Numerical solution and slope field of logistic equation

NSLogis.R

library(deSolve)
library(ggplot2)
library(tidyr)
library(dplyr)
library(foreach)
library(patchwork)

mod_logistic <- function(Time, State, Pars) {
  with(as.list(c(State, Pars)), {
    dX <- a*X*(1.0 - X/N)
    return(list(c(dX)))
  })
}

pars  <- c(a = 0.5, N=1)
ini  <- c(X = 0.1)
times <- seq(0, 10, by = 0.1)
out   <- ode(ini, times, mod_logistic, pars)
plot(out)

sol_ode_from_states <- function(df_state, times, func, parms){
  res_df <- foreach(i=1:nrow(df_state), .packages='deSolve', .combine=rbind) %do% {
    ode_out <- ode(y=c(unlist(df_state[i,,drop=FALSE])), times=times, func=func, parms=parms)
    data.frame(group=i, ode_out)
  }
  return(res_df)
}

df_state <- data.frame(X = seq(-0.001, 1.2, length.out=12))
df_state

res <- sol_ode_from_states(df_state=df_state, times=times, func=mod_logistic, parms=pars)
p1 <- ggplot(res,aes(x=time, y=X, group=group))+
  geom_line()+
  theme_bw()

df_slope <- expand.grid(times=times, X=df_state$X) %>% 
  rowwise() %>% 
  mutate(slope=unlist(mod_logistic(times, c(X=X), pars))) %>% 
  ungroup() %>% 
  mutate(start=X, end = 0.25*slope+X) %>% 
  mutate(i = row_number()) %>% 
  pivot_longer(cols=c(start,end)) %>% 
  mutate(times=if_else(name=="end",times+0.25, times))

p2 <- ggplot(df_slope, aes(x=times, y=value, group=i))+
  geom_path()+
  theme_bw()

p <- p2/p1
ggsave(plot = p,filename = "logistic.png", width = 7, height = 6)