Populations of single-celled algae are amenable to controlled experimentation, and the processes that drive their population dynamics are analytically tractable. Because they have such short generation times (days to weeks) it is appropriate to frame our models as differential equations in continuous time. In the single-species case, there are N individual plankton per unit volume of water, and we model their dynamics by considering the rates of gain and loss of individuals. The change in numbers is written dN/dt and the per-capita rate of change in population size is:

N dt

In the simplest case (Figure 6.1), both the birth rate and death rate are density-independent, and this leads to either exponential growth (when the birth rate exceeds the death rate) or exponential decline (when the death rate exceeds the death rate). Numbers change through time as the integral of the differential equation:

where N(t) is the population size at time t and N0 is the initial population size at time 0. While these patterns of dynamics are important for defining the invasion criterion, they quickly become unrealistic because population growth rate is certain to decline as one or both of the vital rates becomes density-dependent.

In single-celled algae, the most likely cause of density dependence is intraspecific competition for a limiting resource (Tilman, 1982). In some circumstances, phytoplankton might compete for mineral nutrients like nitrogen or phosphorus, or for non-mineral resources like light. In our example, however, we consider that silicate (which is used to construct the ornate cell wall—the frustule—of diatoms) is the single limiting resource. This means that other resources (like light, nitrogen, phosphorus, etc.) are available in non-limiting amounts, reflecting a fundamental rule of plant ecology known as Leibig's Law of the Minimum, which states that the only limiting factor is the most limiting factor. Here it means that because silicate is the most limiting resource, increasing the amount of light, nitrogen, or phosphorus will not increase the algal population growth rate, but decreasing the concentration of silicate will decrease the growth rate.

To keep things a simple as possible, we shall assume that the birth rate of the diatoms is a function of silicate but that the death rate is independent of resource supply. When silicate concentration is greater than 4.4 mM (say) then the birth rate exceeds the death rate and the population increases exponentially. If the silicate concentration falls below 4.4 mM then the death rate will exceed the birth rate and the population will decline exponentially. The silicate concentration, however, is not constant, but depends on the population of plankton. Silicate is removed from the water and tied up in the cell walls of the algae, so as the diatom population grows, the silicate concentration in the water declines. Once the silicate concentration falls below 4.4mM the algal population would begin to decline because its death rate would exceed the birth rate, and silicate would be returned to the water thorough decomposition of algal cell walls. Thus, the silicate concentration of 4.4 mM defines an equilibrium amount of resource at which algal births and deaths are equal. The level to which the diatom population reduces the concentration of its most limiting resource was called R* by Tilman (R means resource and * means equilibrium; see Chapter 7 and Figure 7.2 in this volume). To understand the dynamics of the resource we need to be explicit about the structure of the experimental set-up. For instance, do resources leak out of the system or are they recycled? It turns out that the dynamics of open and closed systems differ in important ways (Daufresne and Hedin, 2005).

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