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Optimal allocation in annual plants with density-dependent fitness

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Abstract

We study optimal two-sector (vegetative and reproductive) allocation models of annual plants in temporally variable environments that incorporate effects of density-dependent lifetime variability and juvenile mortality in a fitness function whose expected value is maximized. Only special cases of arithmetic and geometric mean maximizers have previously been considered in the literature, and we also allow a wider range of production functions with diminishing returns. The model predicts that the time of maturity is pushed to an earlier date as the correlation between individual lifetimes increases, and while optimal schedules are bang-bang at the extremes, the transition is mediated by schedules where vegetative growth is mixed with reproduction for a wide intermediate range. The mixed growth lasts longer when the production function is less concave allowing for better leveraging of plant size when generating seeds. Analytic estimates are obtained for the power means that interpolate between arithmetic and geometric mean and correspond to partially correlated lifetime distributions.

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Notes

  1. For \(\alpha =0,\) this is replaced by \(\tau =\left( 1+\frac{1}{r}\right) \ln (1+r\tau )\), and for \(\alpha =-1\) by \(\tau =\frac{1}{r(1+r)}(1+r\tau )\ln (1+r\tau )\).

  2. For \(\alpha =0,\) the second equation is replaced by \(r=\ln (1+r\tau )\), and for \(\alpha =-1\) by \(r(1+r)=\ln \tau \).

  3. For \(\alpha =0,\) this simplifies to \(q(t)=\frac{1}{C-t}\).

Abbreviations

x, y :

Vegetative, reproductive mass, 1.2

\(T_0,T\) :

Safe, volatile period, 1.2

L(y):

Fitness function, 1.1

P(x):

Production function, 1.2

\(\alpha \) :

Fitness index, 1.1

\(\text {Sw}(\tau ,\xi ,y)\) :

Switching function, 3.1

\(\text {Cut}(\tau ,\xi ,y)\) :

Cutoff function, 3.1

\(\lambda ,\mu \) :

Costate variables for x, y, 3.1

\(\xi \) :

Adjusted vegetative mass, 3.1

\(\tau \) :

Adjusted time-to-go, 3.1

q, r :

Adjusted mass ratios, 4

\(q_c\), \(r_c\), \(\tau _c\) :

Cutoff constants, 4; 4.2

\(t_s\) :

VR switching time, 1.2; 6

\(t_m\) :

Time of maturity, 3.1

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Acknowledgements

This work was conceived during the summer 2019 REU program at the University of Houston-Downtown and is funded by the National Science Foundation Grant 1560401. The authors are grateful to the anonymous reviewers for clarifying the relation between density dependence and fitness and many other helpful suggestions that greatly improved the paper.

Funding

All authors were supported by the National Science Foundation Grant 1560401.

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Authors and Affiliations

  1. University of Houston-Downtown, Houston, TX, USA

    Sergiy Koshkin & Michael F. Tobin

  2. Rice University, Houston, TX, USA

    Zachary Zalles

  3. Emory University, Atlanta, GA, USA

    Nicolas Toumbacaris

  4. Keene State College, Keene, NH, USA

    Cameron Spiess

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  1. Sergiy Koshkin
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  2. Zachary Zalles
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  3. Michael F. Tobin
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  4. Nicolas Toumbacaris
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  5. Cameron Spiess
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Contributions

All authors contributed to the paper’s conception and design. The manuscript was written by SK with contributions from MT. All authors commented on previous versions of the manuscript and approved the final manuscript.

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Correspondence to Sergiy Koshkin.

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Koshkin, S., Zalles, Z., Tobin, M.F. et al. Optimal allocation in annual plants with density-dependent fitness. Theory Biosci. 140, 177–196 (2021). https://doi.org/10.1007/s12064-021-00343-9

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