Optimal allocation of two resources in annual plants

David McMorris; Glenn Ledder; David McMorris; Glenn Ledder

doi:10.3934/mbe.2025055

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 6: 1464-1516. doi: 10.3934/mbe.2025055

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Research article

Optimal allocation of two resources in annual plants

David McMorris ^{1,2
,
,},
Glenn Ledder ¹

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, USA
2.
Department of Mathematics, Christopher Newport University, Newport News, VA, USA

Received: 28 January 2025 Revised: 26 April 2025 Accepted: 30 April 2025 Published: 13 May 2025

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season. Working under the assumption that annual plants grow to maximize fitness, we use optimal control theory to understand this process. We introduce a model for resource allocation in annual plants that extends classical work by Iwasa and Roughgarden to a case where both carbohydrates and mineral nutrients are allocated to shoots, roots, and fruits. We use optimal control theory to determine the optimal resource allocation strategy for the plant throughout its growing season as well as develop a numerical scheme to implement the model. We find that fitness is maximized when the plant undergoes a period of mixed vegetative and reproductive growth prior to switching to reproductive-only growth at the end of the growing season. Our results further suggest that what is optimal for an individual plant is highly dependent on initial conditions, and optimal growth has the effect of driving a wide range of initial conditions toward common configurations of biomass by the end of a growing season.
- plant life history,
- optimal control,
- resource allocation,
- optimal growth
Citation: David McMorris, Glenn Ledder. Optimal allocation of two resources in annual plants[J]. Mathematical Biosciences and Engineering, 2025, 22(6): 1464-1516. doi: 10.3934/mbe.2025055

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Abstract

The fitness of an annual plant can be thought of as how much fruit is produced by the end of its growing season. Working under the assumption that annual plants grow to maximize fitness, we use optimal control theory to understand this process. We introduce a model for resource allocation in annual plants that extends classical work by Iwasa and Roughgarden to a case where both carbohydrates and mineral nutrients are allocated to shoots, roots, and fruits. We use optimal control theory to determine the optimal resource allocation strategy for the plant throughout its growing season as well as develop a numerical scheme to implement the model. We find that fitness is maximized when the plant undergoes a period of mixed vegetative and reproductive growth prior to switching to reproductive-only growth at the end of the growing season. Our results further suggest that what is optimal for an individual plant is highly dependent on initial conditions, and optimal growth has the effect of driving a wide range of initial conditions toward common configurations of biomass by the end of a growing season.

