General Overview

This report is the result of the use of the Python 3.4 package Sympy (for symbolic mathematics), as means to translate published models to a common language. It was created by Verónika Ceballos-Núñez (Orcid ID: 0000-0002-0046-1160) on 15/3/2016, and was last modified on lm.

About the model

The model depicted in this document considers carbon allocation with a process based approach. It was originally described by DeAngelis, Ju, Liu, Bryant, & Gourley (2012).

Abstract

We use modeling to determine the optimal relative plant carbonallocations between foliage, fine roots, anti-herbivoredefense, and reproduction to maximize reproductive output. Themodel treats these plant components and the herbivorecompartment as variables. Herbivory is assumed to be purelyfolivory. Key external factors include nutrient availability,degree of shading, and intensity of herbivory. Threealternative functional responses are used for herbivory, two ofwhich are variations on donor-dependent herbivore (models 1aand 1b) and one of which is a Lotka–Volterra type ofinteraction (model 2). All three were modified to include thenegative effect of chemical defenses on the herbivore. Analysisshowed that, for all three models, two stable equilibria couldoccur, which differs from most common functional responses whenno plant defense component is included. Optimal strategies ofcarbon allocation were defined as the maximum biomass ofreproductive propagules produced per unit time, and found tovary with changes in external factors. Increased intensity ofherbivory always led to an increase in the fractionalallocation of carbon to defense. Decreases in availablelimiting nutrient generally led to increasing importance ofdefense. Decreases in available light had little effect ondefense but led to increased allocation to foliage. Decreasesin limiting nutrient and available light led to decreases inallocation to reproduction in models 1a and 1b but not model 2.Increases in allocation to plant defense were usuallyaccompanied by shifts in carbon allocation away from fineroots, possibly because higher plant defense reduced the lossof nutrients to herbivory.

Space Scale

global

Available parameter values

State Variables

The following table contains the available information regarding this section:

Additional Variables

The following table contains the available information regarding this section:

Photosynthetic Parameters

The following table contains the available information regarding this section:

Nutrient Uptake

The following table contains the available information regarding this section:

Allocation fractions

The following table contains the available information regarding this section:

Cycling Rates

The following table contains the available information regarding this section:

Components

The following table contains the available information regarding this section:

Pool model representation

Flux description

Figure 1: Pool model representation

Input fluxes \(C_{f}: \frac{G_{0}\cdot N_{f}\cdot\eta_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\) \(C_{r}: \frac{G_{0}\cdot N_{f}\cdot\eta_{r}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\) \(C_{w}: \frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right)\) \(N_{f}: \frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right)\) Output fluxes \(C_{f}: \frac{C_{f}}{N_{f}}\cdot\left(F_{i} + N_{f}\cdot\gamma_{f}\right)\) \(C_{r}: C_{r}\cdot\gamma_{r}\) \(C_{w}: C_{w}\cdot\gamma_{w}\) \(N_{f}: F_{i} + N_{f}\cdot\gamma_{f}\)

The right hand side of the ODE

\(\left[\begin{matrix}C_{f}\cdot\left(-\frac{F_{i}}{N_{f}} -\gamma_{f}\right) +\frac{G_{0}\cdot N_{f}\cdot\eta_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\\- C_{r}\cdot\gamma_{r} +\frac{G_{0}\cdot N_{f}\cdot\eta_{r}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\\- C_{w}\cdot\gamma_{w} +\frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right)\\N_{f}\cdot\left(-\frac{F_{i}}{N_{f}} -\gamma_{f}\right) +\frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right)\end{matrix}\right]\)

The Jacobian (derivative of the ODE w.r.t. state variables)

\(\left[\begin{matrix}-\frac{F_{i}}{N_{f}} -\gamma_{f} +\frac{G_{0}\cdot N_{f}\cdot b_{f}\cdot\eta_{f}\cdot k_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot e^{- C_{f}\cdot b_{f}\cdot k_{f}} -\frac{G_{0}\cdot N_{f}\cdot\eta_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) +\frac{G_{0}\cdot N_{f}^{2}\cdot\eta_{f}}{C_{f}^{3}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) & 0 & 0 &\frac{C_{f}}{N_{f}^{2}}\cdot F_{i} +\frac{G_{0}\cdot\eta_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) -\frac{G_{0}\cdot N_{f}\cdot\eta_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\\\frac{G_{0}\cdot N_{f}\cdot b_{f}\cdot\eta_{r}\cdot k_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot e^{- C_{f}\cdot b_{f}\cdot k_{f}} -\frac{G_{0}\cdot N_{f}\cdot\eta_{r}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) +\frac{G_{0}\cdot N_{f}^{2}\cdot\eta_{r}}{C_{f}^{3}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) & -\gamma_{r} & 0 &\frac{G_{0}\cdot\eta_{r}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right) -\frac{G_{0}\cdot N_{f}\cdot\eta_{r}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\\\frac{G_{0}\cdot N_{f}\cdot b_{f}\cdot k_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right)\cdot e^{- C_{f}\cdot b_{f}\cdot k_{f}} -\frac{G_{0}\cdot N_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) +\frac{G_{0}\cdot N_{f}^{2}}{C_{f}^{3}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) & 0 & -\gamma_{w} &\frac{G_{0}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{G_{0}\cdot N_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right)\\\frac{G_{0}\cdot N_{f}\cdot b_{f}\cdot k_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right)\cdot e^{- C_{f}\cdot b_{f}\cdot k_{f}} +\frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(-\frac{C_{f}\cdot N_{pore}\cdot b_{f}\cdot g_{N}\cdot k_{f}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)\cdot e^{- C_{f}\cdot b_{f}\cdot k_{f}}}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)^{2}\cdot\left(N_{pore} + k_{N}\right)} +\frac{N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\frac{N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)}{C_{f}\cdot G_{0}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)}\right) -\frac{G_{0}\cdot N_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right) +\frac{G_{0}\cdot N_{f}^{2}}{C_{f}^{3}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right) &\frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot b_{r}\cdot g_{N}\cdot k_{r}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)\cdot e^{- C_{r}\cdot b_{r}\cdot k_{r}}}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} +\frac{N_{r}}{C_{r}^{2}}\cdot\eta_{r}\right) &\frac{G_{0}\cdot N_{f}\cdot N_{w}}{C_{f}\cdot C_{w}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) & -\gamma_{f} +\frac{G_{0}\cdot N_{f}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(-\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}^{2}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} +\frac{N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)}\right) +\frac{G_{0}}{C_{f}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right) -\frac{G_{0}\cdot N_{f}}{C_{f}^{2}\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)^{2}}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(\frac{C_{f}\cdot N_{pore}\cdot g_{N}\cdot\left(1 - e^{- C_{r}\cdot b_{r}\cdot k_{r}}\right)\cdot\left(v_{0} +\frac{N_{f}}{C_{f}}\right)}{G_{0}\cdot N_{f}\cdot\left(1 - e^{- C_{f}\cdot b_{f}\cdot k_{f}}\right)\cdot\left(N_{pore} + k_{N}\right)} -\eta_{m}\cdot v_{m} -\frac{N_{w}}{C_{w}}\cdot\left(\eta_{f}\cdot s_{f} +\eta_{r}\cdot s_{r}\right) -\frac{N_{r}}{C_{r}}\cdot\eta_{r}\right)\end{matrix}\right]\)

References