the

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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

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

Figure 11. Average pollination distance. Field size 100 � 100 units. Tree density in trees per square area, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

3.4. Average and maximum average pollination distances

far. This decrease is more dramatic at lower densities.

same trees.

of seeds.

Figure 9. Average weighted diversity of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

Figure 10. Average weighted diversity of fathers. Field size 133:2794 � 47:2927 units. Tree density in trees per square unit, ω ¼ 0:071552 (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

The differences in average weighted diversity of fathers as pollen carryover increases are exactly as one would expect, see Figure 10. With higher κmax, the larger the diversity in pollen increases the average weighted diversity in fathers. This increase is small for large δmax due to the increase in the number of multiple visits by pollinators to the same trees.

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

Figure 9. Average weighted diversity of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination

Figure 10. Average weighted diversity of fathers. Field size 133:2794 � 47:2927 units. Tree density in trees per square unit, ω ¼ 0:071552 (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

110 Pollination in Plants

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.

Figure 11. Average pollination distance. Field size 100 � 100 units. Tree density in trees per square area, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

Figure 12. Average maximum pollination distance. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

and do not stay in a small neighborhood. Since the weak clustering coefficient of fathers is an average measure of clustering at a local level, it is natural for Cweak to be low if pollinators do

Figure 13. Weak clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen

At the other extreme, for large δmax, pollinators do not move around enough to increase the value of Cweak. When δmax is not at an extreme value, the displacement of the pollinators is high enough to visit many trees, but low enough so more of the trees that it visits are within a closer

Modeled data on the weak clustering coefficient of fathers, Cweak, from the Rice Center is shown in Figure 14. The Cweak values slowly increase as κmax increases over the entire interval from

the same general area. Thus, in locally dense patches of trees, clustering will naturally be higher. Unexpectedly, varying the tree density, ω, did not have a major effect on Cweak. It would seem that varying ω would have the same quantitative effect on Cweak as varying the maximum pollinator turning angle, δmax. We suspect that the reason for the relative consistency of values for Cweak is due to the having both low and high density regions. The values for Cweak, varying

Figures 16–18 are corresponding plots for strong clustering coefficient of fathers, Cstrong. In these plots we see similar results as with the weak clustering coefficient. Numerically the Cstrong results are a magnitude smaller than the Cweak results since they are a measure of a subset of possible combinations of Cweak. This is however a greater increase in these values

with increasing density, which allows for these fathering triangles to occur.

. As δmax approaches 180∘

, pollinators remain in

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not remain in a small neighborhood.

ω and δmax, are shown in Figure 15.

, mostly flattening out for κmax > 75∘

proximity to one another.

carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

0 to 180∘

Using Rice Center locations (not shown here) if δmax is close to 0∘ , the average pollination distance is greater than that of randomly placed trees. However, if δmax is close to 180∘ , the average pollination distance using Rice Center data is less than that of randomly placed trees. This is again due to the combination of the low and high density distribution of trees at the Rice Center. At low δmax pollinators interact with both densities of trees, whereas at high δmax the interaction between these groups is greatly diminished.

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 κmax increases the average pollination distance. If the pollen carryover increases that allows pollen grains to travel further since pollen can be deposited after visiting several intermediate trees.

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, Cweak, see Figure 13, that occurs between 60 and 90∘ for all maximum pollen carryover values, κmax. This is explained by examining the extreme values of δmax. For small δmax, pollinators travel across the landscape

Figure 13. Weak clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

and do not stay in a small neighborhood. Since the weak clustering coefficient of fathers is an average measure of clustering at a local level, it is natural for Cweak to be low if pollinators do not remain in a small neighborhood.

Using Rice Center locations (not shown here) if δmax is close to 0∘ , the average pollination distance is greater than that of randomly placed trees. However, if δmax is close to 180∘

Figure 12. Average maximum pollination distance. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination

average pollination distance using Rice Center data is less than that of randomly placed trees. This is again due to the combination of the low and high density distribution of trees at the Rice Center. At low δmax pollinators interact with both densities of trees, whereas at high δmax

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 κmax increases the average pollination distance. If the pollen carryover increases that allows pollen grains to

The results are similar with the Rice Center model. The results at the Rice Center are slightly

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, Cweak, see Figure 13, that occurs between 60 and 90∘ for all maximum pollen carryover values, κmax. This is explained by examining the extreme values of δmax. For small δmax, pollinators travel across the landscape

travel further since pollen can be deposited after visiting several intermediate trees.

higher in distances due to the potential longer distances at the edges of the region.

the interaction between these groups is greatly diminished.