References

[1]	Y. Iwasa, J. Roughgarden, Shoot/root balance of plants: Optimal growth of a system with many vegetative organs, Theor. Popul. Biol., 25 (1984), 78–105. https://doi.org/10.1016/0040-5809(84)90007-8 doi: 10.1016/0040-5809(84)90007-8
[2]	K. Velten, O. Richter, Optimal root/shoot-partitioning of carbohydrates in plants, Bull. Math. Biol., 57 (1995), 99–107. https://doi.org/10.1007/BF02458318 doi: 10.1007/BF02458318
[3]	B. J. Enquist, K. J. Niklas, Global allocation rules for patterns of biomass partitioning in seed plants, Science, 295 (2002), 1517–1520. https://doi.org/10.1126/science.1066360 doi: 10.1126/science.1066360
[4]	M. McCarthy, B. J. Enquist, Consistency between an allometric approach and optimal partitioning theory in global patterns of plant biomass allocation, Funct. Ecol., 21 (2007), 713–720. https://doi.org/10.1111/j.1365-2435.2007.01276.x doi: 10.1111/j.1365-2435.2007.01276.x
[5]	R. Dybzinski, C. Farrior, A. Wolf, P. B. Reich, S. W. Pacala, Evolutionarily stable strategy carbon allocation to foliage, wood, and fine roots in trees competing for light and nitrogen: An analytically tractable, individual-based model and quantitative comparisons to data, Am. Nat., 177 (2011), 153–166. https://doi.org/10.1086/657992 doi: 10.1086/657992
[6]	C. Feller, P. Favre, A. Janka, S. C. Zeeman, J.-P. Gabriel, D. Reinhardt, Mathematical modeling of the dynamics of shoot-root interactions and resource partitioning in plant growth, PLoS ONE, 10 (2015), e0127905. https://doi.org/10.1371/journal.pone.0127905 doi: 10.1371/journal.pone.0127905
[7]	G. Ledder, S. E. Russo, E. B. Muller, A. Peace, R. M. Nisbet, Local control of resource allocation is sufficient to model optimal dynamics in syntrophic systems, Theor. Ecol., 13 (2020), 481–501, https://doi.org/10.1007/s12080-020-00464-9 doi: 10.1007/s12080-020-00464-9
[8]	H. Poorter, K. J. Niklas, P. B. Reich, J. Oleksyn, P. Poot, L. Mommer, Biomass allocation to leaves, stems and roots: Meta-analyses of interspecific variation and environmental control, New Phytol., 193 (2012), 30–50. https://doi.org/10.1111/j.1469-8137.2011.03952.x doi: 10.1111/j.1469-8137.2011.03952.x
[9]	A. J. Bloom, F. S. Chapin III, H. A. Mooney, Resource limitation in plants-an economic analogy, Annu. Rev. Ecol. Evol. Syst., 16 (1985), 363–392. https://doi.org/10.1146/annurev.es.16.110185.002051 doi: 10.1146/annurev.es.16.110185.002051
[10]	H. Poorter, O. Nagel, The role of biomass allocation in the growth response of plants to different levels of light, co2, nutrients and water: A quantitative review, Aust. J. Plant Physiol., 27 (2000), 1191–1191. https://doi.org/10.1071/PP99173_CO doi: 10.1071/PP99173_CO
[11]	J. Weiner, Allocation, plasticity and allometry in plants, Pers. Plant Ecol., Evol. And Syst., 6 (2004), 207–215. https://doi.org/10.1078/1433-8319-00083 doi: 10.1078/1433-8319-00083
[12]	G. A. Fox, Annual plant life histories and the paradigm of resource allocation, Evolut. Ecol., 6 (1992), 482–499. https://doi.org/10.1007/BF02270693 doi: 10.1007/BF02270693
[13]	S. Lenhart, J. T. Workman, Optimal control applied to biological models, Chapman and Hall/CRC, 2007. https://doi.org/10.1201/9781420011418
[14]	L. S. Pontryagin, E. F. Mishchenko, V. G. Boltyanskii, R. V. Gamkrelidze, The mathematical theory of optimal processes, Wiley, 1962. https://doi.org/10.1201/9780203749319
[15]	S. Kooijman, Dynamic energy budget theory for metabolic organisation, Cambridge university press, 2010. https://doi.org/10.1017/CBO9780511805400
[16]	M. I. Kamien, N. L. Schwartz, Dynamic optimization: the calculus of variations and optimal control in economics and management, North-Holland, 1991.
[17]	P. A. Hicks, Distribution of carbon/nitrogen ratio in the various organs of the wheat plant at different periods of its life history, New Phytol., 27 (1928), 108–116. https://doi.org/10.1111/j.1469-8137.1928.tb06735.x doi: 10.1111/j.1469-8137.1928.tb06735.x
[18]	V. Minden, M. Kleyer, Internal and external regulation of plant organ stoichiometry, Plant Biol., 16 (2014), 897–907. https://doi.org/10.1111/plb.12155 doi: 10.1111/plb.12155
[19]	S. Koshkin, Z. Zalles, M. F. Tobin, N. Toumbacaris, C. Spiess, Optimal allocation in annual plants with density-dependent fitness, Theory Biosci., 140 (2021), 177–196. https://doi.org/10.1007/s12064-021-00343-9 doi: 10.1007/s12064-021-00343-9
[20]	N. Chiariello, J. Roughgarden, Storage allocation in seasonal races of an annual plant: optimal versus actual allocation, Ecology, 65 (1984), 1290–1301. https://doi.org/10.2307/1938334 doi: 10.2307/1938334
[21]	Y. Iwasa, Dynamic optimization of plant growth, Evol. Ecol. Res., 2, (2000), 437–455.

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