Question

\[\begin{equation} \begin{split} & \frac{dx}{dt} = rx (1- \frac{x}{K}) -a \frac{xy}{1+ H_t a x} \\ \\ & \frac{dy}{dt} = b \frac{xy}{1+ H_t a x} - my \\ \\ & x(0) = 0.1, ~ y(0) = 0.1 \\ \end{split} \end{equation}\]

\[ r = 1.0, a = 1.0, b = 0.5, H_t = 1.0, m = 0.1\]

Plot

library(ggplot2)
col <- scales::hue_pal()(3)
ggplot() + 
    geom_point(data = K_1.0,
               aes(x = X2, y=X3,
                   color="K = 1.0"),
               size = 0.01) +
    geom_point(data = K_1.2,
               aes(x = X2, y=X3, 
                   color="K = 1.2"),
               size = 0.01) +
    geom_point(data = K_2.0,
               aes(x = X2, y=X3,
                   color="K = 2.0"),
               size = 0.01) +
    guides(colour = 
            guide_legend(override.aes =
                             list(size=2)))+
    scale_x_continuous(
        breaks = seq(0,2,0.2))+
    scale_y_continuous(
        breaks = seq(0,2,0.2))+
    labs(x="Prey (x)", y="Predator (y)"
         ,color="Carrying Capacity")