Appendix a Nondimensionalization and Model Simplification

1 Appendix A Nondimensionalization and Model Simplification2 Using the nondimensionalization scheme in Table A1, equations (1-7) in the main text become3 (A1)4 (A2)5 (A3)6 (A4)7 (A5)8 (A6)9 (A7)10 Table A1: Nondimensionalization SchemeState Variables Parameters11 Consider a plant-pollinator system in which the plant population is relatively large, and there 12 exists both a high visitation rate of pollinators to plants and a relatively low rate of pollen loss by1 13 pollinators. This situation appears an apt description of bee-pollinated annual plants (Parrish and 14 Bazzaz 1979, Potts et al. 2003). If these conditions hold, and if pollinator demographic processes15 occur slowly (e.g., on a scale of days to weeks) relative to fast pollination events (e.g., on a scale 16 of minutes to hours), then . In this scenario, equation (A3) can be approximated as17 (A8)18 Solving for gives19 (A9.a)20 (A9.b)21 Substituting equation (A9) into equations (A1, A4-A7), yields22 (A10)23 (A11)24 (A12)25 (A13)26 (A14)27 where we have simplified using the relationships in Table A2. Notice that the new emergence 28 functions are not normalized; specifically29 (A15)30 312 32 Table A2: Simplifying RelationshipsSubstitutions33 References Cited in Appendix A34 Parrish, J. A. D. and F. A. Bazzaz.1979. Difference in pollination niche relationships in early and35 late successional plant communities. Ecology 60:597-610.36 Potts S. G., Vulliamy B, Dafni A., Ne'eman G.and Willmer P. 2003. Linking bees and flowers: 37 how do floral communities structure pollinator communities? Ecology: 84:2628-2642.383 39 Appendix B: A Dirac Delta Function Approximation for Plant and Pollinator Phenology40 Beginning from equations (A10 – A14) in Appendix A we have41 (B1)42 (B2)43 (B3)44 (B4)45 (B5)46 We now consider a simplified functional form for . Specifically, let us assume that47 (B6)48 where is the Dirac delta function and the scaling comes from the fact that (see equation A15 in 49 Appendix A) Substituting into equations (B1) and (B2) gives50 (B7)51 (B8)52 Because equation (B7) is independent of the other state variables it can be solved as follows53 (B9)54 To arrive at a similar analytical expression for requires some level of approximation. First, let 55 us assume that within-season plant death is negligible, thus . Further, let us assume that flower 56 death is fast compared to pollination (i.e., the large majority of flowers die without being 57 pollinated, Ashman et al. 2004). This assumption is suitable for some plants, but not for others, 58 such as orchids that have notoriously long-lived flowers and which can actually increase their 59 floral lifetime if not pollinated. Still, because this is the limit that is of particular concern, it will 60 be the starting point for our approximation. In this case and equation (B8) can be approximated61 (B10)62 Solving equation (B10) gives4 63 (B11)64 which essentially reflects an exponential decay of flowering plants from time . Since we have 65 assumed negligible plant death, this decay is solely a result of the flowers themselves expiring 66 and becoming dead-heads. Applying a similar set of assumptions to equations (B3-B5) we find67 (B12)68 (B13)69 (B14)70 In equation (B12), we cannot assume that , as was done to arrive at equation (B10). This is 71 because, under the assumption of slow pollination, . As a result, the number of pollinated 72 flowers senescing per unit time step is comparable to the number of flowers pollinated per unit 73 time step. In contrast, the number of unpollinated flowers closing per unit time step is 74 significantly greater than the number of flowers pollinated per unit time step, hence the 75 approximation in equation (B10).76 We are primarily interested in . To find this quantity, however, requires additional 77 approximation. Since we have assumed that plant death is negligible, all pollinated flowers will 78 ultimately produce seeds if given sufficient time. Assuming that the season is long compared to 79 the pollination period, we can reasonably ignore transitions between and and assume that is an 80 accurate representation of seed-set. As a result, we can focus on equation (B12), and ignore the 81 flow from pollinated flowers to pollinated dead-heads and on to seeds. Furthermore, since we 82 have already assumed that pollination rate