In this exercise, we will simulate the dynamics of a metapopulation in which the probabilities of local extinction and patch colonization by our virtual species are constant - the model called propagule rain. In this model, there is a constant source of migrants that can colonize any empty patch, and that's where the model name comes from.

In this model, the variation of the fraction of occupied patches in time is described by the following equation:

$$\frac{df}{dt} = I - E$$

here, I is the migrant entry rate, and E the exit rate. From this, we can define a simple model for the dynamics of the patch occupancy of the metapopulation:

$$\frac{df}{dt} = p_i(1 - f)- p_e f$$

here, $p_i$ is the probability of immigration or colonization, $p_e$ is the probability of extinction, and f is the fraction of occupied patches (number of occupied patches / total number of patches). These are the parameters of our model.

Besides it is mathematically the most simple model,it was described more recently in comparison with ohter metapopulation methods that we will present ¹⁾. An important feature of this model is that it is not closed, making it biologically more complex.

Our simple mode, described above, allow us to calculate a very important trait of the system : the equilibrium solution. The equilibrium here is related to the stability of occupied patches in time, defined by:

$$\frac{df}{dt} = 0$$

With little of algebra, we can calculate the fraction of occupied patches in equilibrium:

$$\hat{f} = \frac{p_i}{p_i + p_e}$$

We could go further and analyse the stability of this equilibrium, but we will only assume that this equilibrium is stable. See the tutorial about stability for the definitions of kinds of stability.

Let's imagine a scenario in which a virtual landscape has a certain number of habitat patches that can be either empty or occupied by our species. Just to make things easier, we will represent this landscape having a certain number of columns and rows ($n_{col}=20$ and $n_{row}=20$), totalizing 400 patches. Let's define the occupancy proportion at the initial time step as $f = 0.25$ (or 100 occupied patches). Our virtual species has an immigration probability of 10% ($p_i= 0.1$) and an extinction probability of 5% ($p_e=0.05$).

The game we will play is like we had a board with 400 tiles, representing our landscape, with 100 of them initially occupied by our pieces. Every turn, for each tile, we are drawing a random number from 1 to 100 (a super sized die with 100 sides!). If the patch is occupied, and the number we get is between 1 and 5, (0.05 probability), we take that piece from the board. If the tile is empty and the number is between 1 and 10 (0.10 probability), we put a new piece there. If none of this conditions trigger, we just leave the tile as it were.

Now, we just need to do this several times and observe at each time step what happened, annotating the proportion of occupied patches (which is out variable of interest).

As running a board with 400 tiles by hand would be boring and take a long time, we will use EcoVirtual to run the game for us.

The parameters of the function are:

Run some simulations to understand how the model parameters affect the trajectory and final result (equilibrium) for the population. Below, there are some suggestions of parameter combinations:

Now it's up to you! Try to answer the following questions:

• when you run the function several times with the same parameters, do the results stay the same? Why?²⁾
• What happens with the variation of the occupancy fraction over time when:
  • the number of patches is very large? Or very small?
• the probabilities of colonization and extinction are very large?
• or very small?
• what about when the initial fraction is very different fr the equilibrium $F=\frac{p_i}{p_i-p_e}$? ³⁾
• What conditions lead to population extinction in this model?

• Gotelli, N. 2007. Ecologia. Londrina, Ed. Planta. Capítulo 4.
• Stevens, M. H. 2009. A primer of ecology with R. New York. Springer.Capítulo 4.
• Gotelli, N. 1991. Metapopulation models: the rescue effect, the propagule rain, and the core-satellite hypothesis. The American Naturalist, 138: 768-776. pdf no site do autor