Lotka-Volterra

Solution to Lota-Volterra prey-predator model using Fortran

Lotka Volterra prey-predator model, for an autocatalytic reaction.

File List

• lvpp.f - Main Routine
• derivs.f - ODE Sub Routine
• rk4.f - 4th Order Runge-Kutta Method to solve a system of differential equations

For the reaction

$$A \rightarrow 2X$$

$$X + Y \rightarrow 2Y$$

$$Y \rightarrow B$$

Rate change for X and Y can be defined as:

$${{dX} \over {dt}} = \alpha[A][X] - \beta [X][Y]$$

$${{dY} \over {dt}} = \gamma[X][Y] - \delta [Y]$$

Using the 4th Order Runge-Kutta Method to solve the above system of differential equations:

$$k_1 = hf(t_n,x_n,y_n)$$

$$l_1 = hg(t_n,x_n,y_n)$$

$$k_2 = hf(t_n + {h \over 2}, x_n + {k_1 \over 2}, y_n + {l_1\over2} ) $$

$$l_2 = hf(t_n + {h \over 2}, x_n + {k_1 \over 2}, y_n + {l_1\over2} ) $$

$$k_3 = hf(t_n + {h \over 2}, x_n + {k_2 \over 2}, y_n + {l_2\over2} ) $$

$$l_3 = hf(t_n + {h \over 2}, x_n + {k_2 \over 2}, y_n + {l_2\over2} ) $$

$$k_4 = hf(t_n + h, x_n + k_3, y_n + l_3 ) $$

$$l_4 = hf(t_n + h, x_n + k_3, y_n + l_3 ) $$

$$t_{n+1} = t_n + h$$

$$x_{n+1} = x_n + {1\over6}(k_1 + 2k_2 + 2k_3 + k_4)$$

$$y_{n+1} = y_n + {1\over6}(l_1 + 2l_2 + 2l_3 + l_4)$$

Using the above algorithm, a Fortran program is created and values of $x_i$, $y_i$ are obtained from $x_0 - x_n$ and $y_0 - y_n$ at interval of time $h = 0.01$ for $n = 2000$. Taking the example values as follows:

$\alpha = 2.0$

$\beta = \gamma = 0.02$

$\delta = 1.0$

Initial values as:

$t_0 = 0.0$

$x_0 = 100.0$

$y_0 = 15.0$

Output Result

Output result file lv-res.dat is included here for reference.

Plotting the Result

gnuplot is used for plotting the result obtained from the simulation. After the result is obtained, gnuplot is set to output in svg/png. Sample plot is shown below. You can also check the svg files in images directory.

Settings for gnuplot to output svg files are as follows:

#!/usr/local/bin/gnuplot -persist

set terminal svg enhanced font 'Arial,11'
set output "rk.svg"
set title "[X]/[Y] vs Time Plot"
set xlabel "Time (t)"
set ylabel "[X]/[Y]"
set yrange [0:450]
plot "lv-res.dat" u (column(0)):2 w l lw 3 title "[X]","lv-res.dat" u (column(0)):3 w l lw 3 title "[Y]"

set terminal svg enhanced font 'Arial,11'
set title "[X] vs [Y] Plot"
set xlabel "[X]"
set ylabel "[Y]"
set output "rk2.svg"
plot "lv-res.dat" using 2:3 w l lw 3 title ""

This setting file is included as a shell script file xy_svg.sh

[X],[Y] vs Time (t) plot

[X] vs [Y] Plot

Notes

• The subroutine rk4.f was adapted from Numerical Recipes in Fortran 77, Second Edition (1992)