1 Supplementary Methods 2 3 Historical Background 4 Beyond The
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1 2 Supplementary Methods 3 4 Historical Background 5 Beyond the important role of the phytochemical landscape driving insect herbivore diet breadth, 6 herbivorous insects are subject to a list of general factors that affect the diet breadth of foragers. 7 Originally broadly conceptualized for hunting animals by MacArthur & Pianka7, this list was adapted for 8 application to a pollinator diet by Wasser et al8. These factors are listed in Table S1 with common factors 9 between the two treatments. 10 11 Table S1: Factors favoring specialization in foraging as detailed in MacArthur & Pianka7 and their 12 corresponding construction in Waser et al’s8 study on pollinator specialization. The third column 13 describes the common factor in each treatment. The factor noted in green highlights the work which 14 helped inspire our study. 15 16 17 Baseline model 18 As with recent studies of ecological networks10 31, the makeup of the overall network model used 19 here has two major fundamental components: the structure of the pollination networks themselves and the 20 dynamics occurring on the networks. 21 The structure of a network describes which links (representing species interactions) between 22 plants (푝) and animal pollinators (푎) are present or absent, regardless of the strength of the link. Typical 23 ecological network studies have a collection of unique topological connections across different networks, 24 meaning only certain links of the possible links between plants and pollinators can potentially be realized 25 (Fig S1a). However, the networks we used in the models/simulations here were fully connected, meaning 26 all connections are possible (Fig S1b). The baseline of these networks in our study are sourced from an 27 empirical pollination network32. Using a real-world pollination network as a basis ensured a plausible 28 ratio of flowering plant populations to pollinator populations, which in this case was 58 plant species to 29 100 pollinator species. The network was then fully connected (see Figure S1) such that each pollinator 30 had potential access to all plant species and vis-versa. Fully connecting the plant-pollinator network was 31 done in order to give each pollinator population the maximum within-model range of dietary options. 32 Doing so allows us to test the ability of the phenological mechanism to drive plasticity in pollinator 33 specialization without a priori constraints on pollinator behavior and network structure. The size of this 34 basic network framework was then modified to test effects of species richness, resulting in 3 network size 35 classes (푁푝=plant richness, 푁푎=animal pollinator richness): 푁푝 = 58 & 푁푎 = 100, 푁푝 = 48 & 푁푎 = 71, 36 and 푁푝 = 30 plants & 푁푎 = 50. While only these 3 network frameworks were used, the exact topology of 37 each changed with thousands of permutations of phenological parameters (see below). 38 39 40 Figure S1: Underlying network structure. Example diagram describing the creation of the baseline 41 network structure. Pictures of bees represent populations of pollinators and pictures of flowering plants 42 represent populations of flowering plants. Blue links between them represent a potential pollination 43 interaction between plants and pollinators. a) Example of a starting empirical network. b) Example of 44 fully connecting the empirical network to use in simulations. 45 46 The dynamics occurring on the networks refer to the internal population demographics (birth and 47 death) and species interactions taking place amongst the species in the network. In the case of our model, 48 species interactions refer to competition for resources, consumption of resources, and pollination events. 49 Dynamics were simulated based on the work of Valdovinos et al9 which mechanistically modeled 50 pollination as a consumer-resource interaction by separately accounting for vegetative abundance and 51 floral rewards consumed by pollinators. The model tracks the adaptive dynamics of each plant species’ 52 population dynamics (푝푖), each animal pollinator species population dynamics (푎푗), each plant species’ 53 pool of floral rewards (푅푖), and the adaptive dynamics of the per-capita foraging effort preferences of 54 each pollinator species for each plant species (훼푖푗). Model parameters are described in Table S2. 