Abstract
Many plant species produce excessive flowers but abandon most of them halfway to maturation. Several hypotheses have been proposed to explain adaptive significances of this behavior. To understand this phenomenon, I developed a resource allocation model between flower and fruit/seed production to examine a new hypothesis that excessive flower production is favored to “dilute” predation pressures in plant–pre-dispersal seed predator systems. First, I compared the efficiencies of three abortion strategies: (1) no abortions: the plant matures all pollinated flowers; (2) selective abortions: the plant aborts all flowers oviposited by predators and only intact flowers mature; (3) random abortions: the plant indiscriminately aborts a fraction of the pollinated flowers irrespective of seed-predator oviposition. I assumed that the timing of selective abortions was later than that of random abortions owing to delays in response to feeding damage (the cost of selective abortion). I showed that the reproductive efficiencies of the random-abortion and selective-abortion strategies were much higher than that of the no-abortion strategy when resources were poor, predators were abundant, and the cost of flower production was low. In addition, the reproductive efficiency of the random-abortion strategy was greater than that of the selective-abortion strategy when the cost of selective abortion was high. Second, I examined a mixed-abortion strategy in which plants aborted flowers randomly earlier and selectively later. The proportion of random abortions increased as the amount of resources decreased, density of seed predators increased, flower production cost decreased, and cost of selective abortion increased.
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Acknowledgements
I would like to thank Yusuke Ikegawa for his many valuable comments.
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Appendices
Appendix 1
To find the optimal number of flowers produced n3 = n3* and the optimal abortion rate w3 = w3* for the reproductive success of the random-abortion strategy ϕ3 when q0 = 1 and qi = 0 for i ≥ 1, I applied the Lagrange multiplier method, an optimization technique for finding local maxima of a function with equality constraints (e.g., Traulsen et al. 2007).
We first constructed the following function from Eqs. 1c and 3c:
where λ is a Lagrange multiplier and Q0 = Q0(n3). From Eq. 8 we had:
From Eqs. 10 and 1c, we obtained λ = Q0/s and w3 = 1 − (R − n3f)/{n3s(1 − P0)} respectively, which we substituted into Eq. 9 to numerically calculate n3 = n3*. The optimal abortion rate was w3* = 1 − (R − n3*f)/{n3s(1 − P0)}.
In a similar way, we found the optimal flower production n4 = n4* and the random abortion rate w4 = w4* of the mixed-abortion strategy when q0 = 1 and qi = 0 for i ≥ 1. From Eqs. 4 and 5, we constructed the function:
where λ is a Lagrange multiplier and Q0 = Q0(n4). From Eq. 11 we had:
By substituting Eqs. 14 and 15 into Eq. 12, we had an equation with respect to n4. We could numerically calculate its solution n4 = n4*. The optimal random-abortion rate w4* was calculated with Eq. 14.
Appendix 2
From Eq. 2, we obtained:
where the subscript of n was omitted. When k = 1, ΣqiQi was clearly independent of n. When k > 1, the sign of that derivative depended only on the sign of i − αnk−1. Considering Σ(dQi/dn) = 0 (because ΣQi = 1), there should be a positive integer I such that dQi/dn ≤ 0 and dQi/dn > 0 if i ≤ I and i > I, respectively. Then, we had:
as {qi} was a non-negative decreasing sequence. Thus ΣqiQi was independent of or decreased with the number of flowers n if k ≥ 1.
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Ezoe, H. Excessive flower production as an anti-predator strategy: when is random flower abortion favored?. Popul Ecol 60, 275–286 (2018). https://doi.org/10.1007/s10144-018-0625-6
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DOI: https://doi.org/10.1007/s10144-018-0625-6