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

3.5. Clustering coefficients of fathers

112 Pollination in Plants

, the

At the other extreme, for large δmax, pollinators do not move around enough to increase the value of Cweak. When δmax is not at an extreme value, the displacement of the pollinators is high enough to visit many trees, but low enough so more of the trees that it visits are within a closer proximity to one another.

Modeled data on the weak clustering coefficient of fathers, Cweak, from the Rice Center is shown in Figure 14. The Cweak values slowly increase as κmax increases over the entire interval from 0 to 180∘ , mostly flattening out for κmax > 75∘ . As δmax approaches 180∘ , pollinators remain in the same general area. Thus, in locally dense patches of trees, clustering will naturally be higher.

Unexpectedly, varying the tree density, ω, did not have a major effect on Cweak. It would seem that varying ω would have the same quantitative effect on Cweak as varying the maximum pollinator turning angle, δmax. We suspect that the reason for the relative consistency of values for Cweak is due to the having both low and high density regions. The values for Cweak, varying ω and δmax, are shown in Figure 15.

Figures 16–18 are corresponding plots for strong clustering coefficient of fathers, Cstrong. In these plots we see similar results as with the weak clustering coefficient. Numerically the Cstrong results are a magnitude smaller than the Cweak results since they are a measure of a subset of possible combinations of Cweak. This is however a greater increase in these values with increasing density, which allows for these fathering triangles to occur.

Figure 14. Weak clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

Figure 16. Strong clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen

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Figure 17. Strong clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover

carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

κmax ¼ f g 1; 3; 5; 7; ∞ .

Figure 15. Weak clustering coefficient of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

Figure 16. Strong clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

Figure 14. Weak clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover

Figure 15. Weak clustering coefficient of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

κmax ¼ f g 1; 3; 5; 7; ∞ .

114 Pollination in Plants

Figure 17. Strong clustering coefficient of fathers. Field size 133:2794 � 47:2927 units. Tree density ω ¼ 0:071552 trees per square unit (τ ¼ 451 Rice Center trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ f g 1; 3; 5; 7; ∞ .

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

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 δmax is close to 0∘ and higher when δmax is close to 180∘ , and the clustering coefficient of fathers exhibits both quantitative and qualitative differences. All of these differences highlight the need for specificity in describing the tree locations within a specific ecosystem in order to truly understand how pollen is distributed within that ecosystem. The differences in these graph

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

\* and Rodney J. Dyer2

1 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University,

2 Center for Environmental Studies, Virginia Commonwealth University, Richmond, Virginia,

[1] Slatkin M. Gene flow in natural populations. Annual Review of Ecology and Systematics.

[2] Morris WF. Predicting the consequence of plant spacing and biased movement for pollen

indicators is due to the non-uniform distribution of trees at the Rice Center.

, which could help illuminate some of the

Comparison of Pollination Graphs

117

http://dx.doi.org/10.5772/intechopen.74553

maximized when δmax is between 60 and 90∘

manuscript is VCU Rice Center Manuscript #69.

, David M. Chan1

\*Address all correspondence to<EMAIL_ADDRESS>
dispersal by honey bees. Ecology. 1993;74:493-500

biological processes at work in the system.

Funding

Author details

Richmond, Virginia, USA

1985;16:393-430

James H. Lee1

United States

References

Figure 18. Strong clustering coefficient of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

## 4. Conclusions

From the agent-based, correlated random walk model presented, we observe the effects of varying the maximum pollen carryover, κmax, the maximum pollinator turning area, δmax, and the density of trees, ω, on the distribution of pollen within a population of Cornus florida.

When κmax increases we see that the mean number of fathers per mother, the connectance, the average weighted diversity of fathers, the average and maximum average pollination distances, and the clustering coefficients all increase. The percentage increase varied between the measured, though the largest effect was seen in the connectance of the pollination graph. These increases are due to the increased capability of the pollinators to carry pollen farther from their source.

Changing the pollinator movement by increasing the maximum turning angle, δmax, affected each of the measures as well. With most of the measures decreasing with increasing δmax. These include the connectance, the average weighted diversity of fathers, and the average and maximum average pollination distances. On the other hand, the weak and strong clustering coefficients had maximal values between angles of 45 and 90∘ .

When δmax ¼ 180∘ , pollen is distributed in a purely random walk, and is more representative of pollen dispersal by wind. When δmax ¼ 0∘ , pollinators travel in a straight line, only changing direction when bouncing off of the boundary. This leads to a greater spatial displacement for each pollinator and thus a greater distance that pollen travels, resulting in greater genetic diversity in the C. florida population. While neither of these extremes may be biologically relevant in C. florida populations, we note that the clustering coefficient of fathers, C, is maximized when δmax is between 60 and 90∘ , which could help illuminate some of the biological processes at work in the system.

