Coexistence in Metapopulations - Tutorial in R

In which conditions can species coexist? There are several hypotheses about this, but in this exercise we will investigate the effect of the perturbation regime that an area endures, with differences in the colonization capabilities for the species, as we are still studying the colonization and extinction processes, as described by the metapopulation models.

We will start with the internal colonization model, in which a species dynamics is already familiar to you:

$$\frac{df_1}{dt}=i_1f_1(1-f_1)- p_e f_1 $$

Here:

• $f$ = fraction of occupied patches
• $p_e$ = local extinction probability
• $i$ = increase in the colonization probability as $f$ grows
• The rate of colonization is, then, the product $if$, that varies with the patch occupancy.

Now, we will add another species to the system. The second species will be a weak competitor: it will only persist in patches in which the first species is absent. This means that the patches available for colonization are only the empty patches, and that it is excluded from whichever patch is colonized by the first species. The variation of fraction of patches occupied by the second species is given by:

$$\frac{df_2}{dt}=i_2f_2(1-f_1-f_2)- i_1f_1f_2 - p_e f_2 $$

The equation for species 2 does not add any new coefficient, but only combinations of things we have already seen:

• The term $i_2f_2(1-f_1-f_2)$ indicates that the fraction of patches that can be colonized is proportional to the number of patches that are empty, from both species.
• The term $i_1f_1f_2$ is the expected fraction of patches occupied by species 2 that will be colonized by species 1. Thus, it represents the fraction of the patches in which species 2 will be excluded by species 1.

* The extinction rate from both species is the same, so we didn't add a subscript.

The fraction of patches that are occupied by species 1 in equilibrium remains the same:

$$F_1=1-p_e/i_1$$

And now the fraction of patches occupied by species 2 in equilibrium is:

$$F_2=p_e/i_1- i_1/i_2$$

So, in order to have a viable metapopulation of species 2 ($F_2>0$), it is necessary that the following inequality is satisfied:

$$p_e/i_1 > i_1/i_2$$

Here you can read a tutorial explaining the mathematical deduction of the fraction of patches occupied by species 2 in equilibrium, that you can run in MAXIMA.

The deduction only needs simple algebraic manipulations. If you are scared of maths, keep in mind that you only need to understand the logic behind each step. To do the algebraic manipulation *per se*, there are lots of symbolic mathematics programs that can help you, such as MAXIMA, which is open source and free to use)).

Download and install the program and the graphical interface wxMaxima, open the script linked above and press crtl-R to run.

Let us run a computer simulation to generate a stochastic dynamic following the same rules as the differential model described above. The simulation is very similar to other simulation you have already seen on EcoVirtual :

1. Define a matrix with $r$ rows and $c$ columns. Each cell in the matrix is a habitat patch.
2. Define the fraction of initially occupied patches (fi1 and (fi2), and occupy random patches until you reach the determined fraction.
3. Calculate the colonization probability from each species (using the product pi = i*f).
4. From the occupied patches, draw a number of patches to go extinct, following the extinction probability pe.
5. From the unoccupied patches, draws some to be occupied by species 1, according to the above probability.
6. From the unoccupied patches, draws some to be occupied by species 2, according to the above probability. If the selected patch is already occupied from species 1, the colonization fails and nothing happens.
7. Counts the number of occupied patches from each species to determine the fractions of occupancy,
8. Repeat from step 3 until the desired number of iterations is reached.

The arguments to this function are the number of rows and columns for the habitat matrix, the number of time intervals that should be run, initial fraction of patches occupied by each species, and the model parameters i1, i2 and pe.

The parameters here are almost the same as the internal colonization model, but adding a second species:

Start a simulation with the following parameters:

tmax = 100
cl = 20 
rw = 20,
f01 = 0.1
f02 = 0.4
i1 = 0.4
i2 = 0.5  
pe = 0.25

Calculate the equilibrium value for each species using the formulas shown above. Verify that the theoretical values are close to the simulations results.

We have defined the species 2 as the worse competitor. Does this sentence it to extinction? Examine the effect of the difference between the colonization capabilities over coexistence. To do this, we will vary the colonization parameter for species 2, keeping everything else constant. Identify the conditions for coexistence, and for extinction of each species.

Start simulating the following parameters:

tmax = 100 
cl = 20
rw = 20 
f01 = 0.05 
f02 = 0.05
i1 = 0.1 
i2 = 0.1 
pe = 0.05

Now keep everything unchanged, except for i2. Change i2 to:

1. from 0.2 to 1 with step 0.1
2. 10
3. 100

What is the interpretation of these results?

Let's start a new simulation with a new combination of parameters, in which species 2 has triple the colonization capability as species 1:

tmax = 100, 
cl = 20,
rw = 20, 
f01 = 0.1,
f02 = 0.1,
i1 = 0.1,
i2 = 0.3,
pe = 0.05 

Does this simulation lead to extinction? Vary the extinction probability keeping the other parameters constant, to investigate the effect of the perturbation regime over the results.

1. Increase pe to 0.07
2. Vary the pe from 0.08 to 0.14, with step 0.02

If the equilibrium is not being reached:

Patch occupancy can be defined as the proportion of available patches that are being occupied by a species. In our model, the initial occupancy for each species is given by fi1 and fi2.

1. Run some simulation to check if these parameters affect the long term behavior of the populations
2. Are the trajectories different for different initial occupancies? Does it change the coexistence?

1. Which attributes of the inferior competitor enable the coexistence with the superior competitor? What is the biological interpretation for this?
2. What is the relationship between coexistence and local extinction in this model? Think about the theoretical and applied consequences.
3. What is the effect that species 2 has over species 1 in this model? What is the most straightforward simulation to show this clearly?
4. What is the equilibrium occupancy for species 2 when species 1 is absent from the whole landscape?

• Hastings, A. (1980) Disturbance, coexistence, history and competition for space. Theoretical Population Biology, 18:363–373.
• Stevens, M.H.H. (2009) A primer in ecology with R. New York, Springer.