如何测量等斜线的值或测量给定某个 x 值的 y 值？

问：

此代码生成两个图形，用于查看基于 Lotka-Volterra 模型的捕食者-猎物关系。一张图显示了代表捕食者和猎物种群的两个函数。这些随时间推移，t（横轴）相对于垂直轴上的N（人口增长）起伏。另一张图是周期性的，显示了守恒能量的性质。

也就是说，如何在起伏图上测量时间 t 处的 N 值，以及如何测量第二个周期图的等斜线值？也就是说，如何计算捕食者等斜线的 x 值和猎物等斜线的 y 值？

# Activity 1 
library('deSolve')                      # load the library to be able to integrate

#### THIS SECTION SETS UP THE MODEL. DON'T TOUCH ####

vp_func1 = function(t, N, params){      # this is the function. Don't touch this.
     V = N[1]                           # the population size of Victims
     P = N[2]                           # the population size of Predators
     r = params[1]                      # per capita growth rate of Victims
     a = params[2]                      # capture efficiency of Predators
     c = params[3]                      # conversion efficiency of Predators
     q = params[4]                      # mortality of Predators
     dV = r*V - a*V*P                   # Victim population growth rate
     dP = c*a*V*P - q*P                 # Predator population growth rate
     return(list(c(dV, dP)))
}


#### SET UP INITIAL CONDITIONS AND PARAMETERS ####
V = 4             # Initial Victim population size
P = 6             # Initial Predator population size
N0 = c(V,P)         # Put the initial population sizes into a vector
r = 3.1             # Victim per capita growth rate
a = 0.25             # Capture efficiency
c = 0.5             # Conversion efficiency
q = 0.4             # Predator mortality rate
params = c(r, a, c, q)  # Put parameters into a vector.
t = seq(0, 50)          # run for 50 time steps


#### RUN THE MODEL. DON'T TOUCH! ####
results = round(data.frame(ode(N0, t, vp_func1, params)), 2)


#### TIME SERIES GRAPH. DON'T TOUCH! ####
#### (unless you want to change colors, etc) ####
plot(results$time, results$X1, type='l',
     ylab='N',
     xlab='t',
     col='blue',
     ylim=c(0, max(1.1*results$X1, 1.1*results$X2)))

lines(results$time, results$X2,
      col='red')

legend('topleft',
       lty=c(1,1),
       legend=c('Victims', 'Predators'), 
       col=c('blue', 'red'))


#### STATE-SPACE GRAPH. DON'T TOUCH! ####
#### (unless you want to change colors, etc) ####
plot(results$X1, results$X2, type='l',
     ylab='Predators',
     xlab='Victims',
     ylim=c(0, 1.1*max(results$X2)),
     xlim=c(0, 1.1*max(results$X1)))

points(N0[1], N0[2], pch=16, cex=1.5)

abline(r/a, 0,
       col='blue')

abline(v = q/(c*a),
       col='red')

legend('topright',
       lty=c(1,1,1),
       legend=c('Victim Isocline', 'Predator Isocline', 'Population Trajectory'), 
       col=c('blue', 'red', 'black'))