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 δmax is close to 0∘ and higher when δmax is close to 180∘ , and the clustering coefficient of fathers exhibits both quantitative and qualitative differences. All of these differences highlight the need for specificity in describing the tree locations within a specific ecosystem in order to truly understand how pollen is distributed within that ecosystem. The differences in these graph indicators is due to the non-uniform distribution of trees at the Rice Center.

## Funding

4. Conclusions

116 Pollination in Plants

source.

When δmax ¼ 180∘

From the agent-based, correlated random walk model presented, we observe the effects of varying the maximum pollen carryover, κmax, the maximum pollinator turning area, δmax, and the density of trees, ω, on the distribution of pollen within a population of Cornus florida.

Figure 18. Strong clustering coefficient of fathers. Field size 100 � 100 units. Tree density in trees per square unit, ω ¼ f g 0:025; 0:050; 0:075; 0:100; 0:125; 0:150 (τ ¼ f g 250; 500; 750; 1000; 1250; 1500 randomly placed trees). Pollination

When κmax increases we see that the mean number of fathers per mother, the connectance, the average weighted diversity of fathers, the average and maximum average pollination distances, and the clustering coefficients all increase. The percentage increase varied between the measured, though the largest effect was seen in the connectance of the pollination graph. These increases are due to the increased capability of the pollinators to carry pollen farther from their

Changing the pollinator movement by increasing the maximum turning angle, δmax, affected each of the measures as well. With most of the measures decreasing with increasing δmax. These include the connectance, the average weighted diversity of fathers, and the average and maximum average pollination distances. On the other hand, the weak and strong clustering

ing direction when bouncing off of the boundary. This leads to a greater spatial displacement for each pollinator and thus a greater distance that pollen travels, resulting in greater genetic

.

, pollinators travel in a straight line, only chang-

, pollen is distributed in a purely random walk, and is more representative

coefficients had maximal values between angles of 45 and 90∘

chance diminishing with larger carryover. Maximum pollen carryover κmax ¼ ∞.

of pollen dispersal by wind. When δmax ¼ 0∘

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.

## Author details

James H. Lee1 , David M. Chan1 \* and Rodney J. Dyer2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, Virginia, USA

2 Center for Environmental Studies, Virginia Commonwealth University, Richmond, Virginia, United States

## References

[3] Waples R, Gaggiotti O. What is a population? An empirical evaluation of some genetic methods for identifying the number of gene pools and their degree of connectivity. Molecular Ecology;15:1419-1438

[19] Johnson SD, Nilsson LA. Pollen carryover, geitonogamy, and the evolution of deceptive

Comparison of Pollination Graphs

119

http://dx.doi.org/10.5772/intechopen.74553

[20] deJong TJ, Klinkhamer PGL, Van Staalduinen MJ. The consequences of pollination biology for selection of mass or extended blooming. Functional Ecology. 1992;6:606-615 [21] Foster E. An agent based gene flow model [VCU Thesis and Dissertations]. 2009. Paper

[22] Ramos-Jiliberto R, Albornoz AA, Valdovinos FS, Smith-Ramiriz C, Arim M, Armesto JJ, Marquet PA. A network analysis of plant-pollinator interaction in temperate rain forests

[23] Valdovinos FS, Ramos-Jiliberto R, Flores JD, Espinoza C, Lopez G. Structure and dynamics of pollination networks: The role of alien plants. Oikos. 2009;118:1190-1200

[25] Traveset A, Chamorro S, Olesen JM, Heleno R. Space, time and aliens: Charting the dynamic structure of Galapagos pollination networks. AoB PLANTS 7: plv068. DOI: 10.1093/aobpla/

[26] Lundgren R, Olesen JM. The dense and highly connected world of Greenland's plants and their pollinators. Arctic, Antarctic, and Alpine Research. 2005;37(4):514-520

[27] Borsch J, Gonzalez AMM, Rodrigo A, Navarro D. Plant-pollinator networks: Adding the

[28] Ferrero V, Castro S, Costa J, Acuna P, Navarro L, Loureiro J. Effect of invader removal: Pollinators stay but some native plants miss their new friend. Biological Invasions. 2013;

[29] Bidaux R, Boccara N. Correlated random walks with a finite memory range. International Journal of Modern Physics C: Computational Physics and Physical Computation. 2000;11:

[24] Jørgensen SE. Ecosystem Ecology. Amsterdam, The Netherlands: Elsevier; 2009

pollination systems in orchids. Ecology. 1999;80:2607-2619

pollinator's perspective. Ecology Letters. 2009;12:409-419

of Chiloe, Chile. Oecologia. 2009;160:697-706

No. 1726

plv068

15:2347-2358

921-947

[19] Johnson SD, Nilsson LA. Pollen carryover, geitonogamy, and the evolution of deceptive pollination systems in orchids. Ecology. 1999;80:2607-2619

[3] Waples R, Gaggiotti O. What is a population? An empirical evaluation of some genetic methods for identifying the number of gene pools and their degree of connectivity.

