The Arabidopsis Problem of Zeno

Arabidopsis is a small mustard annual of the family Brassicaceae. It is a favorite amongst plant geneticists for a number of reasons, some of which include a short lifetime to seed and a predisposition to self pollinating. The flowers of the Arabidopsis are borne on an indeterminate raceme, meaning that flowers mature from the base of the inflorescence upwards and there is no distinct number of flowers per inflorescence. The allocation stratagies played by indeterminate annual plants must to some degree be affected by the role of a probibalistic but finite time horizon. Here we are interested in asking an age old question in a rather new dynamical light. That is, how should resource investment in seeds change over the lifetime of an annual plant with an indeterminate inflorescence.

To answer this problem in a life history context we must concern ourselves with how the plant gets resources, how these resources can be allocated, and what role is played by environmental effects on survivability.

We will begin by assuming that the plant has an initial resource pool, $R_{0}$, that is depleted by the production of flowers and seeds. These resources represent all previously acquired nutrient and carbon stores. Since Arabidopsis generally stops producing leaves before it starts flowering, it is safe to assume that the total resources acquired are predetermined before flowering begins.

We will define fitness in the usual way.

$$R_{o} = \int_{0}^{\infty} l(x)m(x)dx$$

Where $l(x)$ is the probability that the plant is alive at age $x$ and $m(x)$ is the plant's fecundity at age $x$. Our general function for $l(x)$,

$$l(x)' = -\mu l(x)$$

where $\mu$ is the death rate at age $x$, is adequate. This gives us

$$l(x) = e^{-\int \mu dx}$$

.

The fecundity function for our plant is the product of seed fitness, $F(s(t))$ with seed production rate, $N(t)$',

$$m(x) = F(s(t))N(t)'$$

.

We expect the seed production rate to be a function of the flower production rate, which itself should be a function of the remaining resources. We can write differential equations for these as follows.

$$N(t)'= f(t)'n$$

where $f'$ is the rate of flower production and $n$ is the number of seeds per flower.

$$f(t)'= cR(t)$$

where c is the rate constant at which resources are turned into flowers.

$$R(t)'= -bcR-cRns(t)$$

where $b$ is the resource cost per flower, and $s(t)$ is the resource cost per seed.

In our fecundity equation, we can replace $N(t)'$ by $cnR(t)$. This leaves us with the simple problem of solving for $R(t)$ from the first order differential equation above. Leaving us with

$$R(t)=e^{-c(bt+n\int s(t) dt)}$$

.

We've now resolved the problem to maximizing $J(s(t))$, by choosing an appropriate seed size over the lifetime of the plant where

$$J(s(t)) = \int_{0}^{\infty} e^{-\int \mu dx} F(s(t)) c n e^{-c(bt+n\int s(t) dt)} dt$$

This problem we can solve using our handy Euler-Lagrange equation. Again, recognizing that the internal components of the integral make up the intermediate functional, $I$, we apply

$$\frac{\partial f}{\partial y}-\frac{d}{dx} \frac{\partial f}{\partial y^{'}} = 0$$

.

When solved, we arrive at

$$cn \left( F(s(t))' \left( \frac{\mu + bc}{cn} + s(t) \right) - F(s(t)) \right) = \frac{dF(s(t))'}{dt}$$

We can solve for $s(t)$ numerically, but first we can make some general observations that give us a feel for whether we got the right answer. Of common interest might be how this problem is related to the similar problem of static seed size optimizition, which has the general solution

$$F(s(t))'s(t)=F(s(t))$$

We will note that if the death rate is constant and the derivative of the fitness function with respect to time is zero, then the problems are practically equivelant, though the seed sizes are then slightly inflated by the probability of death and the cost of of flower production. If we assume the time derivative of $F(s(t))'$ is small, then we make a number of easy observations. If the death rate is increasing, the seed sizes should be larger later in life, and vice versa. Increasing the flower production rate or the seeds per pod leads to a reduction in seed size. Increasing the flower cost leads to increased seed size.

That all these predictions seem to make sense is reassuring. Next problem...