Background

Partridge pea (Chamaecrista fasiculata, Fabaceae) is a small annual vascular plant that interacts with at least four types of mutualists: pollinators (bees), anti-herbivore ants, mycorrhizal fungi, and N-fixing Rhizobia. This document lays out a theoretical approach to describe these interactions

In future iterations of this work, we can add additional biological information including averages and ranges of:

As we think about reality, we can been to think about simulations that could help learn about the consequences of the uncertainty in our estimates.

Interactions

Here we describe the direct pairwise interactions as Jacobian elements (May 1973). These are the effects of a unit of species \(j\) (individual or gram) on the population growth rate of species \(i\) when both species are near their equilibria. We describe them in more detail in a later section.
*Interaction network for partridge pea. Not sure about whether ants benefit from eating herbivores or whether they just kill/remove them. Also, I wonder whether ants are tending aphids.*

Interaction network for partridge pea. Not sure about whether ants benefit from eating herbivores or whether they just kill/remove them. Also, I wonder whether ants are tending aphids.

Interaction strengths: first pass

We can ballpark (spitball?) some relative species interactions. Pimm and Lawton (1977) did this to great effect to investigate general properties of small food webs, and here we follow the same approach.

In our model, we have these nodes:

Current assumptions: - Each effect is direct, and, in that sense only, independent of any other. - Each node is self-limiting. This might be appropriate for both plant parts (e.g., flowers) and populations (bees). - Plant-fungus and plant-rhizobium interactions are symmetric and small (0.1). No idea if this is appropriate. - All other effects are equal (\(\pm 1\)).

##      r    m    v  f  e  d  h  b  a
## r -1.0  0.0  0.1  0  0  0  0  0  0
## m  0.0 -1.0  0.1  0  0  0  0  0  0
## v  0.1  0.1 -1.0  0  0  0  0  0  0
## f  0.0  0.0  1.0 -1  0  0  0 -1  0
## e  0.0  0.0  1.0  0 -1  0  0  0 -1
## d  0.0  0.0  0.0  0  0 -1  0  0  0
## h  0.0  0.0  0.0  0  0  0 -1  0 -1
## b  0.0  0.0  0.0  1  0  0  0 -1  0
## a  0.0  0.0  0.0  0  1  1  0  0 -1

Stability

If there is sufficient self-limitation, \(J_{ii} < 0\), this is a stable system, typically with some oscillations. Below, the real part of this dominant eigenvalue will be negative if the web is stable, while the imaginary part will be non-zero if there are ocillations.

(de <- eigen(J)$values[1])
## [1] -1+1i

A negative real part indicates stability. Any oscillations have the following period.

# Oscillations with period 2 pi / omega (Im)
2*pi / Im(de)
## [1] 6.283185

Reactivity (Neubert and Caswell 1997) is a measure of how much a perturbation grows following the initial perturbation, even if it will eventually return to an equilibrium. It is easily found as the dominant eigenvalue of \[H = \frac{\mathbf{M+M}^T}{2}\] where M is the community matrix. Positive values indicate a reactive system, whereas negative values indicate a system in which a perturbation begins immediately to dissipate or shrink. This is the reactivity of the above matrix:

H <- (J+t(J))/2
eigen(H)$values[1]
## [1] -0.2788897

Estimating interaction strengths

A Jacobian element, \(\mathbf{J}_{ij}\), is the partial derivative of the growth rate species \(i\) with respect to species \(j\), evaluated at the equilibrium . These can be similar, say, to per capita model parameters (e.g., Lotka-Volterra parameters, \(\alpha_{ij}\)) but multiplied by the equilibrium abundance of species \(i\). The units of these are in units of time and the unit used for the non-focal organism (e.g., per year per gram, y-1 g-1).

In contrast to Jacobian elements, per capita interaction strengths are the effect of a unit of consumer on the per capita population growth rate of the resource. Below we start by describing an approach to estimate per capita effects and then derive the Jacobian elements (Laska and Wooton 1998). The units of these are in units of time.

One approach to estimating per capita interaction strengths is to use experimental removals (Laska andd Wooton 1998), or, in our case, thought experiments. Paine (1992) measured per capita interaction strength of predator and prey as \[I = \frac{E-C}{C\cdot M}\] where \(E\) and \(C\) are the biomass (or density) of the focal or prey population in the experimental (with predator) and control (without) conditions. \(M\) is the biomass (or density) of the non-focal species (predator, consumer, or mutualist).

If you prefer to think that the predator (or non-focal species) is part of the ambient “control” condition, you could use, equivalently, \[I=\frac{F_A - F_R}{F_R \cdot N_A}\] where \(F_A\) and \(F_R\) are the biomass of the focal species in the ambient and removal treatments and \(N_A\) is the biomass of the non-focal species.

When data come from short term experiments when prey sizes haven’t changed much, it may be more appropriate to use \[I=\frac{\ln (E/C)/\Delta t}{M}\] where \(\delta t\) is the time period over which we choose to measure the change (Novak and Wootton 2010), or, equivalently \[I=\frac{\ln (F_A/F_R)/ \Delta t}{N_A}\] to measure per capita interaction strength.

With simple predator-prey models we can show that this quantity is the ratio of the attack rate over the intrinsic rate of increase for the prey. Which is really annoying, because the Jacobian element isolates the effect of the predator and depends only on the attack or consumption rate by the predator (times the abundance of the prey without the predator present).