is low, we know that . As a result, equation (B12) can83 be re-written84 (B15)85 In order to solve equation (B15), we write the following approximate piecewise function5 86 (B16)87 Equation (B16) will be a good approximation to equation (B15) when and , and will deviate 88 most when . From equation (B11), we find that occurs at 89 (B17)90 Substituting equations (B17), (B9) and (B11) into equation (B16) gives91 (B18)92 when and93 (B19)94 when . In equations (B18) and (B19),, , . Equation (B18) and (B19) can be solved, giving95 (B20)96 and97 (B21)98 respectively. In equation (B20), . Taking we find the limiting seed-set as 99 (B22)100 Re-writing equation (B22) in terms of the separation between pollinator emergence, , and plant 101 flowering,, we find102 (B23)103 where , , , , and . To test our approximation, we compare predictions from equation (B23) to 104 those from the full model (equations (A10-A14) in Appendix A). Specifically, we set and in 105 equation (B23) to the time of the peak in and from the full model (see equation (A10) in 106 Appendix A). We then compare the predicted seed-set at the end of the season in the full 107 model, , to the suitably rescaled limiting seed-set predicted by the 6 108109 Figure B1 Limit on the number of flowers pollinated as predicted by the approximate 110 solution (red) compared to the number of flowers pollinated at the end of the season as predicted 111 by the full model (black). Insets show an example of the emergence (black) and flowering 112 (green) of the pollinators and plants respectively. Parameters are: , ,,, ,,,, and (a) and (b) .113 approximate model . This is shown in Figure B1 for two gamma functions that differ in the 114 breadth of their respective peaks.115 Clearly equation (B23) reasonably approximates the full solution, at least for systems with a 116 narrow range of emergence/flowering times, low pollination rates () and negligible plant death ().117 As expected, the approximation is noticeably poorer when emergence/flowering occurs over a 118 broader period of time.119 Pollinated Flowers as a Function of the Relative Phenology of Plants and Pollinators120 From Figure B1, it is clear that the separation between pollinator emergence and plant flowering 121 has a significant consequence on flower pollination. To understand the full implications of 122 phenology changes in the context of our approximate model, we take the derivate of with 123 respect to . This gives124 (B24)125 Proof: Case 1 () 126 Rearranging the inequality gives127 (B25)128 Since the right hand side of (B25) lies in the interval for all non-negative values of . Therefore, 129 if we can prove that7 130 (B26)131 then we have proven that the inequality holds over the range specified in equation (B24). 132 Substituting our expressions for ,, and into equation (B26) and rearranging we get133 (B27)134 Simplifying gives135 (B28)136 which will be true for all .137 Proof: Case 2 () 138 Since are positive, this inequality will hold provided 139 (B29)140 Substituting our expression for into equation (B29) and rearranging, we find141 (B30)142 Taking the natural logarithm of both sides and simplifying gives143 (B31)144 Substituting our expression for from Table A2 and simplifying gives145 (B32)146 Equation (B30) holds by virtue of the restriction , which reduces to an identical expression when 147 (i.e. ). The other two inequalities in equation (B24) are obviously true for positive parameters.148 Equation (B24) suggests that is a decreasing function of when and an increasing function of 149 when . In other words, decreases with increasing . This relatively straightforward result implies150 that the limiting number of pollinated flowers decreases as a function of the separation between 151 the timing of insect emergence and the timing of plant flowering. (Notice that this conclusion 152 only holds for the approximate model. In the full model, pollinated flowers appear to peak at 153 slightly negative values of . Even in the full model, however, larger negative values for do 8 154 result in a reduction in pollinated flowers, in accord with predictions from the approximate 155 model). 