Lab#6: A Mechanistic Model of Resource Competition

The intent of this lab is have you be able to predict the outcome of competition between two species based on a knowledge of their individual growth characteristics given varying amounts of limiting resources. The model was originally developed mathematically by David Tilman, who worked initially with algae, cultured in the laboratory under controlled conditions with precisely defined amounts of nutrients. Tilman’s work was converted into a Populus tutorial by Don Alstad. Begin by opening Populus 5.4, and clicking successively on Model/Multi- species Dynamics/Resource Competition/Tilman. This model has relevance to the study of invasive species, which often outcompete native species with similar resource requirements by unusually aggressive resource use and rapid reproduction. Complete this lab assignment by 1) entering data in the appropriate tables, 2) pasting a graph from Populus into each location where a graph is requested, and 3) writing in answers to the 4 questions in bold type. Turn in the entire document below.

The following parameters are defined in the model:

N(0)i = the initial population density of species “i” Rj(0) = the concentration/availability of resource “j” ri = the maximal per capita growth rate of species “i” kij = the concentration of resource “j” at which species “i” achieves half of its maximum per capita growth rate (called the “half saturation constant”) cij = the rate at which individuals of species “i” convert resource “j” into new growth (the per-capita consumption rate)

There are three additional terms that you’ll see but needn’t worry much about: mi = the mortality rate of species “i” (as a proportion of the population) Sj = the continued “supply” of additional resource “j” during the experiment aj = a constant determining the rate at which resource “j” is converted to a form directly available to the consumer.

By growing algal species individually under a range of nutrient concentrations, Tilman confirmed earlier research by Monod that the per individual growth rate for a species “i” given a resource “j” obeys the relationship

dNi/Ndt = [riRj]/[Rj+kij] (1)

Notice that the per individual growth rate (dN/Ndt) is reduced below the maximum (rj) by reducing the resource concentration. The shape of the curve is further modified according to the half saturation constant (the line is initially steeper if the half saturation constant is lower). In effect, r

d N / N d t r / 2

k [ R ]

One Species and One Resource To observe how the Monod model controls population growth, choose 1 species consuming 1 resource at the left side of the Populus window. Set the following parameters: the intrinsic per-individual growth rate (r) = 1.1, initial population size (N0) = 10 individuals, ai = mi = 0.5 and Rj = Sj = 50. Now specify a half-saturation constant of 20, and a per-capita consumption rate of 0.25. For plot type (at the left) choose N vs. t, and set the time to 100 (days). Now click on View at the top left to see the graph of population change over time.

The plot should look rather like a logistic growth curve (to carrying capacity). The mechanism by which carrying capacity is determined, however, is made much more obvious by the Tilman model than in the basic logistic equation. To understand this, hold everything else constant, and change the half-saturation constant (kij) in the table below. Now click on the R vs. t graph option on the left. What happens to the rate of resource depletion? Fill out the table below.

kij Carrying capacity Approximate time to Final Resource Conc. (N*) reach N* (d) (R*) 30 10

1. As the half-saturation constant increases, what happens to the initial rate at which the population approaches its maximum, to the carrying capacity, and to the final resource concentration? Explain kij and its impact on N* and R*.

The per-capita consumption rate (cij) also affects population growth. As you can imagine, a high per-capita resource conversion rate would promote rapid growth if there are few individuals and the resource is abundant. If the population is high, however, more consumption per individual means that fewer individuals can exist given the resource constraints. To draw an analogy to the current United States economy, if all vehicles are SUVs (with high cij), then the available gas can support fewer vehicles. Return the half-saturation constant to 20, and change the conversion rate, filling out the following table.

cij Carrying Capacity Approximate Time Final Resource Conc. (N*) to reach N* (d) (R*) Low (0.05) High (0.50)

2. Explain the effect of per-capita consumption rate on N*, initial growth and time to reach N*, and R*.

Two Species competing for One Resource Species of algae vary fairly widely in their rate of converting the two nutrients N and P to cell tissue (cij), and in their half-saturation constants (species found in nutrient rich waters typically have higher kij values). When two species compete for a single resource, the quantity of the resource, together with the shapes of the Monod curves, can be used to predict the outcome of competition with a high degree of accuracy. The graphs below describe just one possible outcome, given in lecture.