The Jacobian element for the effect of a Lotka-Volterra predator \(P\), is the partial derivative of prey growth rate with respect to \(P\). The prey growth rate is described as \[\dot{N} = rN(1-N/K) - a NP=rN-\frac{r}{K}N^2 - a NP\] so the partial derivative with respect to \(P\), and evaluate it at equilibrium, we get \[\left.\frac{\partial \dot{N} }{\partial P}\right|_{N^*} = a N = a K\] To figure out what Paine’s index gives us, we can solve the predator and prey equations, and substitute back into Paine’s index. Without the predator, the prey will be at \(N=K\). With the predator, we have, \[0=rN-\frac{r}{K}N^2 - a NP=1-\frac{N}{K} - aP/r\] \[\frac{N}{K} = 1-aP/r\] \[N^*=K- \frac{a}{r}PK\] Paine’s index then is \[\frac{K-\frac{a}{r}PK - K}{PK}=a/r\] which is the ratio of predator attack rate and the prey maximum per capita growth rate.

To approximate the Jacobian elements, we need to multiply Paine’s per capita effects by the equilibrium biomass and the intrinsic rate of increase of the focal species. \[J=I \cdot r_F B_F\] which is \[J=\frac{E-C}{M}r_r=\frac{F_A-F_R}{N_A}r_F\] The intrinsic rate of increase is just the maximum per capita growth rate. In this plant with these organisms, “growth” rate will be confounded by immigration rate (Novak and Wootton, n.d.).

A major problem with doing this in practice (in the field) is that the manipulation of any one species may alter any of the other interactions in the network. However, it is easy to conduct the thought experiment….

R functions for these would be,

i_s <- function(x, paine=TRUE, jac=TRUE){
  # elements of x must be in this order
  # 1 - FA (E)
  Fa <- x[1]
  # 2 - FR (C)
  Fr <- x[2]
  # 3 - NA (M)
  Na <- x[3]
  # 4 rF
  rF <- x[4]
  
  # paine=TRUE calculates (E-C)/(C), FALSE calculates ln(E/C)
  # jac=TRUE returns the Jacobian, FALSE returns per capita i.s.
  
  if(paine){
    IS <- (Fa-Fr)/(Na*Fr)
  } else {
    IS <- log(Fa/Fr)/Na
  }
  if(jac==TRUE){
    IS <- IS * Fr * rF
  } else {
    IS <- IS * rF
  }
  names(IS) <- NULL
  IS
}

Examples

Intrinsic rates of growth

We need to estimate \(r\), the maximum per capita growth rate, for each species. We need to consider a few things.

  • Estimate \(r\) from the literature. This would be nice. :-)
  • Take advantage of body size which correlates well with \(r\).
  • Consider the extent to which this rate is actually driven by behavioral aggregation (immigration), rather than population growth. The former is not a per capita rate and thus exponential, but rather fixed and perhaps linearly dependent on a resource.

Today, I’ll let \(r=1\), just to let things run.

Effects of ants on herbivore growth rate

Schmitz, Hamback, and Beckerman (2000) reviewed trophic interaction strengths for a large number of experimental top predator removals. They measured “effect” as the log response ratio, \(\ln (F_A/F_R)\), where \(F_A\) and \(F_R\) are herbivore density with ants present and removed. They found that the effect of ants varied widely (\(0\)-\(-3\)) and tended to be around -0.6-ish. However, we then have to figure out units and abundances/densities/biomass of the populations upon which the number is based.

Based on Keller et al. (2018) the presence of ants on Chamaecrista (w. rhizobia) reduced the abundance of non-aphid herbivores from about 23/plant to about 8/plant. Ants are twice as abundant on as non-aphid herbivores (40/plant vs. 23/plant). We could convert these to biomass, but as long as we assume the average herbivore weighs that same as an ant (1-5 mg), then those units would cancel out. Using our adaptation of Paine’s approach, then the effect of ants on non-aphid herbivores is a…

x <- c(Fa=8, Fr=23, Na=40, r=1)
i_s(x)
## [1] -0.375

…decrease in herbivores per ant (y-1ant-1).

Cool. Using the alternative method (\(\ln E/C\)), we would get

i_s(x, paine=FALSE)
## [1] -0.6072303

but I think the former approach is better. Regardless, same ballpark…?

Effects of Rhizobium on vegetative plant growth

Based on Keller et al. (2018), the effect of Rhizobium nodules on on plant biomass is the following, based 20 nodules without inocula and about 180 with inocula.

x <- c(Fa=1.7, Fr=1.3, Na=160, r=1)
i_s(x)
## [1] 0.0025

This is the instantaneous per-root-nodule effect on plant vegetative growth rate (y-1nodule-1)

References

Keller, Kane R., Sara Carabajal, Felipe Navarro, and Jennifer A. Lau. 2018. “Effects of Multiple Mutualists on Plants and Their Associated Arthropod Communities.” Oecologia 186: 185–94.

Laska, M S, and J T Wooton. 1998. “Theoretical concepts and empirical approaches to measuring interaction strength.” Ecology 79: 461–76.

May, R M. 1973. Stability and Complexity in Model Ecosystems. Vol. 6. Monographs in Population Biology. Princeton, NJ: Princeton University Press.

Neubert, M G, and H Caswell. 1997. “Alternatives to resilience for measuring the responses of ecological systems to perturbations.” Ecology 78: 653–65.

Novak, Mark, and J. Timothy Wootton. n.d. “Using experimental indices to quantify the strength of species interactions.” Oikos 119 (7): 1057–63. https://doi.org/{10.1111/j.1600-0706.2009.18147.x}.

Paine, R T. 1992. “Food-web analysis through field measurement of per capita interation strength.” Nature 355: 73–75.

Pimm, S L, and J H Lawton. 1977. “Number of trophic levels in ecological communities.” Nature 268: 329 331.

Schmitz, O J, P A Hamback, and A P Beckerman. 2000. “Trophic cascades in terrestrial systems: {A} review of the effects of carnivore removals on plants.” American Naturalist 155 (2): 141–53. {\%}3CGo to.