# pop.projection: Calculate population growth rates by projection In popbio: Construction and Analysis of Matrix Population Models

## Description

Calculates the population growth rate and stable stage distribution by repeated projections of the equation `n(t+1)=An(t)`.

## Usage

 `1` ```pop.projection(A,n,iterations=20) ```

## Arguments

 `A` A projection matrix `n` An initial age or stage vector `iterations` Number of iterations

## Details

Eventually, structured populations will convergence to a stable stage distribution where each new stage vector is changing by the same proportion (lambda).

## Value

A list with 5 items

 `lambda ` Estimate of lambda using change between the last two population counts `stable.stage ` Estimate of stable stage distribution using proportions in last stage vector `stage.vector` A matrix with the number of projected individuals in each stage class `pop.sizes` Total number of projected individuals `pop.changes` Proportional change in population size

Chris Stubben

## References

see section 2.2 in Caswell 2001

`stage.vector.plot` to plot stage vectors

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```## mean matrix from Freville et al 2004 stages<-c("seedling", "vegetative", "flowering") A<-matrix(c( 0, 0, 5.905, 0.368, 0.639, 0.025, 0.001, 0.152, 0.051 ), nrow=3, byrow=TRUE, dimnames=list(stages,stages) ) n<-c(5,5,5) p<-pop.projection(A,n, 15) p damping.ratio(A) stage.vector.plot(p\$stage.vectors, col=2:4) #### data(whale) A<-whale #n<-c(4,38,36,22) n<-c(5,5,5,5) p<-pop.projection(A,n, 15) p stage.vector.plot(p\$stage.vectors, col=2:4, ylim=c(0, 0.6)) ## convergence is slow with damping ratio close to 1 damping.ratio(A) pop.projection(A,n, 100)\$pop.changes ```

### Example output

```\$lambda
[1] 0.9969892

\$stable.stage
seedling vegetative  flowering
0.45252933 0.47113978 0.07633089

\$stage.vectors
0      1         2         3         4         5         6         7
seedling   5 29.525  6.023100  5.112933 13.030853 10.841000  8.851612 10.114475
vegetative 5  5.160 14.187940 11.304241  9.160138 10.694580 10.860800 10.240266
flowering  5  1.020  0.865865  2.206749  1.835902  1.499003  1.712866  1.747049
8         9        10        11        12        13
seedling   10.316326  9.777111  9.812795  9.915667  9.809763  9.745986
vegetative 10.309333 10.425465 10.301394 10.235679 10.231096 10.188925
flowering   1.655734  1.661777  1.679198  1.661264  1.650463  1.649110
14
seedling    9.737995
vegetative 10.138473
flowering   1.642567

\$pop.sizes
[1] 15.00000 35.70500 21.07691 18.62392 24.02689 23.03458 21.42528 22.10179
[9] 22.28139 21.86435 21.79339 21.81261 21.69132 21.58402 21.51904

\$pop.changes
[1] 2.3803333 0.5903068 0.8836175 1.2901091 0.9587000 0.9301353 1.0315755
[8] 1.0081262 0.9812831 0.9967542 1.0008820 0.9944395 0.9950533 0.9969892

[1] 1.739809
\$lambda
[1] 1.027321

\$stable.stage
yearling   juvenile     mature postreprod
0.03839089 0.34401197 0.33595865 0.28163850

\$stage.vectors
0      1         2         3         4         5        6         7
yearling   5 0.5875 0.6218869 0.6723333 0.7184122 0.7608606 0.800382 0.8375661
juvenile   5 9.4430 9.1777986 8.9697867 8.8295785 8.7468769 8.713021 8.7208067
mature     5 5.1350 5.5907138 6.0056725 6.3859845 6.7382546 7.068022 7.3799306
postreprod 5 5.1280 5.2595932 5.4092054 5.5746414 5.7540249 5.945815 6.1487518
8        9        10        11       12       13       14
yearling   0.8729076 0.906822 0.9396585 0.9717115 1.003229 1.034422 1.065469
juvenile   8.7642478 8.838373 8.9390604 9.0628941 9.207051 9.369201 9.547427
mature     7.6778772 7.965137 8.2444656 8.5181884 8.788270 9.056375 9.323921
postreprod 6.3618091 6.584158 6.8151324 7.0542056 7.300965 7.555096 7.816364

\$pop.sizes
[1] 20.00000 20.29350 20.64999 21.05700 21.50862 22.00002 22.52724 23.08706
[9] 23.67684 24.29449 24.93832 25.60700 26.29952 27.01509 27.75318

\$pop.changes
[1] 1.014675 1.017567 1.019710 1.021447 1.022847 1.023965 1.024851 1.025546
[9] 1.026087 1.026501 1.026813 1.027044 1.027209 1.027321

[1] 1.04594
[1] 1.014675 1.017567 1.019710 1.021447 1.022847 1.023965 1.024851 1.025546
[9] 1.026087 1.026501 1.026813 1.027044 1.027209 1.027321 1.027392 1.027430
[17] 1.027442 1.027435 1.027412 1.027377 1.027333 1.027283 1.027229 1.027171
[25] 1.027112 1.027052 1.026992 1.026932 1.026873 1.026815 1.026758 1.026703
[33] 1.026650 1.026599 1.026549 1.026501 1.026455 1.026411 1.026369 1.026328
[41] 1.026289 1.026252 1.026216 1.026182 1.026149 1.026118 1.026088 1.026059
[49] 1.026032 1.026006 1.025981 1.025957 1.025934 1.025912 1.025891 1.025871
[57] 1.025852 1.025834 1.025817 1.025800 1.025784 1.025769 1.025755 1.025741
[65] 1.025728 1.025715 1.025703 1.025691 1.025680 1.025670 1.025660 1.025650
[73] 1.025641 1.025632 1.025624 1.025616 1.025608 1.025601 1.025594 1.025587
[81] 1.025580 1.025574 1.025568 1.025563 1.025557 1.025552 1.025547 1.025543
[89] 1.025538 1.025534 1.025530 1.025526 1.025522 1.025519 1.025515 1.025512
[97] 1.025509 1.025506 1.025503
```

popbio documentation built on May 4, 2018, 1:04 a.m.