B.Pichugin, S.Klokov, P.Haccou, V.Topchii
Supported by grants NWO 047.016.013, RFBR 06-01-00127.
This is a modification of the model considered in [1].
There is a population of individuals. It is assumed that there is female demographic dominance, and dynamics of males and females are equal. So, it suffice to follow only female dynamics.
There are two habitats, A and B. Each female in the population is characterized by its genotype (gAgA, gAgB, or gBgB), cultural trait (cA or cB) related to the habitat the female was born in, and two real-valued parameters, alpha and p. Parameter alpha determines mating preferences of the female. The value 1-p is the habitat preference.
Let N1, N2, N3 be the number of females with cultural trait cA and genotypes gAgA, gAgB, gBgB, and let N4, N5, N6 be the number of females with cultural trait cB and genotypes gAgA, gAgB, gBgB. The total number of females with cultural trait cA is denoted by NA = N1+N2+N3. The value of NB = N4+N5+N6 is introduced similarly. The total population size is N = NA+NB.
The population has non-overlapping generations. Each generation goes through the following 4 stages.
Mating. Each female born in habitat X mates with a male from the same habitat with probability rho(X, alpha), and with a male from another habitat with probability 1-rho(X, alpha), where rho(A, alpha) = alpha*NA/(alpha*NA+(1-alpha)*NB), rho(B, alpha) = alpha*NB/((1-alpha)*NA+alpha*NB). The value of alpha is between 0 and 1.
Producing youngs. Each female with probability 1-p produces its young in the same habitat it was born in, otherwise it chooses another habitat. Denote the female fertility, i.e. the number of young produced, by E. Each young genotype is inherited from both parents accordingly the laws of genetics. The values of alpha and p are inherited from the female with small random mutations.
Carrying capacity restrictions. Each young in habitat A surives with probability QA = K/(K+(E-1)*NA'), where K is the carrying capacity of habitat A and NA' is the number of females producing their youngs in habitat A. Note that NA' equals NA minus the number of females from habitat A that prefer habitat B plus the number of females from habitat B that prefer habitat A. Thus, if NA' = K, then one young survives, in average. If NA' is less/more than K, more/less than one young survives in average. Similarly QB = K/(K+(E-1)*NB').
Viability. Each young's viability is determined by its genotype. Usually, viability w of youngs in habitat A is higher if they have more genes gA, and viability of youngs in habitat B is higher if they have more genes gB. A young with viability w becomes an adult individual, a female or a male, with probability w.
The next generation of the population is formed by youngs that become adult.
A complete model description is available as a pdf-file.
The model is implemented as a java-applet. Visit www.java.com to get a free virtual java-machine (Java 2 Platform Standard Edition J2SE 5.0) for displaying the applet on your computer platform.
Applet's window has four panels, simulations start/stop/pause panel, model parameters panel, graph options panel and graph result panel. There are 4 steps to use the applet.
Step 1. Set the values of model parameters by double clicking a value and changing it.
Step 2. Choose the graphs you would like to draw by checking the options and setting the graph type.
Step 3. Press the button "START" to begin simulations. Press the button "PAUSE" to pause/resume simulations.
Step 4. Press the button "STOP" to finish simulations.
When simulations are running, paused or stopped you can explore the graphs. The upper graph shows simulation results for one iteration. The lower graph displays average simulation results over all iterations that have been made so far. The graphs are coupled together.
The graphs can be scaled along the horizontal axis by pressing buttons "X+" and "X-". Similarly, buttons "Y+" and "Y-" resize the graphs along the vertical axis. The graphs are zoomed in and out by pressing buttons "G+" and "G-". The original size can be restored by pressing the button "Normal size".
Both graphs can be moved simultaneously by pressing the right button of the mouse and gragging it.
To explore a graph in detail, select a rectangular area by pressing the left button of the mouse, dragging it and releasing the left button. The area selected will be resized and fit into the graphs.
When simulations are running, paused or stopped you can choose the graphs to display.
Model paramemeters can be changed only if the simulations are stopped.
Pressing the button "Tabular view" changes the graphical view to the tabular view. The tabular view shows the results of simulations. In order to export the data to other applications, select the data by pressing Ctrl+A and copy to the clipboard by pressing Ctrl+C. To switch back to the graphical view, press the button "Graphical view".
Note. Leaving this page, you abandon simulation results.
[1] J.B.Beltman, P.Haccou. Speciation Through the Learning of Habitat Features.// Theoretical Population Biology, 67 (2005), p.189-202.