Population Models

This repo contains the scripts to simulate a population model of 3 species with 3 different methods:

1. Using a deterministic ODE solver
2. Using an stochastic method (Gillespie Algorithm)
3. Using a agent based approach

System model

The ODE system is:

The system models a food chain as follows:

The food chain begins with the plant. The plant is eaten by the rabbit. The rabbit is then eaten by a larger animal, the fox.

Being G the amount of grass, R the number of rabbits, and F the number of foxes. The deltas are the death rates of each specie according to their subindex....

Cellular Automata

The algorith for the Cellular Automata (C.A.) is as follows:

FOR T steps

FOR j individuals in the population list P

choose a random neighbor i

IF j=prey and i=empty

j reproduce with probability r

ELSEIF j=predator 1 and i=empty

j move with probability m_r

ELSEIF j=predator 1 and i=prey

j eats i with probability e_r and i becomes of the type j

ELSEIF j=predator 1 and i=predator 2

j dies with probability d_r

ELSEIF j=predator 2 and i=empty

j move with probability m_f

ELSEIF j=predator 2 and i=predator 1

j eats $i$ with probability e_f and i becomes of the type j

ELSEIF j=predator 2 and i=predator 2

j dies with probability d_f

ENDFOR

ENDFOR

The scripts are organized in the next way: prey_pred_2.m contains the ODE system stated above. Model_Lotka_analytic.m solves the model using a ode45 solver. lotca_stochastic_3species.m solves the Lotka-Volterra model using Gillespie Algorithm, The simulation depends on the parameters size, alpha, beta_gr, beta_r, gamma_r, gamma_f, delta_r, delta_f, k_g, k_r, s_f and s_r as stated in the equation system, and the initial populations: g0, r0 and f0 which can be changed in the files Model_Lotka_analytic.m and lotca_stochastic_3species.m.

2species_CA.pywill simulate the system as a Cellular Automata as stated above. It will create a file with 4 columns indicating

Simulation Step	Population G	Population F	Population R

The initial populations can be changed in the lines

spec1_i=XXX #grass
spec2_i=XXX #rabbit
spec3_i=XXX #foxes

If there are several simulations which were run under the same conditions you can plot all of them in a single graph by using plot_data.m you just need to change the line

name='results/XXXXXX_';

being XXXXX the base name of the simulatons you want to plot

If you want to change the ODE system to your own you can do it by modifying the fifle prey_pred_2.m

1. Determinisit Results

2. Stochastic Results

3. C.A. Results