Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 219

plants, the vector would look like this:
⎛
⎝

⎞
⎠

800
90
56
23
31
11

. (7.3)

Here we go through the steps of conducting the calculations for predicting the number
of two-leaved plants in the population at time t 0 + 1, and then we present the matrix

algebra to calculate the number of individuals in all life stages simultaneously.
Using the projection matrix ( 7.2 ) that we constructed above, we can predict the
number of two-leaved plants at time t 0 +

1 (the next year) by applying Eq. ( 7.1 ).

Recall that n j ( ) i , j is

the average number of members that a single member in stage j produces in stage i
between t 0 and t 0 +

t 0 ) is the number of members in stage j at time t 0 , the entry ( A

1, and that two-leaved plants are in stage 4. So

n 4 ( t 0 +

) =

n 1 ( t 0 ) × A 4 , 1 + n 2 ( t 0 ) × A 4 , 2 + n 3 ( t 0 ) × A 4 , 3 + n 4 ( t 0 ) 1

× A 4 , 4 + n 5 ( t 0 ) × A 4 , 5 + n 6 ( t 0 ) × A 4 , 6 .

Written out in words, this becomes:

(number of seeds at time t 0 ) × (probability that a seed becomes a two-leaved

plant between t 0 and t 0 + 1) +

(number of seedlings at time t 0 ) × (probability that a seedling becomes a two-

leaved plant between t 0 and t 0 + 1) +

(number of one-leaved plants at time t 0 ) (the probability that a one-leaved

plant becomes a two-leaved plant between t 0 and t 0 +

×

1) +

(number of two-leaved plants at time t 0 ) (the probability that a two-leaved

plant stays a two-leaved plant between t 0 and t 0 +

×

1) +

(number of three-leaved plants at time t 0 ) (the probability that a three-leaved

plant becomes a two-leaved plant between t 0 and t 0 +

×

1) +

(number of (the probability that

a four-leaved plant becomes a two-leaved plant between t 0 and t 0

four-leaved plants at time t 0 ) ×

+ 1)

= 800 × 0 + 90 × 0 + 56 × 0 . 35 + 23 × 0 . 45 + 31 × 0 + 11 × 0 = 29 . 95.

Therefore, we predict that in the next year the population will have 29.95 two-leaved
individuals.
To expand our predictions to include every life stage in the following year, we
will make full use of matrix algebra. Note that what we have done in the previous
paragraph is multiply the vector ( 7.3 ) by the fourth row of the projection matrix. To
determine the number of individuals in every stage at time t 0 + 1, we can multiply