N 1 N 2 s p . 1 R

d N / N d t N s p . 2

[ R ] t

Suppose the two species are competing for phosphorus (P). Change the number of species to two, and enter the following values for each species. Parameter Species 1 Species 2 N(0) 10 10

ri 1.1 3.2

mi 0.2 0.2

ki 4 20

cij 0.1 0.3

ai 0.2 R(0) 100

Si 20

3. In the case where two species are competing for a single resource, which species has the higher half-saturation constant? ______Which uses up more P to divide and create new cells? ______Which has the higher intrinsic growth rate under optimal conditions? ______Given the data, which species do you predict will be the better competitor? ______

Now click on the N vs. t option and click on View at the upper left. Sketch the results in the graph below, adding units to the left-hand Y-axis. Then switch to R vs. t and add the change in available resource to the graph; add units for R on the right-hand axis.

N R

t (days)

Finally, switch the output type to N vs. N and view the results as a “phase plane” diagram. The “ball” at the end of the line represents the combination of N1 and N2 at the end of the time period; if the ball has reached one of the axes then one of the two species has gone extinct. Sketch the result in the graph below, including units, and make sure you understand its interpretation. N 2

N 1

4. Note that one of the two species did better initially, but was ultimately outcompeted by the other. Why?

Confirm your understanding of the effect of cij, kij and ri on the outcome by changing the values for these parameters one at a time and observing the effect on resource depletion and competitive outcome (space reserved below for notes or doodling). Two Species Competing for Two Resources Now we’ll consider two “essential” nutrients, nitrogen (N) and phosphorus (P), either of which may potentially limit growth. They are considered “essential” because neither can be substituted for the other. Whichever nutrient is in least supply relative to need is considered the “limiting nutrient”, and directly controls growth regardless of the concentration of other non-limiting nutrients. As shown in lecture, algae differ in terms of their need for N and P, and thus in their “zero net growth isoclines” (ZNGI). You can think of the ZNGIs as being determined by the half saturation constants for each nutrient; a low half-saturation constant for a given nutrient means that the ZNGI will be closer to the X- or Y-axis. As shown below, if the ZNGI of one species is completely below that of another (in “a” at the left), it should reduce concentrations of both nutrients below levels needed by its competitor to survive, effectively driving it extinct. If, on the other hand, the ZNGIs cross, such that one species is a better for N and the other a better competitor for P, then stable coexistence may be possible depending on the initial concentrations of the two nutrients. a . b . s t a b l e s p . 1 c o e x i s t e n c e R 2 R 2

s p . 2 s p . 1 s p . 2

R 1 R 1

First make sure “Essential”, “2” species and “2” resources are specified in the model. To simulate the effects of changing half-saturation constants (k), instantaneous growth rates (r) and resource consumption rates (c) on the outcome, return to the Populus window, add a second resource (Nitrogen). As a starting point, enter the following values for the model parameters.

Parameter Species 1 Species 2 N(0) 10 10

ri 1.0 3.2

mi 0.2 0.2

ki1 4 20

ki2 10 8

ci1 0.1 0.3

ci2 0.2 0.1

ai 0.4 0.4 R(0) 15 15

Si 12 12 Once the values are entered, click on R vs. R, and sketch the plot using the axes below. Populus is a bit unclear regarding its choice of color for the two ZNGI lines, so if you’re not sure which species is which, try removing species 2 (the remaining line represents species 1). You can change the colors of the two ZNGI lines if you wish by clicking on Options at the upper left in the Output window. The green line describes the joint decline of both nutrients, ending at the large dot. Where is the large dot in this instance, and what does that mean about the form of nutrient limitation for species 1? For species 2?

5. Based on the R1 vs. R2 graph below, do you predict that one species will outcompete the other (which one will win out?) or will a stable equilibrium result? Why? Now click on N vs. t and then on N vs. N to observe the population trends, entering them in the templates below the R1 vs. R2 graph. Was your prediction correct? Explain. R

2

R

1

N

t

N2

N1 As your final task, modify the values of cij and kij for the two species, so that species 1 is considered invasive, and outcompetes species 2. Enter your values in the table below, and plot the phase plane diagram R1 vs. R2 in the figure that follows.

Parameter Species 1 Species 2 N(0) 10 10

ri 1.0 3.2

mi 0.2 0.2

ki1

ki2

ci1

ci2

ai 0.4 0.4 R(0) 15 15

Si 12 12

R 2

R 1

6. Explain what happened in terms of population trends and resource availability over time, and state why species 1 “took over” given the specifications for half-saturation constants and per- capita consumption rates that you provided.