The previous article introduced the sensitivity and elasticity to seasonal matrix model of imaginary annual plant. Both sensitivity and elasticity are partial derivatives. This means the values can only predict a change of λ with respect to a small change of a element.

To know how λ will affected by a large shift or changes of multiple elements, the simplest way is to calculate each λ for each case.

R can easily do this.

The λ can also be solved analytically, because this example is very simple. Let’s check whether both results match.

We have four elements:

seed <- 0.9^4 # Seed surviving rate; annual germ <- 0.3 # Germination rate; spring plant <- 0.05 # Plant surviving rate; from germination to mature yield <- 100 # Seed production number; per matured plant

The function `lambda`

and `A.spring`

were defined in the previous article:

lambda <- function(A) eigen(A)$values[1] # and so on.

n <- 100 lambdas <- numeric(n) seeds <- seq(from=0, to=1, length.out=n)^2 for(i in 1:n) { seed <- seeds[i] lambdas[i] <- lambda(A.spring()) } seed <- 0.9^4 # restore the initial value plot(seeds, lambdas, ylab='Population growth rate λ', xlab='Seed surviving rate; annual', col='blue') abline(a=1, b=0)

Blue plots indicate the result of simulation.

From analytic solution:

# λ = 0.7 * seed + 1.5 * seed^(1/4)

Drawing this curve with red line on the blue plots.

curve(0.7 * x + 1.5 * x^(1/4), add=T, from=min(seeds), to=max(seeds), col='red')

Both results met very well.

germs <- seq(from=0, to=1, length.out=n) for(i in 1:n) { germ <- germs[i] lambdas[i] <- lambda(A.spring()) } germ <- 0.3 # restore the initial value plot(germs, lambdas, ylab='Population growth rate λ', xlab='Germination rate; spring', col='blue') abline(a=1, b=0)

From analytic solution:

# λ = 0.6561 + 3.8439 * germ curve(0.6561 + 3.8439 * x, add=T, from=min(germs), to=max(germs), col='red')

Both results met very well.

plants <- seq(from=0, to=0.1, length.out=n) for(i in 1:n) { plant <- plants[i] lambdas[i] <- lambda(A.spring()) } plant <- 0.05 # restore the initial value plot(plants, lambdas, ylab='Population growth rate λ', xlab='Plant surviving rate; from germination to mature', col='blue') abline(a=1, b=0)

From analytic solution:

# λ = 0.45927 + 27 * plant curve(0.45927 + 27 * x, add=T, from=min(plants), to=max(plants), col='red')

Both results met very well.

yields <- seq(from=0, to=200, length.out=n) for(i in 1:n) { yield <- yields[i] lambdas[i] <- lambda(A.spring()) } yield <- 100 # restore the initial value plot(yields, lambdas, ylab='Population growth rate λ', xlab='Seed production number; per matured plant', col='blue') abline(a=1, b=0)

From analytic solution:

# λ = 0.45927 + 0.0135 * yield curve(0.45927 + 0.0135 * x, add=T, from=min(yields), to=max(yields), col='red')

Both results met very well.

n <- 64 lambdas <- matrix(nrow=n, ncol=n) plant <- 0.05 # Plant surviving rate; from germination to mature yield <- 100 # Seed production number; per matured plant seeds <- seq(from=0, to=1, length.out=n)^2 germs <- seq(from=0, to=1, length.out=n)^2 for(ro in 1:n) for(co in 1:n) { seed <- seeds[ro] germ <- germs[co] lambdas[ro, co] <- lambda(A.spring()) } contour(x=seeds, y=germs, z=lambdas, main='λ', xlab='Seed surviving rate; annual', ylab='Germination rate; spring')

n <- 64 lambdas <- matrix(nrow=n, ncol=n) germ <- 0.3 # Germination rate; spring yield <- 100 # Seed production number; per matured plant seeds <- seq(from=0, to=1, length.out=n)^2 plants <- seq(from=0, to=0.3, length.out=n)^2 for(ro in 1:n) for(co in 1:n) { seed <- seeds[ro] plant <- plants[co] lambdas[ro, co] <- lambda(A.spring()) } contour(x=seeds, y=plants, z=lambdas, xlab='Seed surviving rate; annual', main='λ', ylab='Plant surviving rate; from germination to mature')

n <- 64 lambdas <- matrix(nrow=n, ncol=n) germ <- 0.3 # Germination rate; spring plant <- 0.05 # Plant surviving rate; from germination to mature seeds <- seq(from=0, to=1, length.out=n)^2 yields <- seq(from=0, to=sqrt(100), length.out=n)^2 for(ro in 1:n) for(co in 1:n) { seed <- seeds[ro] yield <- yields[co] lambdas[ro, co] <- lambda(A.spring()) } contour(x=seeds, y=yields, z=lambdas, main='λ', xlab='Seed surviving rate; annual', ylab='Seed production number; per matured plant')

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