156 Appendix B References Cited:157 Ashman, T. L., Knight, T. M., Steets, J. A., Amarasekare, P., Burd, M., Campbell, D. R., ... & 158 Wilson, W. G. (2004). Pollen limitation of plant reproduction: ecological and 159 evolutionary causes and consequences. Ecology, 85: 2408-2421.160161162163164165166167168169170171 Appendix C. Supplemental Figures9 172173 Figure C1 Scaled population growth rates as a function of initial population size, , assuming174 (a) broad plant and pollinator emergence/emigration functions ( 2, , 10.5) or (b) narrow plant and175 pollinator emergence/emigration functions ( 5, , 28.2). Phenology shifts of the plants relative to 176 the pollinator are denoted as follows:  = 0 (solid), (long dashes), - (short dashes). Systems 177 with a large number of pollinators and a high plant-pollinator contact rate ( are shown in black. 178 Systems with a small number of pollinators and a low plant-pollinator contact rate ( are shown in179 grey. As expected, the Allee effect is negligible for systems with high levels of pollinator 180 visitation, but increases as pollinators or pollinator contact rates become limiting. Parameter 181 values for all plotted scenarios are: .182183184185186 Figure C2 Full system dynamics at three different phenology values ( -15 (blue), 0 (purple), 187 15 (red)) for the broad flower opening / pollinator emergence scenario (, ()) and a system with 188 long-lived pollinators and long-lived flowers (see (a), which is a reproduction of Figure 1.b from 189 the main text). In (a) colored dots are used to show the locations of the three different phenology190 scenarios. The remainder of the panels show (b) pollinator abundance, (c) abundance of 191 unpollinated flowers (inset shows an enlargement for -15,0), (d) abundance of pollinated flowers,192 (e) abundance of dead-heads, and (f) seed-set. Parameter values are:, , , and .19310 194195196 Figure C3 Full system dynamics at three different phenology values ( -15 (blue), 0 (purple), 197 15 (red)) for the broad flower opening / pollinator emergence scenario (, ()) and a system with 198 long-lived pollinators and short-lived flowers (see (a), which is a reproduction of Figure 1.c from199 the main text). In (a) colored dots are used to show the locations of the three different phenology200 scenarios. The remainder of the panels show (b) pollinator abundance, (c) abundance of 201 unpollinated flowers (inset shows an enlargement for -15,0), (d) abundance of pollinated 202 flowers, (e) abundance of dead-heads, and (f) seed-set. Parameter values are:, , , and .203204205206207208 Figure C4 Full system dynamics at three different phenology values ( -15 (blue), 0 (purple), 209 15 (red)) for the broad flower opening / pollinator emergence scenario (, ()) and a system with 210 short-lived pollinators and short-lived flowers (see (a), which is a reproduction of Figure 1.d 211 from the main text). In (a) colored dots are used to show the locations of the three different 212 phenology scenarios. The remainder of the panels show (b) pollinator abundance, (c) abundance 213 of unpollinated flowers (inset shows an enlargement for ), (d) abundance of pollinated flowers, 214 (e) abundance of dead-heads, and (f) seed-set. Parameter values are:, , , and .11 215216 Figure C5 Contours for the population growth / decay threshold () as a function of plant 217 death rate and pollination rate for narrow or broad flower opening windows and several values 218 of . Panels are: (a) 2, 10.5; (b) , 10.5; (c) 5, 28.2; and (d) 5, 28.2. Parameters are: 60, 1, 15,219 0.075, 0.05, 1000, 150, 0.05, and 1.5. 220221222223 Appendix D. Pollinator Demographics Dependent on Flower Abundance224 Throughout the main text, we assume that seed-set, which determines plant population growth 225 rates, depends on the level of pollination received throughout flower life-time. We do not, 226 however, assume that there is any dependence of the pollinator population on flower availability.227 Here we explore one scenario wherein pollinator abundance is a function of flower abundance. 