[4] R. Dyer landscape and plant population genetics. In: Balkenhol N, Cushman S, Storfer A, editors. Landscape Genetics: Concepts,Methods, and Applications. JohnWiley and Sons; 2015

[6] Dyer R, RJ VS. Pollen pool heterogeneity in shortleaf pine, Pinus echinata mill. Molecular

[7] Manel S, Schwartz MK, Luikart G, Taberlet P. Landscape genetics: Combining landscape ecology and population genetics. Trends in Ecology & Evolution. 2003;18:189-197

[8] Dyer R. Is there such a thing as landscape genetics? Molecular Ecology. 2015;24:3518-3528 [9] Brunet J, Holmquist KGA. The influence of distinct pollinators on female and male reproductive success in the Rocky Mountain columbine. Molecular Ecology. 2009;18:3745-3758

[10] Foster E, Chan DM, Dyer RJ. Model comparison for abiotic versus biotic pollen dispersal.

[11] Dyer RJ, Chan DM, Gardiakos VA, Meadows CA. Pollination networks: Quantifying pollen pool covariance networks and the influence of intervening landscape on genetic connectivity in the North American understory tree, Cornus florida L. Landscape Ecology.

[12] DiLeo MF, Siu JC, Rhodes MK, López-Villalobos A, Redwine A, Ksiazek K, Dyer RJ. RJ. The gravity of pollination: Integrating at-site features into spatial analyses of contempo-

[14] Bartumeus F, Catalan J, Viswanathan GM, Raposo EP, da Luz MGE. The influence of turning angles on the success of non-oriented animal searches. Journal of Theoretical Biol-

[15] Bovet P, Benhamou S. Spatial analysis of animal's movements using a correlated random

[16] Byers JA. Correlated random walk equations of animal dispersal resolved by simulation.

[17] Codling EA, Plank MJ, Benhamou S. Random walk models in biology. Journal of The

[18] Prasad BRG, Borges RM. Searching on patch networks using correlated random walks: Space usage and optimal foraging predictions using Markov chain models. Journal of

[13] R. J. Dyer, Coordinates of C. florida at the VCU Rice Center, (2014) . Unpublished

Nonlinear Dynamics, Psychology, and Life Sciences. (20):471-484

rary pollen movement. Molecular Ecology. 2014;23:3793-3982

walk model. Journal of Theoretical Biology. 1988;131:419-433

[5] Levin D, Kerster H. Gene flow in seed plants. Evolutionary Biology;7:139-220

Molecular Ecology;15:1419-1438

118 Pollination in Plants

Ecology. 2001;10:859-866

2012;27:239-251

ogy. 2008;252:43-55

Ecology. 2008;82:1620-1690

Royal Society Interface. 2008;5:813-834

Theoretical Biology. 2006;240:241-249

## *Edited by Phatlane William Mokwala*

Plants are the basic source of food for both humans and animals. Most of the food is made of fruits and seeds. For these to be formed, pollination must first take place. This process is the transfer of pollen grains from the anther, which is the male structure of the flower, to the sigma on the female structure of the flower. The transfer process requires agents to be carried out. The agents can be either biotic or abiotic. Nature perfected this arrangement between the pollination agents and the plants. As ecosystems and agricultural systems are changing, this balanced arrangement becomes disturbed. This makes it necessary that pollination systems be studied so that necessary measures can be undertaken to ensure productivity.

The chapters of this book present results in research undertaken to improve productivity in crops such as *Actinidia chinensis* (the kiwifruit), *Theobroma cacao*  (cocoa), and *Manicaria saccifera* (a tropical forest palm). Some results are presented on tests to check the viability of pollen grains and the delivery of sperm cells through pollen tubes to the embryo sac. These results can serve as guidelines to any person seeking to improve pollination and productivity or to check the efficiency on pollination in ecosystems or agricultural production systems.

Published in London, UK © 2018 IntechOpen © paulbein / iStock

Pollination in Plants

Pollination in Plants

*Edited by Phatlane William Mokwala*