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 Table S2: Baseline model state variables and parameters. In Mean Value column, * indicate initial 86 conditions, kaj is the number of interactions of animal j Definition Symbol Dimension Mean Value -1 Density of plant population i 푝푖(푡) individuals area 0.7* -1 Density of animal population j 푎푗(푡) individuals area 0.7* -1 Density of floral resources of plant 푅푖(푡) mass area 0.7* population i ∗ Foraging Effort of pollinator j on plant i 훼푖푗(푡) none 1 푘훼푗 -1 -1 Visitation Efficiency 휏푖푗 visits area time individuals 1 individuals-1 -1 Expected number of seeds produced per 푒푖푗 individuals visits 0.8 pollination event 푃 -1 Per capita mortality rate of plants 휇푖 time 0.002 -1 Conversion efficiency of floral resources 푐푖푗 individuals mass 0.2 to pollinator births 퐴 -1 Per capita mortality rate for pollinators 휇푗 time 0.003 -1 Pollinator extraction efficiency of 푏푖푗 individuals visits 0.4 resource 푅푖 in each visit Max fraction of total seeds that recruit to 푔푖 none 0.4 plants -1 Intra-specific competition coefficient for 푢푖 area individuals 1.2 plants -1 Inter-specific competition coefficient for 푤푖 area individuals 0.002 plants -1 -1 Production rate of floral resources 훽푖 mass individuals time 0.2 -1 Self-limitation parameter for resource 휙푖 time 0.04 production Adaptation rate of pollinator foraging 퐺푗 none 2 effort 87 88 Changes in population density of plant species (푝푖) are calculated through: 푑푝 89 푖 = 훾 ∑ 푒 휎 푉 − 휇푝푖푝 Eq (1) 푑푡 푖 푗∈퐴푗 푖푗 푖푗 푖푗 푖 푖 푝 90 The second term in the equation describes background mortality, where 휇푖 is the constant 91 density-independent per-capita mortality rate of plant 푖.The first term describes plant population growth 92 where 훾푖 describes the realized fraction of seeds that successfully recruit to adults: 93 훾푖 = 푔푖(1 − ∑푙≠푖∈푃 푢푙푝푙 − 푤푖푝푖) Eq (2) 94 where 푔푖 is the maximum fraction of seeds that can potentially recruit to fecund adulthood. The 95 recruitment is subject to both interspecific (푢푙) and intraspecific (푤푖) competition with 푢푙 < 푤푖. The 96 parameter 푒푖 in Equation (1) is the constant max seed set induced by a pollination event between plant 97 and pollinator. The quality of each pollination event is determined by the fraction of pollen from 98 conspecific plants on a pollinator compared to other plant species pollen. This fraction is proportional to 99 the visits each animal pollinator (푎푗) makes on each plant species (푝푖). We label this term, 휎푖푗 and define 100 it as follows: 푉 101 휎 = 푖푗 Eq (3) 푖푗 ∑ 푉 푘∈푃푗 푘푗 102 where 푉푖푗 is the frequency of visits by animal species 푗 to plant species 푖 and it defined by: 103 푉푖푗 = 훼푖푗휏푖푗푇푎푗(푡)푎푗푝푖 Eq (4) 104 Visits from pollinator 푗 to plant species 푖 are zero (푉푖푗 = 0) if the two do not interact. Pollinator 105 푗’s visitation efficiency on plant 푖 is determined by the parameter 휏푖푗 and is fixed at 1 for this study so as 106 not to bias any pollination interaction over others and affect visitation preferences a priori. The function ( ) 107 푇푎푗 푡 is the phenological determinant of activity of pollinator 푎푗, in this case controlling the flight period 108 of 푎푗. See Phenology section below for more. The dimensionless function 0 ≤ 훼푖푗 ≤ 1 is the foraging 109 preference of pollinator 푗 on plant 푖 and changes over time as defined by: 푑훼 110 푖푗 = 푇 (푡)퐺 훼 (푐 휏 푏 푅 − ∑ 훼 푐 휏 푏 푅 ) Eq (5) 푑푡 푎푗 푗 푖푗 푖푗 푖푗 푖푗 푖 푘∈푃푗 푘푗 푘푗 푘푗 푘푗 푘 111 where 퐺푗 is the basal adaptation rate of foraging preference. Higher or lower rates of 퐺푗 produce faster or 112 slower rates of adaptation based on changes seen inside the parentheses of Eq (5). 푐푖푗 represents the 113 constant per-capita conversion efficiency of pollinator 푗 converting plant 푖’s floral resources into 푗’s 114 births. 푏푖푗 is the constant efficiency of pollinator 푗 extracting plant 푖’s floral resources (푅푖). Pollinator 푗 115 allocates more foraging effort to plant species 푖 whenever such reallocation increases 푗’s food intake. 116 Such reallocation causes a commensurate reduce in foraging effort from other plant species. For every ∑ ( ) 117 animal pollinator species 푗, 훼푖푗 = 1 for all plant species. Finally, the function 푇푎푗 푡 appears here to 118 limit the adaptation of foraging preference only to periods when pollinators are actively flying. 119 Each plant 푖’s floral resources, 푅푖, changes over time as defined by: 푑푅푖 푅푖 120 = 푇푝 (푡)훽푖푝푖 − 휙푖푅푖 − ∑푗∈퐴 푉푖푗푏푖푗 ( ) Eq (6) 푑푡 푖 푝푖 121 where 훽푖 is plant 푖’s per-capita resource production rate and 휙푖 is a constant self-limitation parameter. 122 Rewards of plant 푖 are removed with an efficiency 푏푖푗 by pollinator 푗 in proportion to the amount of visits, ( ) 123 푉푖푗. The function 푇푝푖 푡 controls the phenological expression of resource production in each flowering 124 plant 푖. Further details are provided below in Phenology section. The population dynamics of the animal 125 pollinators are then defined by: 푑푎푗 푅푖 퐴 126 = ∑푖∈푃 푐푖푗푉푖푗푏푖푗 ( ) − 휇푗 푎푗 Eq (7) 푑푡 푝푖 127 where 푐푖푗, 푉푖푗, and 푏푖푗 are as defined above. Pollinator population growth is driven by the sum of 퐴 128 resources gathered from pollination visits while death occurs at a constant rate 휇푗 .