228 Specifically, we assume that pollinator death and/or departure is reduced when there is a large 229 population of flowering plants. To do this, we modify Eqs. (1-3) from the main text as follows:230 (D1) 231 (D2)232 (D3)233234 We leave the remainder of the system, i.e. Eqs. (4-7), unchanged. Eqs. (D1-D3) can be 235 nondimensionalized as follows:236 (D4)237 (D5)12 238 (D6)239 where and all other terms are as defined in Table A1. For , we can again assume that both and 240 are approximately constant, allowing us to replace Eqs. (D4-D6) with241 (D10)242 (D11)243 (D12)244 where is defined in Table A2. The approximate dynamical system is then described by Eq. 245 (D12) along with Eqs. (A11-A14). In Figure D1 we show how dependence of the pollinator 246 population on flower availability can alter the predictions in Figure 1.c of the main text.247 In Figure D1, curves, shown in black, correspond to scenarios with no dependence of the248 pollinator population on flower availability (i.e. our assumption from the main text). When the 249 presence of open flowers can reduce pollinator loss (grey curves, , e.g. flowers prevent pollinator250 starvation or the decision to forage elsewhere), plant growth rates fall off more slowly with early 251 pollinator arrival. This effect is strong for scenarios with pollinators arriving first, but small to 252 non-existent for scenarios with plants blooming first. Similarly, the effect is stronger when 253 emergence windows are broad as compared to narrow.254 These results make intuitive sense. When plants bloom first, phenological mismatch 255 causes a reduction in plant seed-set because most flowers close before pollinators emerge. Even 256 if the presence of flowers can extend the lifespan (or residence time) of pollinators, this has little 257 effect on plant growth rates, because it is the plant lifespan, not the pollinator lifespan that 258 impedes seed-set. In contrast, when pollinators arrive first, phenological mismatch causes a 259 reduction in plant seed-set because most pollinators die or depart before plants bloom. In this 260 case, if the presence of flowers extends the pollinator lifespan (or residence time) this can have a 13 261 strong effect on plant growth rates, since it is the pollinator lifespan, not the plant lifespan that 262 limits seed-set. Open flowers have a greater effect when emergence functions are broad, because263 broad emergence functions result in a larger population of open flowers earlier in the year. 264 These early blooming flowers extend pollinator lifespan (or residence time) such that pollinators 265 are still available at peak flowering. As a result, plant growth rates can become almost 266 independent of the level of mismatch between pollinator arrival and plant blooming (see Figure 267 D1). In contrast, for narrow emergence functions, even early flowers may be too late to prevent 268 death or departure of the majority of the pollinator population, minimizing the effect of flower 269 availability on the pollinator population.270271 Figure D1 Plant population growth rates,, as a function of the separation between peak 272 pollinator emergence and peak plant emergence for short-lived pollinators and short-lived 273 flowers (, see Figure 1.c. from main text), assuming three different levels of dependence of the 274 pollinator population on flower availability ( 0 (black), 0.01 (dark grey), 0.1 (light grey)) and 275 assuming either broad (, , dotted) or narrow (,, , solid) plant and pollinator emergence functions. 276 The vertical grey line separates situations where pollinators arrive first from those where flowers 277 appear first. The dotted horizontal grey line 1000 = 1 indicates the population growth rate 278 corresponding to a constant population size. All other parameters are held constant as: , , and .279 Other forms of feedback between the plant and pollinator population are possible as well. For 280 example, flower availability in year may influence pollinator abundance in year . While this 281 could be studied using our Zonneveld model approach, we would have to select a different 282 metric for understanding implications to the plant population, since is a single-generation 283 metric, and thus cannot capture longer-scale feedbacks on the pollinator population. For species 14 284 with multiple generations per year and/or continuous breeding, similar feedback between flower 285 availability and pollinator abundance may occur within a single season. In this case, it would be 286 necessary to consider a pollinator model with multivoltinism or else continuous breeding. 287 Particularly for multivoltinism, this could involve multiple emergence functions and could 288 become quite complex. Nevertheless, with some modification, the same basic approach that we 289 present can be used for these scenarios as well. 15

Appendix a Nondimensionalization and Model Simplification

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